1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
|
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<script type="text/javascript"
src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (CTblLibXpls) - Chapter 9: Ambiguous Class Fusions in the GAP Character Table Library</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap9" onload="jscontent()">
<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chap3_mj.html">3</a> <a href="chap4_mj.html">4</a> <a href="chap5_mj.html">5</a> <a href="chap6_mj.html">6</a> <a href="chap7_mj.html">7</a> <a href="chap8_mj.html">8</a> <a href="chap9_mj.html">9</a> <a href="chap10_mj.html">10</a> <a href="chap11_mj.html">11</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap8_mj.html">[Previous Chapter]</a> <a href="chap10_mj.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap9.html">[MathJax off]</a></p>
<p><a id="X7A03A83E87FB1189" name="X7A03A83E87FB1189"></a></p>
<div class="ChapSects"><a href="chap9_mj.html#X7A03A83E87FB1189">9 <span class="Heading">Ambiguous Class Fusions in the <strong class="pkg">GAP</strong> Character Table Library</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X784492877DB04FE9">9.1 <span class="Heading">Some <strong class="pkg">GAP</strong> Utilities</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7EA839057D3AD3B4">9.2 <span class="Heading">Fusions Determined by Factorization through Intermediate Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X78DCEEFD85FF1EE2">9.2-1 <span class="Heading"><span class="SimpleMath">\(Co_3N5 \rightarrow Co_3\)</span> (September 2002)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X86BCEA907EC4C833">9.2-2 <span class="Heading"><span class="SimpleMath">\(31:15 \rightarrow B\)</span> (March 2003)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7C719F527831F35A">9.2-3 <span class="Heading"><span class="SimpleMath">\(SuzN3 \rightarrow Suz\)</span> (September 2002)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X828879F481EF30DD">9.2-4 <span class="Heading"><span class="SimpleMath">\(F_{{3+}}N5 \rightarrow F_{{3+}}\)</span> (March 2002)</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7981579278F81AC6">9.3 <span class="Heading">Fusions Determined Using Commutative Diagrams Involving Smaller
Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7F5186E28201B027">9.3-1 <span class="Heading"><span class="SimpleMath">\(BN7 \rightarrow B\)</span> (March 2002)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X79710B137B5BB1B8">9.3-2 <span class="Heading"><span class="SimpleMath">\((A_4 \times O_8^+(2).3).2 \rightarrow Fi_{24}^{\prime}\)</span> (November 2002)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X85822C647B29117B">9.3-3 <span class="Heading"><span class="SimpleMath">\(A_6 \times L_2(8).3 \rightarrow Fi_{24}^{\prime}\)</span> (November 2002)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X81A607758682D9A9">9.3-4 <span class="Heading"><span class="SimpleMath">\((3^2:D_8 \times U_4(3).2^2).2 \rightarrow B\)</span> (June 2007)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7962DD4387D63675">9.3-5 <span class="Heading"><span class="SimpleMath">\(7^{1+4}:(3 \times 2.S_7) \rightarrow M\)</span> (May 2009)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X860B6C30812DE3FC">9.3-6 <span class="Heading"><span class="SimpleMath">\(3^7.O_7(3):2 \rightarrow Fi_{24}\)</span> (November 2010)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7C3AC42F8342EE2E">9.3-7 <span class="Heading"><span class="SimpleMath">\({}^2E_6(2)N3C \rightarrow {}^2E_6(2)\)</span> (January 2019)</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X84F966E2824F5D52">9.4 <span class="Heading">Fusions Determined Using Commutative Diagrams Involving Factor
Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7F2B104686509CAA">9.4-1 <span class="Heading"><span class="SimpleMath">\(3.A_7 \rightarrow 3.Suz\)</span> (December 2010)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X82FB71647D37F4FD">9.4-2 <span class="Heading"><span class="SimpleMath">\(S_6 \rightarrow U_4(2)\)</span> (September 2011)</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7CFBC41B818A318C">9.5 <span class="Heading">Fusions Determined Using Commutative Diagrams Involving
Automorphic Extensions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7E91F8707BA93081">9.5-1 <span class="Heading"><span class="SimpleMath">\(U_3(8).3_1 \rightarrow {}^2E_6(2)\)</span> (December 2010)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X81B37EF378E89E00">9.5-2 <span class="Heading"><span class="SimpleMath">\(L_3(4).2_1 \rightarrow U_6(2)\)</span> (December 2010)</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X85E2A6F480026C95">9.6 <span class="Heading">Conditions Imposed by Brauer Tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7ACC7F588213D5D5">9.6-1 <span class="Heading"><span class="SimpleMath">\(L_2(16).4 \rightarrow J_3.2\)</span> (January 2004)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7ACB86CB82ED49D1">9.6-2 <span class="Heading"><span class="SimpleMath">\(L_2(17) \rightarrow S_8(2)\)</span> (July 2004)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7DED4C437D479226">9.6-3 <span class="Heading"><span class="SimpleMath">\(L_2(19) \rightarrow J_3\)</span> (April 2003)</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X8225D9FA80A7D20F">9.7 <span class="Heading">Fusions Determined by Information about the Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7AE2962E82B4C814">9.7-1 <span class="Heading"><span class="SimpleMath">\(U_3(3).2 \rightarrow Fi_{24}^{\prime}\)</span> (November 2002)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X83061094871EE241">9.7-2 <span class="Heading"><span class="SimpleMath">\(L_2(13).2 \rightarrow Fi_{24}^{\prime}\)</span> (September 2002)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7E9C203C7C4D709D">9.7-3 <span class="Heading"><span class="SimpleMath">\(M_{11} \rightarrow B\)</span> (April 2009)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X85821D748716DC7E">9.7-4 <span class="Heading"><span class="SimpleMath">\(L_2(11):2 \rightarrow B\)</span> (April 2009)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X828D81487F57D612">9.7-5 <span class="Heading"><span class="SimpleMath">\(L_3(3) \rightarrow B\)</span> (April 2009)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7B4E13337D66020F">9.7-6 <span class="Heading"><span class="SimpleMath">\(L_2(17).2 \rightarrow B\)</span> (March 2004)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X8528432A84851F7B">9.7-7 <span class="Heading"><span class="SimpleMath">\(L_2(49).2_3 \rightarrow B\)</span> (June 2006)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7EAD52AA7A28D956">9.7-8 <span class="Heading"><span class="SimpleMath">\(2^3.L_3(2) \rightarrow G_2(5)\)</span> (January 2004)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X79617107849A6CEA">9.7-9 <span class="Heading"><span class="SimpleMath">\(5^{{1+4}}.2^{{1+4}}.A_5.4 \rightarrow B\)</span> (April 2009)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X85C48EEB7B711C09">9.7-10 <span class="Heading">The fusion from the character table of <span class="SimpleMath">\(7^2:2L_2(7).2\)</span>
into the table of marks (January 2004)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7B1C689C7EFD07CB">9.7-11 <span class="Heading"><span class="SimpleMath">\(3 \times U_4(2) \rightarrow 3_1.U_4(3)\)</span> (March 2010)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7A94F78C792122D5">9.7-12 <span class="Heading"><span class="SimpleMath">\(2.3^4.2^3.S_4 \rightarrow 2.A12\)</span> (September 2011)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7E2AF30C7E8F89F9">9.7-13 <span class="Heading"><span class="SimpleMath">\(127:7 \rightarrow L_7(2)\)</span> (January 2012)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7E7B2AD67ACD27AE">9.7-14 <span class="Heading"><span class="SimpleMath">\(L_2(59) \rightarrow M\)</span> (May 2009)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X8409DA2E83A41ABE">9.7-15 <span class="Heading"><span class="SimpleMath">\(L_2(71) \rightarrow M\)</span> (May 2009)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X78B3B1BE7A2CA4D1">9.7-16 <span class="Heading"><span class="SimpleMath">\(L_2(41) \rightarrow M\)</span> (April 2012)</span></a>
</span>
</div></div>
</div>
<h3>9 <span class="Heading">Ambiguous Class Fusions in the <strong class="pkg">GAP</strong> Character Table Library</span></h3>
<p>Date: January 11th, 2004</p>
<p>This is a collection of examples showing how class fusions between character tables can be determined using the <strong class="pkg">GAP</strong> system <a href="chapBib_mj.html#biBGAP">[GAP21]</a>. In each of these examples, the fusion is <em>ambiguous</em> in the sense that the character tables do not determine it up to table automorphisms. Our strategy is to compute first all possibilities with the <strong class="pkg">GAP</strong> function <code class="func">PossibleClassFusions</code> (<a href="../../../doc/ref/chap73_mj.html#X7883271F7F26356E"><span class="RefLink">Reference: PossibleClassFusions</span></a>), and then to use either other character tables or information about the groups for excluding some of these candidates until only one (orbit under table automorphisms) remains.</p>
<p>The purpose of this writeup is twofold. On the one hand, the computations are documented this way. On the other hand, the <strong class="pkg">GAP</strong> code shown for the examples can be used as test input for automatic checking of the data and the functions used; therefore, each example ends with a comparison of the result with the fusion that is actually stored in the <strong class="pkg">GAP</strong> Character Table Library <a href="chapBib_mj.html#biBCTblLib">[Bre24]</a>.</p>
<p>The examples use the <strong class="pkg">GAP</strong> Character Table Library, so we first load this package.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "ctbllib", false );</span>
true
</pre></div>
<p><a id="X784492877DB04FE9" name="X784492877DB04FE9"></a></p>
<h4>9.1 <span class="Heading">Some <strong class="pkg">GAP</strong> Utilities</span></h4>
<p>The function <code class="code">SetOfComposedClassFusions</code> takes two list of class fusions, where the first list consists of fusions between the character tables of the groups <span class="SimpleMath">\(H\)</span> and <span class="SimpleMath">\(G\)</span>, say, and the second list consists of class fusions between the character tables of the groups <span class="SimpleMath">\(U\)</span> and <span class="SimpleMath">\(H\)</span>, say; the return value is the set of compositions of each map in the first list with each map in the second list (via <code class="func">CompositionMaps</code> (<a href="../../../doc/ref/chap73_mj.html#X8740C1397C6A96C8"><span class="RefLink">Reference: CompositionMaps</span></a>)).</p>
<p>Note that the returned list may be a proper subset of the set of all possible class fusions between <span class="SimpleMath">\(U\)</span> and <span class="SimpleMath">\(G\)</span>, which can be computed with <code class="func">PossibleClassFusions</code> (<a href="../../../doc/ref/chap73_mj.html#X7883271F7F26356E"><span class="RefLink">Reference: PossibleClassFusions</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetOfComposedClassFusions:= function( hfusg, ufush )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> local result, map1, map2;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> result:= [];;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> for map2 in hfusg do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> for map1 in ufush do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> AddSet( result, CompositionMaps( map2, map1 ) );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> return result;</span>
<span class="GAPprompt">></span> <span class="GAPinput">end;;</span>
</pre></div>
<p><a id="X7EA839057D3AD3B4" name="X7EA839057D3AD3B4"></a></p>
<h4>9.2 <span class="Heading">Fusions Determined by Factorization through Intermediate Subgroups</span></h4>
<p>This situation clearly occurs only for nonmaximal subgroups. Interesting examples are Sylow normalizers.</p>
<p><a id="X78DCEEFD85FF1EE2" name="X78DCEEFD85FF1EE2"></a></p>
<h5>9.2-1 <span class="Heading"><span class="SimpleMath">\(Co_3N5 \rightarrow Co_3\)</span> (September 2002)</span></h5>
<p>Let <span class="SimpleMath">\(H\)</span> be the Sylow <span class="SimpleMath">\(5\)</span> normalizer in the sporadic simple group <span class="SimpleMath">\(Co_3\)</span>. The class fusion of <span class="SimpleMath">\(H\)</span> into <span class="SimpleMath">\(Co_3\)</span> is not uniquely determined by the character tables of the two groups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">co3:= CharacterTable( "Co3" );</span>
CharacterTable( "Co3" )
<span class="GAPprompt">gap></span> <span class="GAPinput">h:= CharacterTable( "Co3N5" );</span>
CharacterTable( "5^(1+2):(24:2)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">hfusco3:= PossibleClassFusions( h, co3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( h, hfusco3, co3 ) );</span>
2
</pre></div>
<p>As <span class="SimpleMath">\(H\)</span> is not maximal in <span class="SimpleMath">\(Co_3\)</span>, we look at those maximal subgroups of <span class="SimpleMath">\(Co_3\)</span> whose order is divisible by that of <span class="SimpleMath">\(H\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mx:= Maxes( co3 );</span>
[ "McL.2", "HS", "U4(3).(2^2)_{133}", "M23", "3^5:(2xm11)",
"2.S6(2)", "U3(5).3.2", "3^1+4:4s6", "2^4.a8", "L3(4).D12",
"2xm12", "2^2.[2^7*3^2].S3", "s3xpsl(2,8).3", "a4xs5" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">maxes:= List( mx, CharacterTable );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( maxes, x -> Size( x ) mod Size( h ) = 0 );</span>
[ CharacterTable( "McL.2" ), CharacterTable( "HS" ),
CharacterTable( "U3(5).3.2" ) ]
</pre></div>
<p>According to the <strong class="pkg">Atlas</strong> (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, pp. 34 and 100]</a>), <span class="SimpleMath">\(H\)</span> occurs as the Sylow <span class="SimpleMath">\(5\)</span> normalizer in <span class="SimpleMath">\(U_3(5).3.2\)</span> and in <span class="SimpleMath">\(McL.2\)</span>; however, <span class="SimpleMath">\(H\)</span> is not a subgroup of <span class="SimpleMath">\(HS\)</span>, since otherwise <span class="SimpleMath">\(H\)</span> would be contained in subgroups of type <span class="SimpleMath">\(U_3(5).2\)</span> (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 80]</a>), but the only possible subgroups in these groups are too small (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 34]</a>).</p>
<p>We compute the possible class fusions from <span class="SimpleMath">\(H\)</span> into <span class="SimpleMath">\(McL.2\)</span> and from <span class="SimpleMath">\(McL.2\)</span> to <span class="SimpleMath">\(Co_3\)</span>, and then form the compositions of these maps.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">max:= filt[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">hfusmax:= PossibleClassFusions( h, max );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">maxfusco3:= PossibleClassFusions( max, co3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( maxfusco3, hfusmax );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( comp );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">reps:= RepresentativesFusions( h, comp, co3 );</span>
[ [ 1, 2, 3, 4, 8, 8, 7, 9, 10, 11, 17, 17, 19, 19, 22, 23, 27, 27,
30, 33, 34, 40, 40, 40, 40, 42 ] ]
</pre></div>
<p>So factoring through a maximal subgroup of type <span class="SimpleMath">\(McL.2\)</span> determines the fusion from <span class="SimpleMath">\(H\)</span> to <span class="SimpleMath">\(Co_3\)</span> uniquely up to table automorphisms.</p>
<p>Alternatively, we can use the group <span class="SimpleMath">\(U_3(5).3.2\)</span> as intermediate subgroup, which leads to the same result.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">max:= filt[3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">hfusmax:= PossibleClassFusions( h, max );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">maxfusco3:= PossibleClassFusions( max, co3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( maxfusco3, hfusmax );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">reps2:= RepresentativesFusions( h, comp, co3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">reps2 = reps;</span>
true
</pre></div>
<p>Finally, we compare the result with the map that is stored on the library table of <span class="SimpleMath">\(H\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( h, co3 ) in reps;</span>
true
</pre></div>
<p><a id="X86BCEA907EC4C833" name="X86BCEA907EC4C833"></a></p>
<h5>9.2-2 <span class="Heading"><span class="SimpleMath">\(31:15 \rightarrow B\)</span> (March 2003)</span></h5>
<p>The Sylow <span class="SimpleMath">\(31\)</span> normalizer <span class="SimpleMath">\(H\)</span> in the sporadic simple group <span class="SimpleMath">\(B\)</span> has the structure <span class="SimpleMath">\(31:15\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CharacterTable( "B" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:= CharacterTable( "31:15" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">hfusb:= PossibleClassFusions( h, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( h, hfusb, b ) );</span>
2
</pre></div>
<p>We determine the correct fusion using the fact that <span class="SimpleMath">\(H\)</span> is contained in a (maximal) subgroup of type <span class="SimpleMath">\(Th\)</span> in <span class="SimpleMath">\(B\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">th:= CharacterTable( "Th" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">hfusth:= PossibleClassFusions( h, th );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">thfusb:= PossibleClassFusions( th, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( thfusb, hfusth );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( comp );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">reps:= RepresentativesFusions( h, comp, b );</span>
[ [ 1, 145, 146, 82, 82, 19, 82, 7, 19, 82, 82, 19, 7, 82, 19, 82, 82
] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( h, b ) in reps;</span>
true
</pre></div>
<p><a id="X7C719F527831F35A" name="X7C719F527831F35A"></a></p>
<h5>9.2-3 <span class="Heading"><span class="SimpleMath">\(SuzN3 \rightarrow Suz\)</span> (September 2002)</span></h5>
<p>The class fusion from the Sylow <span class="SimpleMath">\(3\)</span> normalizer into the sporadic simple group <span class="SimpleMath">\(Suz\)</span> is not uniquely determined by the character tables of these groups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:= CharacterTable( "SuzN3" );</span>
CharacterTable( "3^5:(3^2:SD16)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">suz:= CharacterTable( "Suz" );</span>
CharacterTable( "Suz" )
<span class="GAPprompt">gap></span> <span class="GAPinput">hfussuz:= PossibleClassFusions( h, suz );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( h, hfussuz, suz ) );</span>
2
</pre></div>
<p>Since <span class="SimpleMath">\(H\)</span> is not maximal in <span class="SimpleMath">\(Suz\)</span>, we try to factorize the fusion through a suitable maximal subgroup.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">maxes:= List( Maxes( suz ), CharacterTable );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( maxes, x -> Size( x ) mod Size( h ) = 0 );</span>
[ CharacterTable( "3_2.U4(3).2_3'" ), CharacterTable( "3^5:M11" ),
CharacterTable( "3^2+4:2(2^2xa4)2" ) ]
</pre></div>
<p>The group <span class="SimpleMath">\(3_2.U_4(3).2_3^{\prime}\)</span> does not admit a fusion from <span class="SimpleMath">\(H\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PossibleClassFusions( h, filt[1] );</span>
[ ]
</pre></div>
<p>Definitely <span class="SimpleMath">\(3^5:M_{11}\)</span> contains a group isomorphic with <span class="SimpleMath">\(H\)</span>, because the Sylow <span class="SimpleMath">\(3\)</span> normalizer in <span class="SimpleMath">\(M_{11}\)</span> has the structure <span class="SimpleMath">\(3^2:SD_{16}\)</span>; using <span class="SimpleMath">\(3^{2+4}:2(2^2 \times A_4)2\)</span> would lead to the same result as we get below. We compute the compositions of possible class fusions.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">max:= filt[2];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">hfusmax:= PossibleClassFusions( h, max );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">maxfussuz:= PossibleClassFusions( max, suz );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( maxfussuz, hfusmax );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( h, comp, suz );</span>
[ [ 1, 2, 2, 4, 5, 4, 5, 5, 5, 5, 5, 6, 9, 9, 14, 15, 13, 16, 16, 14,
15, 13, 13, 13, 16, 15, 14, 16, 16, 16, 21, 21, 23, 22, 29, 29,
29, 38, 39 ] ]
</pre></div>
<p>So the factorization determines the fusion map up to table automorphisms. We check that this map is equal to the stored one.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( h, suz ) in repr;</span>
true
</pre></div>
<p><a id="X828879F481EF30DD" name="X828879F481EF30DD"></a></p>
<h5>9.2-4 <span class="Heading"><span class="SimpleMath">\(F_{{3+}}N5 \rightarrow F_{{3+}}\)</span> (March 2002)</span></h5>
<p>The class fusion from the table of the Sylow <span class="SimpleMath">\(5\)</span> normalizer <span class="SimpleMath">\(H\)</span> in the sporadic simple group <span class="SimpleMath">\(F_{{3+}}\)</span> into <span class="SimpleMath">\(F_{{3+}}\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f3p:= CharacterTable( "F3+" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:= CharacterTable( "F3+N5" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">hfusf3p:= PossibleClassFusions( h, f3p );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( h, hfusf3p, f3p ) );</span>
2
</pre></div>
<p><span class="SimpleMath">\(H\)</span> is not maximal in <span class="SimpleMath">\(F_{{3+}}\)</span>, so we look for tables of maximal subgroups that can contain <span class="SimpleMath">\(H\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">maxes:= List( Maxes( f3p ), CharacterTable );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( maxes, x -> Size( x ) mod Size( h ) = 0 );</span>
[ CharacterTable( "Fi23" ), CharacterTable( "2.Fi22.2" ),
CharacterTable( "(3xO8+(3):3):2" ), CharacterTable( "O10-(2)" ),
CharacterTable( "(A4xO8+(2).3).2" ), CharacterTable( "He.2" ),
CharacterTable( "F3+M14" ), CharacterTable( "(A5xA9):2" ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">possfus:= List( filt, x -> PossibleClassFusions( h, x ) );</span>
[ [ ], [ ], [ ], [ ],
[ [ 1, 69, 110, 12, 80, 121, 4, 72, 113, 11, 11, 79, 79, 120, 120,
3, 71, 11, 79, 23, 91, 112, 120, 132, 29, 32, 97, 100, 37,
37, 105, 105, 139, 140, 145, 146, 155, 155, 156, 156, 44,
44, 167, 167, 48, 48, 171, 171, 57, 57, 180, 180, 66, 66,
189, 189 ],
[ 1, 69, 110, 12, 80, 121, 4, 72, 113, 11, 11, 79, 79, 120,
120, 3, 71, 11, 79, 23, 91, 112, 120, 132, 29, 32, 97, 100,
37, 37, 105, 105, 140, 139, 146, 145, 156, 156, 155, 155,
44, 44, 167, 167, 48, 48, 171, 171, 57, 57, 180, 180, 66,
66, 189, 189 ] ], [ ], [ ], [ ] ]
</pre></div>
<p>We see that from the eight possible classes of maximal subgroups in <span class="SimpleMath">\(F_{{3+}}\)</span> that might contain <span class="SimpleMath">\(H\)</span>, only the group of type <span class="SimpleMath">\((A_4 \times O_8^+(2).3).2\)</span> admits a class fusion from <span class="SimpleMath">\(H\)</span>. Hence we can compute the compositions of the possible fusions from <span class="SimpleMath">\(H\)</span> into this group with the possible fusions from this group into <span class="SimpleMath">\(F_{{3+}}\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">max:= filt[5];</span>
CharacterTable( "(A4xO8+(2).3).2" )
<span class="GAPprompt">gap></span> <span class="GAPinput">hfusmax:= possfus[5];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">maxfusf3p:= PossibleClassFusions( max, f3p );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( maxfusf3p, hfusmax );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( comp );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( h, comp, f3p );</span>
[ [ 1, 2, 4, 12, 35, 54, 3, 3, 16, 9, 9, 11, 11, 40, 40, 2, 3, 9, 11,
35, 36, 13, 40, 90, 7, 22, 19, 20, 43, 43, 50, 50, 8, 8, 23,
23, 46, 46, 47, 47, 10, 10, 9, 9, 10, 10, 11, 11, 26, 26, 28,
28, 67, 67, 68, 68 ] ]
</pre></div>
<p>Finally, we check whether the map stored in the table library is correct.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( h, f3p ) in repr;</span>
true
</pre></div>
<p>Note that we did <em>not</em> determine the class fusion from the maximal subgroup <span class="SimpleMath">\((A_4 \times O_8^+(2).3).2\)</span> into <span class="SimpleMath">\(F_{{3+}}\)</span> up to table automorphisms (see Section <a href="chap9_mj.html#X79710B137B5BB1B8"><span class="RefLink">9.3-2</span></a> for this problem), since also the ambiguous result was enough for computing the fusion from <span class="SimpleMath">\(H\)</span> into <span class="SimpleMath">\(F_{{3+}}\)</span>.</p>
<p><a id="X7981579278F81AC6" name="X7981579278F81AC6"></a></p>
<h4>9.3 <span class="Heading">Fusions Determined Using Commutative Diagrams Involving Smaller
Subgroups</span></h4>
<p>In each of the following examples, the class fusion of a (not necessarily maximal) subgroup <span class="SimpleMath">\(M\)</span> of a group <span class="SimpleMath">\(G\)</span> into <span class="SimpleMath">\(G\)</span> is determined by considering a proper subgroup <span class="SimpleMath">\(U\)</span> of <span class="SimpleMath">\(M\)</span> whose class fusion into <span class="SimpleMath">\(G\)</span> can be computed, perhaps using another subgroup <span class="SimpleMath">\(S\)</span> of <span class="SimpleMath">\(G\)</span> that also contains <span class="SimpleMath">\(U\)</span>.</p>
<p><center> <img src="ambigfus1.png" alt="setup: some subgroups of G"/> </center></p>
<p><a id="X7F5186E28201B027" name="X7F5186E28201B027"></a></p>
<h5>9.3-1 <span class="Heading"><span class="SimpleMath">\(BN7 \rightarrow B\)</span> (March 2002)</span></h5>
<p>Let <span class="SimpleMath">\(H\)</span> be a Sylow <span class="SimpleMath">\(7\)</span> normalizer in the sporadic simple group <span class="SimpleMath">\(B\)</span>. The class fusion of <span class="SimpleMath">\(H\)</span> into <span class="SimpleMath">\(B\)</span> is not uniquely determined by the character tables of the two groups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CharacterTable( "B" );</span>
CharacterTable( "B" )
<span class="GAPprompt">gap></span> <span class="GAPinput">h:= CharacterTable( "BN7" );</span>
CharacterTable( "BN7" )
<span class="GAPprompt">gap></span> <span class="GAPinput">hfusb:= PossibleClassFusions( h, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( h, hfusb, b ) );</span>
2
</pre></div>
<p>Let us consider a maximal subgroup of the type <span class="SimpleMath">\(Th\)</span> in <span class="SimpleMath">\(B\)</span> (cf. <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 217]</a>). By <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 177]</a>, the Sylow <span class="SimpleMath">\(7\)</span> normalizers in <span class="SimpleMath">\(Th\)</span> are maximal subgroups of <span class="SimpleMath">\(Th\)</span> and have the structure <span class="SimpleMath">\(7^2:(3 \times 2S_4)\)</span>. Let <span class="SimpleMath">\(U\)</span> be such a subgroup.</p>
<p>Note that the only maximal subgroups of <span class="SimpleMath">\(Th\)</span> whose order is divisible by the order of a Sylow <span class="SimpleMath">\(7\)</span> subgroup of <span class="SimpleMath">\(B\)</span> have the types <span class="SimpleMath">\({}^3D_4(2).3\)</span> and <span class="SimpleMath">\(7^2:(3 \times 2S_4)\)</span>, and the Sylow <span class="SimpleMath">\(7\)</span> normalizers in the former groups have the structure <span class="SimpleMath">\(7^2:(3 \times 2A_4)\)</span>, cf. <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 89]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Number( Factors( Size( b ) ), x -> x = 7 );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">th:= CharacterTable( "Th" );</span>
CharacterTable( "Th" )
<span class="GAPprompt">gap></span> <span class="GAPinput">Filtered( Maxes( th ), x -> Size( CharacterTable( x ) ) mod 7^2 = 0 );</span>
[ "3D4(2).3", "7^2:(3x2S4)" ]
</pre></div>
<p>The class fusion of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(B\)</span> via <span class="SimpleMath">\(Th\)</span> is uniquely determined by the character tables of these groups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">thn7:= CharacterTable( "ThN7" );</span>
CharacterTable( "7^2:(3x2S4)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( PossibleClassFusions( th, b ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PossibleClassFusions( thn7, th ) );</span>
[ [ 1, 31, 7, 7, 5, 28, 28, 17, 72, 72, 6, 6, 7, 28, 27, 27, 109,
109, 17, 45, 45, 72, 72, 127, 127, 127, 127 ] ]
</pre></div>
<p>The condition that the class fusion of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(B\)</span> factors through <span class="SimpleMath">\(H\)</span> determines the class fusion of <span class="SimpleMath">\(H\)</span> into <span class="SimpleMath">\(B\)</span> up to table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">thn7fush:= PossibleClassFusions( thn7, h );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( hfusb, x -></span>
<span class="GAPprompt">></span> <span class="GAPinput"> ForAny( thn7fush, y -> CompositionMaps( x, y ) in comp ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( h, filt, b ) );</span>
1
</pre></div>
<p>Finally, we compare the result with the map that is stored on the library table of <span class="SimpleMath">\(H\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( h, b ) in filt;</span>
true
</pre></div>
<p><a id="X79710B137B5BB1B8" name="X79710B137B5BB1B8"></a></p>
<h5>9.3-2 <span class="Heading"><span class="SimpleMath">\((A_4 \times O_8^+(2).3).2 \rightarrow Fi_{24}^{\prime}\)</span> (November 2002)</span></h5>
<p>The class fusion of the maximal subgroup <span class="SimpleMath">\(M \cong (A_4 \times O_8^+(2).3).2\)</span> of <span class="SimpleMath">\(G = Fi_{24}^{\prime}\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= CharacterTable( "(A4xO8+(2).3).2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "F3+" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mfust:= PossibleClassFusions( m, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( m, mfust, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( repr );</span>
2
</pre></div>
<p>We first observe that the elements of order three in the normal subgroup of type <span class="SimpleMath">\(A_4\)</span> in <span class="SimpleMath">\(M\)</span> lie in the class <code class="code">3A</code> of <span class="SimpleMath">\(Fi_{24}^{\prime}\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a4inm:= Filtered( ClassPositionsOfNormalSubgroups( m ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> n -> Sum( SizesConjugacyClasses( m ){ n } ) = 12 );</span>
[ [ 1, 69, 110 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( m ){ a4inm[1] };</span>
[ 1, 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( repr, map -> map[110] );</span>
[ 4, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( t ){ [ 1 .. 4 ] };</span>
[ 1, 2, 2, 3 ]
</pre></div>
<p>Let us take one such element <span class="SimpleMath">\(g\)</span>, say. Its normalizer <span class="SimpleMath">\(S\)</span> in <span class="SimpleMath">\(G\)</span> has the structure <span class="SimpleMath">\((3 \times O_8^+(3).3).2\)</span>; this group is maximal in <span class="SimpleMath">\(G\)</span>, and its character table is available in <strong class="pkg">GAP</strong>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "F3+N3A" );</span>
CharacterTable( "(3xO8+(3):3):2" )
</pre></div>
<p>The intersection <span class="SimpleMath">\(N_M(g) = S \cap M\)</span> contains a subgroup <span class="SimpleMath">\(U\)</span> of the type <span class="SimpleMath">\(3 \times O_8^+(2).3\)</span>, and in the following we compute the class fusions of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(S\)</span> and <span class="SimpleMath">\(M\)</span>, and then utilize the fact that only those class fusions from <span class="SimpleMath">\(M\)</span> into <span class="SimpleMath">\(G\)</span> are possible whose composition with the class fusion from <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(M\)</span> equals a composition of class fusions from <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(S\)</span> and from <span class="SimpleMath">\(S\)</span> into <span class="SimpleMath">\(G\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( "Cyclic", 3 ) * CharacterTable( "O8+(2).3" );</span>
CharacterTable( "C3xO8+(2).3" )
<span class="GAPprompt">gap></span> <span class="GAPinput">ufuss:= PossibleClassFusions( u, s );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusm:= PossibleClassFusions( u, m );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( sfust, ufuss );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( comp );</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( mfust,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x -> ForAny( ufusm, map -> CompositionMaps( x, map ) in comp ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( m, filt, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( repr );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( m, t ) in repr;</span>
true
</pre></div>
<p>So the class fusion from <span class="SimpleMath">\(M\)</span> into <span class="SimpleMath">\(G\)</span> is determined up to table automorphisms by the commutative diagram.</p>
<p><a id="X85822C647B29117B" name="X85822C647B29117B"></a></p>
<h5>9.3-3 <span class="Heading"><span class="SimpleMath">\(A_6 \times L_2(8).3 \rightarrow Fi_{24}^{\prime}\)</span> (November 2002)</span></h5>
<p>The class fusion of the maximal subgroup <span class="SimpleMath">\(M \cong A_6 \times L_2(8).3\)</span> of <span class="SimpleMath">\(G = Fi_{24}^{\prime}\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= CharacterTable( "A6xL2(8):3" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "F3+" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mfust:= PossibleClassFusions( m, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( m, mfust, t ) );</span>
2
</pre></div>
<p>We will use the fact that the direct factor of the type <span class="SimpleMath">\(A_6\)</span> in <span class="SimpleMath">\(M\)</span> contains elements in the class <code class="code">3A</code> of <span class="SimpleMath">\(G\)</span>. This fact can be shown as follows.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">dppos:= ClassPositionsOfDirectProductDecompositions( m );</span>
[ [ [ 1, 12 .. 67 ], [ 1 .. 11 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( dppos[1], l -> Sum( SizesConjugacyClasses( t ){ l } ) );</span>
[ 17733424133316996808705, 4545066196775803392 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( dppos[1], l -> Sum( SizesConjugacyClasses( m ){ l } ) );</span>
[ 360, 1512 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">3Apos:= Position( OrdersClassRepresentatives( t ), 3 );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">3Ainm:= List( mfust, map -> Position( map, 3Apos ) );</span>
[ 23, 23, 23, 23, 34, 34, 34, 34 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( 3Ainm, x -> x in dppos[1][1] );</span>
true
</pre></div>
<p>Since the normalizer of an element of order three in <span class="SimpleMath">\(A_6\)</span> has the form <span class="SimpleMath">\(3^2:2\)</span>, such a <code class="code">3A</code> element in <span class="SimpleMath">\(M\)</span> contains a subgroup <span class="SimpleMath">\(U\)</span> of the structure <span class="SimpleMath">\(3^2:2 \times L_2(8).3\)</span> which is contained in the <code class="code">3A</code> normalizer <span class="SimpleMath">\(S\)</span> in <span class="SimpleMath">\(G\)</span>, which has the structure <span class="SimpleMath">\((3 \times O_8^+(3).3).2\)</span>.</p>
<p>(Note that all classes in the <span class="SimpleMath">\(3^2:2\)</span> type group are rational, and its character table is available in the <strong class="pkg">GAP</strong> Character Table Library with the identifier <code class="code">"3^2:2"</code>.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( "3^2:2" ) * CharacterTable( "L2(8).3" );</span>
CharacterTable( "3^2:2xL2(8).3" )
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "F3+N3A" );</span>
CharacterTable( "(3xO8+(3):3):2" )
<span class="GAPprompt">gap></span> <span class="GAPinput">ufuss:= PossibleClassFusions( u, s );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( sfust, ufuss );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusm:= PossibleClassFusions( u, m );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( mfust,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map -> ForAny( ufusm,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map2 -> CompositionMaps( map, map2 ) in comp ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( m, filt, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( repr );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( m, t ) in repr;</span>
true
</pre></div>
<p><a id="X81A607758682D9A9" name="X81A607758682D9A9"></a></p>
<h5>9.3-4 <span class="Heading"><span class="SimpleMath">\((3^2:D_8 \times U_4(3).2^2).2 \rightarrow B\)</span> (June 2007)</span></h5>
<p>Let <span class="SimpleMath">\(G\)</span> be a maximal subgroup of the type <span class="SimpleMath">\((3^2:D_8 \times U_4(3).2^2).2\)</span> in the sporadic simple group <span class="SimpleMath">\(B\)</span>, cf. <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 217]</a>. Computing the class fusion of <span class="SimpleMath">\(G\)</span> into <span class="SimpleMath">\(B\)</span> just from the character tables of the two groups takes extremely long. So we use additional information.</p>
<p>According to <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 217]</a>, <span class="SimpleMath">\(G\)</span> is the normalizer in <span class="SimpleMath">\(B\)</span> of an elementary abelian group <span class="SimpleMath">\(\langle x, y \rangle\)</span> of order <span class="SimpleMath">\(9\)</span>, with <span class="SimpleMath">\(x, y\)</span> in the class <code class="code">3A</code> of <span class="SimpleMath">\(B\)</span>, and <span class="SimpleMath">\(N = N_B(\langle x \rangle)\)</span> has the structure <span class="SimpleMath">\(S_3 \times Fi_{22}.2\)</span>. The intersection <span class="SimpleMath">\(G \cap N\)</span> has the structure <span class="SimpleMath">\(S_3 \times S_3 \times U_4(3).2^2\)</span>, which is the direct product of <span class="SimpleMath">\(S_3\)</span> and the normalizer in <span class="SimpleMath">\(Fi_{22}.2\)</span> of a <code class="code">3A</code> element of <span class="SimpleMath">\(Fi_{22}.2\)</span>, see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 163]</a>. Thus we may use that the class fusions from <span class="SimpleMath">\(G \cap N\)</span> into <span class="SimpleMath">\(B\)</span> through <span class="SimpleMath">\(G\)</span> or <span class="SimpleMath">\(N\)</span> coincide.</p>
<p>The class fusion from <span class="SimpleMath">\(N\)</span> into <span class="SimpleMath">\(B\)</span> is uniquely determined by the character tables.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CharacterTable( "B" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:= CharacterTable( "BN3A" );</span>
CharacterTable( "S3xFi22.2" )
<span class="GAPprompt">gap></span> <span class="GAPinput">nfusb:= PossibleClassFusions( n, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( nfusb );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">nfusb:= nfusb[1];;</span>
</pre></div>
<p>The computation of the class fusion from <span class="SimpleMath">\(G \cap N\)</span> into <span class="SimpleMath">\(N\)</span> is sped up by computing first the class fusion modulo the direct factor <span class="SimpleMath">\(S_3\)</span>, and then lifting these fusion maps.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">fi222:= CharacterTable( "Fi22.2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fi222n3a:= CharacterTable( "S3xU4(3).(2^2)_{122}" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s3:= CharacterTable( "S3" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">inter:= s3 * fi222n3a;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">intermods3fusnmods3:= PossibleClassFusions( fi222n3a, fi222 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( intermods3fusnmods3 );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( fi222n3a, intermods3fusnmods3, fi222 ) );</span>
1
</pre></div>
<p>We get two equivalent possibilities, and need to consider only one of them. For lifting it to a map between <span class="SimpleMath">\(G \cap N\)</span> and <span class="SimpleMath">\(N\)</span>, the safe way is to use the fusion map between the two factors for computing an approximation. (Additionally, we could interpret the known maps as fusions between two subgroups, and use this for improving the approximation, but in this case the speedup is not worth the effort.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">interfusn:= CompositionMaps( InverseMap( GetFusionMap( n, fi222 ) ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> CompositionMaps( intermods3fusnmods3[1],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GetFusionMap( inter, fi222n3a ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">interfusn:= PossibleClassFusions( inter, n,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> rec( fusionmap:= interfusn, quick:= true ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( interfusn );</span>
1
</pre></div>
<p>The lift is unique. Since we lift a class fusion to direct products, we could also "extend" the fusion directly. But note that this would assume the ordering of classes in character tables of direct products. This alternative would work as follows.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">nccl:= NrConjugacyClasses( fi222 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">interfusn[1] = Concatenation( List( [ 0 .. 2 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> i -> intermods3fusnmods3[1] + i * nccl ) );</span>
true
</pre></div>
<p>Next we compute the class fusions from <span class="SimpleMath">\(G \cap N\)</span> to <span class="SimpleMath">\(G\)</span>. We get two equivalent solutions.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tblg:= CharacterTable( "BM14" );</span>
CharacterTable( "(3^2:D8xU4(3).2^2).2" )
<span class="GAPprompt">gap></span> <span class="GAPinput">interfusg:= PossibleClassFusions( inter, tblg );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( interfusg );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( inter, interfusg, tblg ) );</span>
1
</pre></div>
<p>The approximation of the class fusion from <span class="SimpleMath">\(G\)</span> to <span class="SimpleMath">\(B\)</span> is computed by composing the known maps. Because we have chosen one of the two possible maps from <span class="SimpleMath">\(G \cap N\)</span> to <span class="SimpleMath">\(N\)</span>, here we consider the two possibilities. From these approximations, we compute the possible class fusions.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">interfusb:= CompositionMaps( nfusb, interfusn[1] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">approx:= List( interfusg,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map -> CompositionMaps( interfusb, InverseMap( map ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gfusb:= Set( Concatenation( List( approx,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map -> PossibleClassFusions( tblg, b,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> rec( fusionmap:= map ) ) ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( gfusb );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( tblg, gfusb, b ) );</span>
1
</pre></div>
<p>Finally, we compare the result with the class fusion that is stored on the library table.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( tblg, b ) in gfusb;</span>
true
</pre></div>
<p><a id="X7962DD4387D63675" name="X7962DD4387D63675"></a></p>
<h5>9.3-5 <span class="Heading"><span class="SimpleMath">\(7^{1+4}:(3 \times 2.S_7) \rightarrow M\)</span> (May 2009)</span></h5>
<p>The class fusion of the maximal subgroup <span class="SimpleMath">\(U\)</span> of type <span class="SimpleMath">\(7^{1+4}:(3 \times 2.S_7)\)</span> of the Monster group <span class="SimpleMath">\(M\)</span> into <span class="SimpleMath">\(M\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tblu:= CharacterTable( "7^(1+4):(3x2.S7)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= CharacterTable( "M" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusm:= PossibleClassFusions( tblu, m );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( tblu, ufusm, m ) );</span>
2
</pre></div>
<p>The subgroup <span class="SimpleMath">\(U\)</span> contains a Sylow <span class="SimpleMath">\(7\)</span>-subgroup of <span class="SimpleMath">\(M\)</span>, and the only maximal subgroups of <span class="SimpleMath">\(M\)</span> with this property are the class of <span class="SimpleMath">\(U\)</span> and another class of subgroups, of the type <span class="SimpleMath">\(7^{2+1+2}:GL_2(7)\)</span>. Moreover, it turns out that the Sylow <span class="SimpleMath">\(7\)</span> normalizers in the subgroups in both classes have the same order, hence they are the Sylow <span class="SimpleMath">\(7\)</span> normalizers in <span class="SimpleMath">\(M\)</span>.</p>
<p>For that, we use representations from the <strong class="pkg">Atlas</strong> of Group Representations <a href="chapBib_mj.html#biBAGRv3">[WWT+]</a>, and access these representations via the <strong class="pkg">GAP</strong> package <strong class="pkg">AtlasRep</strong> (<a href="chapBib_mj.html#biBAtlasRep">[WPN+22]</a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "atlasrep", false );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">g1:= AtlasGroup( "7^(2+1+2):GL2(7)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s1:= SylowSubgroup( g1, 7 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">n1:= Normalizer( g1, s1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g2:= AtlasGroup( "7^(1+4):(3x2.S7)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s2:= SylowSubgroup( g2, 7 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">n2:= Normalizer( g2, s2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( n1 ) = Size( n2 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">( Size( m ) / Size( s1 ) ) mod 7 <> 0;</span>
true
</pre></div>
<p>So let <span class="SimpleMath">\(N\)</span> be a Sylow <span class="SimpleMath">\(7\)</span> normalizer in <span class="SimpleMath">\(U\)</span>, and choose a subgroup <span class="SimpleMath">\(S\)</span> of the type <span class="SimpleMath">\(7^{2+1+2}:GL_2(7)\)</span> that contains <span class="SimpleMath">\(N\)</span>.</p>
<p>We compute the character table of <span class="SimpleMath">\(N\)</span>. Computing the possible class fusions of <span class="SimpleMath">\(N\)</span> into <span class="SimpleMath">\(M\)</span> directly yields two possibilities, but the class fusion of <span class="SimpleMath">\(N\)</span> into <span class="SimpleMath">\(M\)</span> via <span class="SimpleMath">\(S\)</span> is uniquely determined by the character tables.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbln:= CharacterTable( Image( IsomorphismPcGroup( n1 ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbls:= CharacterTable( "7^(2+1+2):GL2(7)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">nfusm:= PossibleClassFusions( tbln, m );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( tbln, nfusm, m ) );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">nfuss:= PossibleClassFusions( tbln, tbls );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfusm:= PossibleClassFusions( tbls, m );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">nfusm:= SetOfComposedClassFusions( sfusm, nfuss );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( nfusm );</span>
1
</pre></div>
<p>Now we use the condition that the class fusions from <span class="SimpleMath">\(N\)</span> into <span class="SimpleMath">\(M\)</span> factors through <span class="SimpleMath">\(U\)</span>. This determines the class fusion of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(M\)</span> up to table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">nfusu:= PossibleClassFusions( tbln, tblu );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusm:= Filtered( ufusm, map2 -> ForAny( nfusu, </span>
<span class="GAPprompt">></span> <span class="GAPinput"> map1 -> CompositionMaps( map2, map1 ) in nfusm ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( tblu, ufusm, m ) );</span>
1
</pre></div>
<p>Let <span class="SimpleMath">\(C\)</span> be the centralizer in <span class="SimpleMath">\(U\)</span> of the normal subgroup of order <span class="SimpleMath">\(7\)</span>; note that <span class="SimpleMath">\(C\)</span> is the <code class="code">7B</code> centralizer on <span class="SimpleMath">\(M\)</span>. We can use the information about the class fusion of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(M\)</span> for determining the class fusion of <span class="SimpleMath">\(C\)</span> into <span class="SimpleMath">\(M\)</span>. The class fusion of <span class="SimpleMath">\(C\)</span> into <span class="SimpleMath">\(M\)</span> is not determined by the character tables, but the class fusion of <span class="SimpleMath">\(C\)</span> into <span class="SimpleMath">\(U\)</span> is determined up to table automorphisms, so the same holds for the class fusion of <span class="SimpleMath">\(C\)</span> into <span class="SimpleMath">\(M\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tblc:= CharacterTable( "MC7B" ); </span>
CharacterTable( "7^1+4.2A7" )
<span class="GAPprompt">gap></span> <span class="GAPinput">cfusm:= PossibleClassFusions( tblc, m );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( tblc, cfusm, m ) );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">cfusu:= PossibleClassFusions( tblc, tblu );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cfusm:= SetOfComposedClassFusions( ufusm, cfusu );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( tblc, cfusm, m ) );</span>
1
</pre></div>
<p><a id="X860B6C30812DE3FC" name="X860B6C30812DE3FC"></a></p>
<h5>9.3-6 <span class="Heading"><span class="SimpleMath">\(3^7.O_7(3):2 \rightarrow Fi_{24}\)</span> (November 2010)</span></h5>
<p>The class fusion of the maximal subgroup <span class="SimpleMath">\(M \cong 3^7.O_7(3):2\)</span> of <span class="SimpleMath">\(G = Fi_{24} = F_{{3+}}.2\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= CharacterTable( "3^7.O7(3):2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "F3+.2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mfust:= PossibleClassFusions( m, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( m, mfust, t ) );</span>
2
</pre></div>
<p>We will use the fact that the elementary abelian normal subgroup of order <span class="SimpleMath">\(3^7\)</span> in <span class="SimpleMath">\(M\)</span> contains an element <span class="SimpleMath">\(x\)</span>, say, in the class <code class="code">3A</code> of <span class="SimpleMath">\(G\)</span>. This fact can be shown as follows.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">nsg:= ClassPositionsOfNormalSubgroups( m );</span>
[ [ 1 ], [ 1 .. 4 ], [ 1 .. 158 ], [ 1 .. 291 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Sum( SizesConjugacyClasses( m ){ nsg[2] } );</span>
2187
<span class="GAPprompt">gap></span> <span class="GAPinput">3^7;</span>
2187
<span class="GAPprompt">gap></span> <span class="GAPinput">rest:= Set( mfust, map -> map{ nsg[2] } );</span>
[ [ 1, 4, 5, 6 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( rest, l -> ClassNames( t, "Atlas" ){ l } );</span>
[ [ "1A", "3A", "3B", "3C" ] ]
</pre></div>
<p>The normalizer <span class="SimpleMath">\(S\)</span> of <span class="SimpleMath">\(\langle x \rangle\)</span> in <span class="SimpleMath">\(G\)</span> has the form <span class="SimpleMath">\(S_3 \times O_8^+(3):S_3\)</span>, and the order of <span class="SimpleMath">\(U = S \cap M = N_M( \langle x \rangle)\)</span> is <span class="SimpleMath">\(53059069440\)</span>, so <span class="SimpleMath">\(U\)</span> has index <span class="SimpleMath">\(3360\)</span> in <span class="SimpleMath">\(S\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "F3+.2N3A" );</span>
CharacterTable( "S3xO8+(3):S3" )
<span class="GAPprompt">gap></span> <span class="GAPinput">PowerMap( m, 2 )[4];</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">size_u:= 2 * SizesCentralizers( m )[ 2 ];</span>
53059069440
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( s ) / size_u;</span>
3360
</pre></div>
<p>Using the list of maximal subgroups of <span class="SimpleMath">\(O_8^+(3)\)</span>, we see that only the maximal subgroups of the type <span class="SimpleMath">\(3^6:L_4(3)\)</span> have index dividing <span class="SimpleMath">\(3360\)</span> in <span class="SimpleMath">\(O_8^+(3)\)</span>. (There are three classes of such subgroups.) This implies that <span class="SimpleMath">\(U\)</span> contains a subgroup of the type <span class="SimpleMath">\(S_3 \times 3^6:L_4(3)\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">o8p3:= CharacterTable( "O8+(3)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mx:= List( Maxes( o8p3 ), CharacterTable );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( mx, x -> 3360 mod Index( o8p3, x ) = 0 );</span>
[ CharacterTable( "3^6:L4(3)" ), CharacterTable( "O8+(3)M8" ),
CharacterTable( "O8+(3)M9" ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( filt, x -> Index( o8p3, x ) );</span>
[ 1120, 1120, 1120 ]
</pre></div>
<p>We compute the possible class fusions from <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(S\)</span> in two steps, because this is faster. First the possible class fusions from <span class="SimpleMath">\(U^{\prime\prime} \cong 3^6:L_4(3)\)</span> into <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(S\)</span> are computed, and then these fusions are used to derive approximations for the fusions from <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(S\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">uu:= filt[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( "Symmetric", 3 ) * uu;</span>
CharacterTable( "Sym(3)x3^6:L4(3)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">uufusm:= PossibleClassFusions( uu, m );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( uufusm );</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">approx:= List( uufusm, map -> CompositionMaps( map,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> InverseMap( GetFusionMap( uu, u ) ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusm:= Concatenation( List( approx, map -></span>
<span class="GAPprompt">></span> <span class="GAPinput"> PossibleClassFusions( u, m, rec( fusionmap:= map ) ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( ufusm );</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">uufuss:= PossibleClassFusions( uu, s );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( uufuss );</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">approx:= List( uufuss, map -> CompositionMaps( map,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> InverseMap( GetFusionMap( uu, u ) ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufuss:= Concatenation( List( approx, map -></span>
<span class="GAPprompt">></span> <span class="GAPinput"> PossibleClassFusions( u, s, rec( fusionmap:= map ) ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( ufuss );</span>
8
</pre></div>
<p>Now we compute the possible class fusions from <span class="SimpleMath">\(S\)</span> into <span class="SimpleMath">\(G\)</span>, and the compositions of these maps with the possible class fusions from <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(S\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( sfust, ufuss );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( comp );</span>
8
</pre></div>
<p>It turns out that only one orbit of the possible class fusions from <span class="SimpleMath">\(M\)</span> to <span class="SimpleMath">\(G\)</span> is compatible with these possible class fusions from <span class="SimpleMath">\(U\)</span> to <span class="SimpleMath">\(G\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( mfust, map2 -> ForAny( ufusm, map1 -></span>
<span class="GAPprompt">></span> <span class="GAPinput"> CompositionMaps( map2, map1 ) in comp ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( filt );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( m, filt, t ) );</span>
1
</pre></div>
<p>The class fusion stored in the <strong class="pkg">GAP</strong> Character Table Library is one of them.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( m, t ) in filt;</span>
true
</pre></div>
<p><a id="X7C3AC42F8342EE2E" name="X7C3AC42F8342EE2E"></a></p>
<h5>9.3-7 <span class="Heading"><span class="SimpleMath">\({}^2E_6(2)N3C \rightarrow {}^2E_6(2)\)</span> (January 2019)</span></h5>
<p>Let <span class="SimpleMath">\(G = {}^2E_6(2)\)</span>, and <span class="SimpleMath">\(g \in G\)</span> in the conjugacy class <code class="code">3C</code>. Using a permutation representation of <span class="SimpleMath">\(G\)</span>, Frank Lübeck has computed a representation and the character table of the maximal subgroup <span class="SimpleMath">\(N = N_G(\langle g \rangle)\)</span> of <span class="SimpleMath">\(G\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "2E6(2)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos3CinG:= Position( ClassNames( t ), "3c" );</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">n:= CharacterTable( "2E6(2)N3C" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">nclasses:= SizesConjugacyClasses( n );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos3CinN:= Filtered( [ 1 .. NrConjugacyClasses( n ) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> i -> nclasses[i] = 2 );</span>
[ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">nfust:= PossibleClassFusions( n, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( nfust, x -> x[ pos3CinN[1] ] = pos3CinG );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( n ) = 2 * SizesCentralizers( t )[ pos3CinG ];</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( Irr( n ), x -> IsInt( x[ pos3CinN[1] ] ) );</span>
true
</pre></div>
<p>The class fusion of <span class="SimpleMath">\(N\)</span> in <span class="SimpleMath">\(G\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">rep:= RepresentativesFusions( n, nfust, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( rep );</span>
4
</pre></div>
<p>We use the fact that <span class="SimpleMath">\(g\)</span> is contained in a subgroup <span class="SimpleMath">\(S \cong Fi_{22}\)</span> of <span class="SimpleMath">\(G\)</span>, <span class="SimpleMath">\(\ldots\)</span></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "Fi22" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( sfust, x -> x[6] = pos3CinG );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">pos3CinS:= 6;;</span>
</pre></div>
<p><span class="SimpleMath">\(\ldots\)</span> and that <span class="SimpleMath">\(U = N_S(\langle g \rangle) \cong 3^{1+6}:2^{3+4}:3^2:2\)</span> is a maximal subgroup of <span class="SimpleMath">\(S\)</span> whose character table is available. Thus <span class="SimpleMath">\(U \leq N\)</span>, of index four.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( Maxes( s )[11] );</span>
CharacterTable( "3^(1+6):2^(3+4):3^2:2" )
<span class="GAPprompt">gap></span> <span class="GAPinput">uclasses:= SizesConjugacyClasses( u );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos3CinU:= Filtered( [ 1 .. NrConjugacyClasses( u ) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> i -> uclasses[i] = 2 );</span>
[ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ufuss:= PossibleClassFusions( u, s );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( ufuss, x -> x[ pos3CinU[1] ] = pos3CinS );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( n ) / Size( u );</span>
4
</pre></div>
<p>Composing the class fusions of <span class="SimpleMath">\(U\)</span> in <span class="SimpleMath">\(N\)</span> and <span class="SimpleMath">\(N\)</span> in <span class="SimpleMath">\(G\)</span> must be equal to the composition of the class fusions of <span class="SimpleMath">\(U\)</span> in <span class="SimpleMath">\(S\)</span> and <span class="SimpleMath">\(S\)</span> in <span class="SimpleMath">\(G\)</span>. This reduces the number of candidates for the fusion of <span class="SimpleMath">\(N\)</span> in <span class="SimpleMath">\(G\)</span> from four to two.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusn:= PossibleClassFusions( u, n );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( sfust, ufuss );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">good:= Filtered( nfust, map2 -> ForAny( ufusn,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map1 -> CompositionMaps( map2, map1 ) in comp ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( good );</span>
1728
<span class="GAPprompt">gap></span> <span class="GAPinput">goodrep:= RepresentativesFusions( n, good, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( goodrep );</span>
2
</pre></div>
<p>Next we use the fact that <span class="SimpleMath">\(g\)</span> and thus <span class="SimpleMath">\(N\)</span> is invariant under an outer automorphism <span class="SimpleMath">\(\alpha\)</span>, say, of order three of <span class="SimpleMath">\(G\)</span>. Note that such an automorphism acts nontrivially on the conjugacy classes of <span class="SimpleMath">\(G\)</span>, for example because the class fusion of <span class="SimpleMath">\(G\)</span> into <span class="SimpleMath">\(G.3 = \langle G, \alpha \rangle\)</span> shows the existence of orbits of length three, and that the permutation action of <span class="SimpleMath">\(\alpha\)</span> on the classes of <span class="SimpleMath">\(G\)</span> is given by the unique subgroup of order three in the group of table automorphisms of <span class="SimpleMath">\(G\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tfust3:= GetFusionMap( t, CharacterTable( "2E6(2).3" ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Number( InverseMap( tfust3 ), IsList );</span>
14
<span class="GAPprompt">gap></span> <span class="GAPinput">autt:= AutomorphismsOfTable( t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ord3:= Filtered( autt, x -> Order( x ) = 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( ord3 );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha:= ord3[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos3CinG ^ alpha = pos3CinG;</span>
true
</pre></div>
<p>The character table of <span class="SimpleMath">\(N\)</span> has <span class="SimpleMath">\(26\)</span> table automorphisms of order three. We do not know which of them (or perhaps the identity permutation) is induced by the restriction <span class="SimpleMath">\(\alpha_N\)</span> of <span class="SimpleMath">\(\alpha\)</span> to <span class="SimpleMath">\(N\)</span>, but the embedding <span class="SimpleMath">\(\iota\colon N \rightarrow G\)</span> satisfies <span class="SimpleMath">\(\alpha \circ \iota = \iota \circ \alpha_N\)</span>, and we can check each fusion candidate for the existence of a candidate for <span class="SimpleMath">\(\alpha_N\)</span> such that this relation holds.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">autn:= AutomorphismsOfTable( n );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ord3:= Filtered( autn, x -> Order( x ) = 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( ord3 );</span>
26
<span class="GAPprompt">gap></span> <span class="GAPinput">Add( ord3, () );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( rep, map -> ForAny( ord3, beta -></span>
<span class="GAPprompt">></span> <span class="GAPinput">OnTuples( map, alpha ) = Permuted( map, beta ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( filt );</span>
2
</pre></div>
<p>Again, the number of candidates for the fusion of <span class="SimpleMath">\(N\)</span> in <span class="SimpleMath">\(G\)</span> is reduced from four to two. Moreover, we are lucky because only one candidate satifies also the first criterion we have checked.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">inter:= Intersection( good, filt );</span>
[ [ 1, 7, 5, 6, 7, 2, 3, 4, 27, 30, 24, 32, 25, 26, 9, 11, 12, 13,
10, 14, 19, 19, 19, 16, 17, 18, 21, 58, 61, 62, 67, 68, 69, 57,
72, 59, 75, 76, 77, 78, 79, 80, 64, 65, 66, 60, 81, 82, 5, 6,
7, 6, 7, 7, 7, 7, 6, 7, 6, 7, 7, 24, 25, 27, 26, 28, 30, 29,
31, 32, 31, 32, 32, 32, 32, 31, 32, 31, 32, 51, 52, 52, 52, 52,
74, 76, 77, 77, 75, 74, 76, 74, 75, 99, 100, 101, 102, 4, 20,
29, 31, 32, 36, 36, 42, 42, 39, 40, 41, 49, 49, 49, 49, 49, 49,
71, 112, 112, 114, 115, 116 ] ]
</pre></div>
<p>The class fusion stored in the <strong class="pkg">GAP</strong> Character Table Library is this candidate.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( n, t ) = inter[1];</span>
true
</pre></div>
<p><em>Remark:</em></p>
<p>Note that the structure of <span class="SimpleMath">\(N\)</span> is <span class="SimpleMath">\(3^{1+6}:2^{3+6}:3^2:2\)</span>, as is stated in <a href="chapBib_mj.html#biBAtlasImpII">[Nor]</a>. The structure <span class="SimpleMath">\(3^{1+6}.2^{3+6}.(S_3 \times 3)\)</span> claimed in the <strong class="pkg">Atlas</strong> <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 191]</a> is wrong, as we can read off for example from the fact that <span class="SimpleMath">\(N\)</span> has exactly two linear characters.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( LinearCharacters( n ) );</span>
2
</pre></div>
<p><a id="X84F966E2824F5D52" name="X84F966E2824F5D52"></a></p>
<h4>9.4 <span class="Heading">Fusions Determined Using Commutative Diagrams Involving Factor
Groups</span></h4>
<p><a id="X7F2B104686509CAA" name="X7F2B104686509CAA"></a></p>
<h5>9.4-1 <span class="Heading"><span class="SimpleMath">\(3.A_7 \rightarrow 3.Suz\)</span> (December 2010)</span></h5>
<p>The maximal subgroups of type <span class="SimpleMath">\(A_7\)</span> in the sporadic simple Suzuki group <span class="SimpleMath">\(Suz\)</span> lift to groups of the type <span class="SimpleMath">\(3.A_7\)</span> in <span class="SimpleMath">\(3.Suz\)</span>. This can be seen from the fact that <span class="SimpleMath">\(3.Suz\)</span> does not admit a class fusion from <span class="SimpleMath">\(A_7\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "Suz" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">3t:= CharacterTable( "3.Suz" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "A7" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">3s:= CharacterTable( "3.A7" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PossibleClassFusions( s, 3t );</span>
[ ]
</pre></div>
<p>The class fusion of <span class="SimpleMath">\(3.A_7\)</span> into <span class="SimpleMath">\(3.Suz\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">3sfus3t:= PossibleClassFusions( 3s, 3t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( 3sfus3t );</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativesFusions( 3s, 3sfus3t, 3t );</span>
[ [ 1, 2, 3, 7, 8, 9, 16, 16, 26, 27, 28, 32, 33, 34, 47, 47, 47, 48,
49, 50, 48, 49, 50 ],
[ 1, 11, 12, 4, 36, 37, 13, 16, 23, 82, 83, 32, 100, 101, 44, 38,
41, 48, 112, 116, 48, 115, 113 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfCentre( 3t );</span>
[ 1, 2, 3 ]
</pre></div>
<p>We see that the possible fusions in the second orbit avoid the centre of <span class="SimpleMath">\(3.Suz\)</span>. Since the preimages in <span class="SimpleMath">\(3.Suz\)</span> of the <span class="SimpleMath">\(A_7\)</span> type subgroups of <span class="SimpleMath">\(Suz\)</span> contain the centre of <span class="SimpleMath">\(3.Suz\)</span>, we know that the class fusion of these preimages belong to the first orbit. This can be formalized by checking the commutativity of the diagram of fusions between <span class="SimpleMath">\(3.A_7\)</span>, <span class="SimpleMath">\(3.Suz\)</span>, and their factors <span class="SimpleMath">\(A_7\)</span> and <span class="SimpleMath">\(Suz\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( sfust );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( 3sfus3t, map -> CompositionMaps( GetFusionMap( 3t, t ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> = CompositionMaps( sfust[1], GetFusionMap( 3s, s ) ) );</span>
[ [ 1, 2, 3, 7, 8, 9, 16, 16, 26, 27, 28, 32, 33, 34, 47, 47, 47, 48,
49, 50, 48, 49, 50 ],
[ 1, 3, 2, 7, 9, 8, 16, 16, 26, 28, 27, 32, 34, 33, 47, 47, 47, 48,
50, 49, 48, 50, 49 ] ]
</pre></div>
<p>So the class fusion of maximal <span class="SimpleMath">\(3.A_7\)</span> type subgroups of <span class="SimpleMath">\(3.Suz\)</span> is determined up to table automorphisms. One of these fusions is stored on the table of <span class="SimpleMath">\(3.A_7\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativesFusions( 3s, filt, 3t );</span>
[ [ 1, 2, 3, 7, 8, 9, 16, 16, 26, 27, 28, 32, 33, 34, 47, 47, 47, 48,
49, 50, 48, 49, 50 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( 3s, 3t ) in filt;</span>
true
</pre></div>
<p>Also the class fusions in the other orbit belong to subgroups of type <span class="SimpleMath">\(3.A_7\)</span> in <span class="SimpleMath">\(3.Suz\)</span>. Note that <span class="SimpleMath">\(Suz\)</span> contains maximal subgroups of the type <span class="SimpleMath">\(3_2.U_4(3).2_3^{\prime}\)</span> (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 131]</a>), and the <span class="SimpleMath">\(A_7\)</span> type subgroups of <span class="SimpleMath">\(U_4(3)\)</span> (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 52]</a>) lift to groups of the type <span class="SimpleMath">\(3.A_7\)</span> in <span class="SimpleMath">\(3_2.U_4(3)\)</span> because <span class="SimpleMath">\(3_2.U_4(3)\)</span> does not admit a class fusion from <span class="SimpleMath">\(A_7\)</span>. The preimages in <span class="SimpleMath">\(3.Suz\)</span> of the <span class="SimpleMath">\(3.A_7\)</span> type subgroups of <span class="SimpleMath">\(Suz\)</span> have the structure <span class="SimpleMath">\(3 \times 3.A_7\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( "3_2.U4(3)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PossibleClassFusions( s, u );</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( PossibleClassFusions( 3s, u ) );</span>
8
</pre></div>
<p><a id="X82FB71647D37F4FD" name="X82FB71647D37F4FD"></a></p>
<h5>9.4-2 <span class="Heading"><span class="SimpleMath">\(S_6 \rightarrow U_4(2)\)</span> (September 2011)</span></h5>
<p>The simple group <span class="SimpleMath">\(G = U_4(2)\)</span> contains a maximal subgroup <span class="SimpleMath">\(U\)</span> of type <span class="SimpleMath">\(S_6\)</span>. The class fusion from <span class="SimpleMath">\(U\)</span> to <span class="SimpleMath">\(G\)</span> is unique up to table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "S6" );</span>
CharacterTable( "A6.2_1" )
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "U4(2)" );</span>
CharacterTable( "U4(2)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t );</span>
[ [ 1, 3, 6, 7, 9, 10, 3, 2, 9, 16, 15 ],
[ 1, 3, 7, 6, 9, 10, 2, 3, 9, 15, 16 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( s, sfust, t ) );</span>
1
</pre></div>
<p>In the double cover <span class="SimpleMath">\(2.G\)</span> of <span class="SimpleMath">\(G\)</span>, <span class="SimpleMath">\(U\)</span> lifts to the double cover <span class="SimpleMath">\(2.U\)</span> of <span class="SimpleMath">\(U\)</span> (which is unique up to isomorphism). Also the class fusion from <span class="SimpleMath">\(2.U\)</span> to <span class="SimpleMath">\(2.G\)</span> is unique up to table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">2t:= CharacterTable( "2.U4(2)" );</span>
CharacterTable( "2.U4(2)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">2s:= CharacterTable( "2.A6.2_1" );</span>
CharacterTable( "2.A6.2_1" )
<span class="GAPprompt">gap></span> <span class="GAPinput">2sfus2t:= PossibleClassFusions( 2s, 2t );</span>
[ [ 1, 2, 4, 11, 12, 9, 10, 15, 16, 17, 3, 4, 15, 24, 25, 26, 26 ],
[ 1, 2, 4, 11, 12, 9, 10, 15, 16, 17, 3, 4, 15, 25, 24, 26, 26 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( 2s, 2sfus2t, 2t ) );</span>
1
</pre></div>
<p>However, the two possible fusions from <span class="SimpleMath">\(2.U\)</span> to <span class="SimpleMath">\(2.G\)</span> are lifts of the same class fusion from <span class="SimpleMath">\(U\)</span> to <span class="SimpleMath">\(G\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">2sfuss:= GetFusionMap( 2s, s );</span>
[ 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 11 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">2tfust:= GetFusionMap( 2t, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">induced:= Set( 2sfus2t, x -> CompositionMaps( 2tfust,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> CompositionMaps( x, InverseMap( 2sfuss ) ) ) );</span>
[ [ 1, 3, 7, 6, 9, 10, 2, 3, 9, 15, 16 ] ]
</pre></div>
<p>The point is that the outer automorphism of <span class="SimpleMath">\(S_6\)</span> that makes the two fusions from <span class="SimpleMath">\(U\)</span> to <span class="SimpleMath">\(G\)</span> equivalent does not lift to <span class="SimpleMath">\(2.U\)</span>, and that we have silently assumed a fixed factor fusion from <span class="SimpleMath">\(2.U\)</span> to <span class="SimpleMath">\(U\)</span>. Note that composing this factor fusion with the automorphism of <span class="SimpleMath">\(U\)</span> would also yield a factor fusion, and w. r. t. the commutative diagram involving this factor fusion, the other possible class fusion from <span class="SimpleMath">\(U\)</span> to <span class="SimpleMath">\(G\)</span> is induced by the possible fusions from <span class="SimpleMath">\(2.U\)</span> to <span class="SimpleMath">\(2.G\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">auts:= AutomorphismsOfTable( s );</span>
Group([ (3,4)(7,8)(10,11) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">other:= OnTuples( 2sfuss, GeneratorsOfGroup( auts )[1] );</span>
[ 1, 1, 2, 4, 4, 3, 3, 5, 6, 6, 8, 7, 9, 11, 11, 10, 10 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( 2sfus2t, x -> CompositionMaps( 2tfust,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> CompositionMaps( x, InverseMap( other ) ) ) );</span>
[ [ 1, 3, 6, 7, 9, 10, 3, 2, 9, 16, 15 ] ]
</pre></div>
<p>The library table of <span class="SimpleMath">\(U\)</span> stores the class fusion to <span class="SimpleMath">\(G\)</span> that is compatible with the stored factor fusion from <span class="SimpleMath">\(2.U\)</span> to <span class="SimpleMath">\(U\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( s, t ) in induced;</span>
true
</pre></div>
<p><a id="X7CFBC41B818A318C" name="X7CFBC41B818A318C"></a></p>
<h4>9.5 <span class="Heading">Fusions Determined Using Commutative Diagrams Involving
Automorphic Extensions</span></h4>
<p><a id="X7E91F8707BA93081" name="X7E91F8707BA93081"></a></p>
<h5>9.5-1 <span class="Heading"><span class="SimpleMath">\(U_3(8).3_1 \rightarrow {}^2E_6(2)\)</span> (December 2010)</span></h5>
<p>According to the <strong class="pkg">Atlas</strong> (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 191]</a>), the group <span class="SimpleMath">\(G = {}^2E_6(2)\)</span> contains a maximal subgroup <span class="SimpleMath">\(U\)</span> of the type <span class="SimpleMath">\(U_3(8).3_1\)</span>. The class fusion of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(G\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "U3(8).3_1" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "2E6(2)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( sfust );</span>
24
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( s, sfust, t ) );</span>
2
</pre></div>
<p>In the automorphic extension <span class="SimpleMath">\(G.2 = {}^2E_6(2).2\)</span> of <span class="SimpleMath">\(G\)</span>, the subgroup <span class="SimpleMath">\(U\)</span> extends to a group <span class="SimpleMath">\(U.2\)</span> of the type <span class="SimpleMath">\(U_3(8).6\)</span> (again, see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 191]</a>). The class fusion of <span class="SimpleMath">\(U.2\)</span> into <span class="SimpleMath">\(G.2\)</span> is unique up to table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s2:= CharacterTable( "U3(8).6" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t2:= CharacterTable( "2E6(2).2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s2fust2:= PossibleClassFusions( s2, t2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( s2fust2 );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( s2, s2fust2, t2 ) );</span>
1
</pre></div>
<p>Only half of the possible class fusions from <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(G\)</span> are compatible with the embeddings of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(G.2\)</span> via <span class="SimpleMath">\(U.2\)</span> and <span class="SimpleMath">\(G\)</span>, and the compatible maps form one orbit under table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfuss2:= PossibleClassFusions( s, s2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( s2fust2, sfuss2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tfust2:= PossibleClassFusions( t, t2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( sfust, map -> ForAny( tfust2,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map2 -> CompositionMaps( map2, map ) in comp ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( filt );</span>
12
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( s, filt, t ) );</span>
1
</pre></div>
<p>Let us see which classes of <span class="SimpleMath">\(U\)</span> and <span class="SimpleMath">\(G\)</span> are involved in the disambiguation of the class fusion. The "good" fusion candidates differ from the excluded ones on the classes at the positions <span class="SimpleMath">\(31\)</span> to <span class="SimpleMath">\(36\)</span>: Under all possible class fusions, two pairs of classes are mapped to the classes <span class="SimpleMath">\(81\)</span> and <span class="SimpleMath">\(82\)</span> of <span class="SimpleMath">\(G\)</span>; from these classes, the excluded maps fuse classes at odd positions with classes at even positions, whereas the "good" class fusions do not have this property.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( filt, x -> x{ [ 31 .. 36 ] } );</span>
[ [ 74, 74, 81, 82, 81, 82 ], [ 74, 74, 82, 81, 82, 81 ],
[ 81, 82, 74, 74, 81, 82 ], [ 81, 82, 81, 82, 74, 74 ],
[ 82, 81, 74, 74, 82, 81 ], [ 82, 81, 82, 81, 74, 74 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( Difference( sfust, filt ), x -> x{ [ 31 .. 36 ] } );</span>
[ [ 74, 74, 81, 82, 82, 81 ], [ 74, 74, 82, 81, 81, 82 ],
[ 81, 82, 74, 74, 82, 81 ], [ 81, 82, 82, 81, 74, 74 ],
[ 82, 81, 74, 74, 81, 82 ], [ 82, 81, 81, 82, 74, 74 ] ]
</pre></div>
<p>None of the possible class fusions from <span class="SimpleMath">\(U\)</span> to <span class="SimpleMath">\(U.2\)</span> fuses classes at odd positions in the interval from <span class="SimpleMath">\(31\)</span> to <span class="SimpleMath">\(36\)</span> with classes at even positions.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( sfuss2, x -> x{ [ 31 .. 36 ] } );</span>
[ [ 28, 29, 30, 31, 30, 31 ], [ 29, 28, 31, 30, 31, 30 ],
[ 30, 31, 28, 29, 30, 31 ], [ 30, 31, 30, 31, 28, 29 ],
[ 31, 30, 29, 28, 31, 30 ], [ 31, 30, 31, 30, 29, 28 ] ]
</pre></div>
<p>This suffices to exclude the "bad" fusion candidates because no further fusion of the relevant classes of <span class="SimpleMath">\(G\)</span> happens in <span class="SimpleMath">\(G.2\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( tfust2, x -> x{ [ 74, 81, 82 ] } );</span>
[ [ 65, 70, 71 ], [ 65, 70, 71 ], [ 65, 71, 70 ], [ 65, 71, 70 ],
[ 65, 70, 71 ], [ 65, 70, 71 ], [ 65, 71, 70 ], [ 65, 71, 70 ],
[ 65, 70, 71 ], [ 65, 70, 71 ], [ 65, 71, 70 ], [ 65, 71, 70 ] ]
</pre></div>
<p>(The same holds for the fusion of the relevant classes of <span class="SimpleMath">\(U.2\)</span> in <span class="SimpleMath">\(G.2\)</span>.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( s2fust2, x -> x{ [ 28 .. 31 ] } );</span>
[ [ 65, 65, 70, 71 ], [ 65, 65, 71, 70 ] ]
</pre></div>
<p>Finally, we check that a correct map is stored on the library table.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( s, t ) in filt;</span>
true
</pre></div>
<p><a id="X81B37EF378E89E00" name="X81B37EF378E89E00"></a></p>
<h5>9.5-2 <span class="Heading"><span class="SimpleMath">\(L_3(4).2_1 \rightarrow U_6(2)\)</span> (December 2010)</span></h5>
<p>According to the <strong class="pkg">Atlas</strong> (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 115]</a>), the group <span class="SimpleMath">\(G = U_6(2)\)</span> contains a maximal subgroup <span class="SimpleMath">\(U\)</span> of the type <span class="SimpleMath">\(L_3(4).2_1\)</span>. The class fusion of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(G\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "L3(4).2_1" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "U6(2)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( sfust );</span>
27
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( s, sfust, t ) );</span>
3
</pre></div>
<p>In the automorphic extension <span class="SimpleMath">\(G.3 = U_6(2).3\)</span> of <span class="SimpleMath">\(G\)</span>, the subgroup <span class="SimpleMath">\(U\)</span> extends to a group <span class="SimpleMath">\(U.3\)</span> of the type <span class="SimpleMath">\(L_3(4).6\)</span> (again, see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 115]</a>). The class fusion of <span class="SimpleMath">\(U.3\)</span> into <span class="SimpleMath">\(G.3\)</span> is unique up to table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s3:= CharacterTable( "L3(4).6" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t3:= CharacterTable( "U6(2).3" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s3fust3:= PossibleClassFusions( s3, t3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( s3fust3 );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( s3, s3fust3, t3 ) );</span>
1
</pre></div>
<p>Here the argument used in Section <a href="chap9_mj.html#X7E91F8707BA93081"><span class="RefLink">9.5-1</span></a> does not work, because all possible class fusions from <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(G\)</span> are compatible with the embeddings of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(G.3\)</span> via <span class="SimpleMath">\(U.3\)</span> and <span class="SimpleMath">\(G\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfuss3:= PossibleClassFusions( s, s3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= SetOfComposedClassFusions( s3fust3, sfuss3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tfust3:= PossibleClassFusions( t, t3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust = Filtered( sfust, map -> ForAny( tfust3,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map2 -> CompositionMaps( map2, map ) in comp ) );</span>
true
</pre></div>
<p>Consider the elements of order four in <span class="SimpleMath">\(U\)</span>. There are three such classes inside <span class="SimpleMath">\(U^{\prime} \cong L_3(4)\)</span>, which fuse to one class of <span class="SimpleMath">\(U.3\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( s );</span>
[ 1, 2, 3, 4, 4, 4, 5, 7, 2, 4, 6, 8, 8, 8 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">sfuss3;</span>
[ [ 1, 2, 3, 4, 4, 4, 5, 6, 7, 8, 9, 10, 10, 10 ] ]
</pre></div>
<p>These classes of <span class="SimpleMath">\(U\)</span> fuse into some of the classes <span class="SimpleMath">\(10\)</span> to <span class="SimpleMath">\(12\)</span> of <span class="SimpleMath">\(G\)</span>. In <span class="SimpleMath">\(G.3\)</span>, these three classes fuse into one class.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( sfust, map -> map{ [ 4 .. 6 ] } );</span>
[ [ 10, 10, 10 ], [ 10, 10, 11 ], [ 10, 10, 12 ], [ 10, 11, 10 ],
[ 10, 11, 11 ], [ 10, 11, 12 ], [ 10, 12, 10 ], [ 10, 12, 11 ],
[ 10, 12, 12 ], [ 11, 10, 10 ], [ 11, 10, 11 ], [ 11, 10, 12 ],
[ 11, 11, 10 ], [ 11, 11, 11 ], [ 11, 11, 12 ], [ 11, 12, 10 ],
[ 11, 12, 11 ], [ 11, 12, 12 ], [ 12, 10, 10 ], [ 12, 10, 11 ],
[ 12, 10, 12 ], [ 12, 11, 10 ], [ 12, 11, 11 ], [ 12, 11, 12 ],
[ 12, 12, 10 ], [ 12, 12, 11 ], [ 12, 12, 12 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( tfust3, map -> map{ [ 10 .. 12 ] } );</span>
[ [ 10, 10, 10 ] ]
</pre></div>
<p>This means that the automorphism <span class="SimpleMath">\(\alpha\)</span> of <span class="SimpleMath">\(G\)</span> that is induced by the action of <span class="SimpleMath">\(G.3\)</span> permutes the classes <span class="SimpleMath">\(10\)</span> to <span class="SimpleMath">\(12\)</span> of <span class="SimpleMath">\(G\)</span> transitively. The fact that <span class="SimpleMath">\(U\)</span> extends to <span class="SimpleMath">\(U.3\)</span> in <span class="SimpleMath">\(G.3\)</span> means that <span class="SimpleMath">\(U\)</span> is invariant under <span class="SimpleMath">\(\alpha\)</span>. This implies that <span class="SimpleMath">\(U\)</span> contains either no elements from the classes <span class="SimpleMath">\(10\)</span> to <span class="SimpleMath">\(12\)</span> or elements from all of these classes. The possible class fusions from <span class="SimpleMath">\(U\)</span> to <span class="SimpleMath">\(G\)</span> satisfying this condition form one orbit under table automprhisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Filtered( sfust, map -> Intersection( map, [ 10 .. 12 ] ) = [] );</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( sfust, map -> IsSubset( map, [ 10 .. 12 ] ) );</span>
[ [ 1, 3, 7, 10, 11, 12, 15, 24, 4, 14, 23, 26, 27, 28 ],
[ 1, 3, 7, 10, 12, 11, 15, 24, 4, 14, 23, 26, 28, 27 ],
[ 1, 3, 7, 11, 10, 12, 15, 24, 4, 14, 23, 27, 26, 28 ],
[ 1, 3, 7, 11, 12, 10, 15, 24, 4, 14, 23, 27, 28, 26 ],
[ 1, 3, 7, 12, 10, 11, 15, 24, 4, 14, 23, 28, 26, 27 ],
[ 1, 3, 7, 12, 11, 10, 15, 24, 4, 14, 23, 28, 27, 26 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( s, filt, t ) );</span>
1
</pre></div>
<p>Finally, we check that a correct map is stored on the library table.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( s, t ) in filt;</span>
true
</pre></div>
<p><a id="X85E2A6F480026C95" name="X85E2A6F480026C95"></a></p>
<h4>9.6 <span class="Heading">Conditions Imposed by Brauer Tables</span></h4>
<p>The examples in this section show that symmetries can be broken as soon as the class fusions between two ordinary tables shall be compatible with the corresponding Brauer character tables. More precisely, we assume that the class fusion from each Brauer table to its ordinary table is already fixed; choosing these fusions consistently can be a nontrivial task, solving so-called "generality problems" may require the construction of certain modules, similar to the arguments used in <a href="chap9_mj.html#X7DED4C437D479226"><span class="RefLink">9.6-3</span></a> below.</p>
<p><a id="X7ACC7F588213D5D5" name="X7ACC7F588213D5D5"></a></p>
<h5>9.6-1 <span class="Heading"><span class="SimpleMath">\(L_2(16).4 \rightarrow J_3.2\)</span> (January 2004)</span></h5>
<p>It can happen that Brauer tables decide ambiguities of class fusions between the corresponding ordinary tables. An easy example is the class fusion of <span class="SimpleMath">\(L_2(16).4\)</span> into <span class="SimpleMath">\(J_3.2\)</span>. The ordinary tables admit four possible class fusions, of which two are essentially different.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "L2(16).4" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "J3.2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= PossibleClassFusions( s, t );</span>
[ [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 5, 5, 8, 8, 13, 13 ],
[ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ],
[ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 5, 5, 8, 8, 13, 13 ],
[ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativesFusions( s, fus, t );</span>
[ [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 5, 5, 8, 8, 13, 13 ],
[ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ]
</pre></div>
<p>Using Brauer tables, we will see that just one fusion is admissible.</p>
<p>We can exclude two possible fusions by the fact that their images all lie inside the normal subgroup <span class="SimpleMath">\(J_3\)</span>, but <span class="SimpleMath">\(J_3\)</span> does not contain a subgroup of type <span class="SimpleMath">\(L_2(16).4\)</span>; so still one orbit of length two remains.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">j3:= CharacterTable( "J3" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PossibleClassFusions( s, j3 );</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( j3, t );</span>
[ 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16,
17, 17 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( fus,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x -> not IsSubset( ClassPositionsOfDerivedSubgroup( t ), x ) );</span>
[ [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ],
[ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ]
</pre></div>
<p>Now the remaining wrong fusion is excluded by the fact that the table automorphism of <span class="SimpleMath">\(J_3.2\)</span> that swaps the two classes of element order <span class="SimpleMath">\(17\)</span> –which swaps two of the possible class fusions– does not live in the <span class="SimpleMath">\(2\)</span>-modular table.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">smod2:= s mod 2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tmod2:= t mod 2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">admissible:= [];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for map in filt do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> modmap:= CompositionMaps( InverseMap( GetFusionMap( tmod2, t ) ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> CompositionMaps( map, GetFusionMap( smod2, s ) ) );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if not fail in Decomposition( Irr( smod2 ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> List( Irr( tmod2 ), chi -> chi{ modmap } ), "nonnegative" ) then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> AddSet( admissible, map );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">admissible;</span>
[ [ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ]
</pre></div>
<p>The test of all available Brauer tables is implemented in the function <code class="code">CTblLib.Test.Decompositions</code> of the <strong class="pkg">GAP</strong> Character Table Library (<a href="chapBib_mj.html#biBCTblLib">[Bre24]</a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CTblLib.Test.Decompositions( s, fus, t ) = admissible;</span>
true
</pre></div>
<p>We see that <span class="SimpleMath">\(p\)</span>-modular tables alone determine the class fusion uniquely; in fact the primes <span class="SimpleMath">\(2\)</span> and <span class="SimpleMath">\(3\)</span> suffice for that.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( s, t ) in admissible;</span>
true
</pre></div>
<p><em>Remark:</em></p>
<p>In May 2015, the <span class="SimpleMath">\(19\)</span>-modular character table of <span class="SimpleMath">\(J_3\)</span> has been corrected, by swapping the two classes of element order <span class="SimpleMath">\(17\)</span>. Since the class fusion of <span class="SimpleMath">\(L_2(16).4\)</span> into <span class="SimpleMath">\(J_3.2\)</span> is uniquely determined by the <span class="SimpleMath">\(2\)</span>-modular tables of <span class="SimpleMath">\(L_2(16).4\)</span> and <span class="SimpleMath">\(J_3.2\)</span> and since this class fusion has been compatible with the previous version of the <span class="SimpleMath">\(19\)</span>-modular table of <span class="SimpleMath">\(J_3\)</span>, the correction does not affect the above arguments.</p>
<p><a id="X7ACB86CB82ED49D1" name="X7ACB86CB82ED49D1"></a></p>
<h5>9.6-2 <span class="Heading"><span class="SimpleMath">\(L_2(17) \rightarrow S_8(2)\)</span> (July 2004)</span></h5>
<p>The class fusion of the maximal subgroup <span class="SimpleMath">\(M \cong L_2(17)\)</span> of <span class="SimpleMath">\(G = S_8(2)\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= CharacterTable( "L2(17)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "S8(2)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mfust:= PossibleClassFusions( m, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( m, mfust, t ) );</span>
4
</pre></div>
<p>The Brauer tables for <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(G\)</span> determine the class fusion up to table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= CTblLib.Test.Decompositions( m, mfust, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( m, filt, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( repr );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( m, t ) in repr;</span>
true
</pre></div>
<p><a id="X7DED4C437D479226" name="X7DED4C437D479226"></a></p>
<h5>9.6-3 <span class="Heading"><span class="SimpleMath">\(L_2(19) \rightarrow J_3\)</span> (April 2003)</span></h5>
<p>It can happen that Brauer tables impose conditions such that ambiguities arise which are not visible if one considers only ordinary tables.</p>
<p>The class fusion between the ordinary character tables of <span class="SimpleMath">\(L_2(19)\)</span> and <span class="SimpleMath">\(J_3\)</span> is unique up to table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "L2(19)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "J3" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t );</span>
[ [ 1, 2, 4, 6, 7, 10, 11, 12, 13, 14, 20, 21 ],
[ 1, 2, 4, 6, 7, 10, 11, 12, 13, 14, 21, 20 ],
[ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 20, 21 ],
[ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 21, 20 ],
[ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 20, 21 ],
[ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 21, 20 ],
[ 1, 2, 4, 7, 6, 10, 11, 12, 14, 13, 20, 21 ],
[ 1, 2, 4, 7, 6, 10, 11, 12, 14, 13, 21, 20 ],
[ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 20, 21 ],
[ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 21, 20 ],
[ 1, 2, 4, 7, 6, 12, 10, 11, 14, 13, 20, 21 ],
[ 1, 2, 4, 7, 6, 12, 10, 11, 14, 13, 21, 20 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">fusreps:= RepresentativesFusions( s, sfust, t );</span>
[ [ 1, 2, 4, 6, 7, 10, 11, 12, 13, 14, 20, 21 ] ]
</pre></div>
<p>The Galois automorphism that permutes the three classes of element order <span class="SimpleMath">\(9\)</span> in the tables of (<span class="SimpleMath">\(L_2(19)\)</span> and) <span class="SimpleMath">\(J_3\)</span> does not live in characteristic <span class="SimpleMath">\(19\)</span>. For example, the unique irreducible Brauer character of degree <span class="SimpleMath">\(110\)</span> in the <span class="SimpleMath">\(19\)</span>-modular table of <span class="SimpleMath">\(J_3\)</span> is <span class="SimpleMath">\(\varphi_3\)</span>, and the value of this character on the class <code class="code">9A</code> is <code class="code">-1+2y9+&4</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tmod19:= t mod 19;</span>
BrauerTable( "J3", 19 )
<span class="GAPprompt">gap></span> <span class="GAPinput">deg110:= Filtered( Irr( tmod19 ), phi -> phi[1] = 110 );</span>
[ Character( BrauerTable( "J3", 19 ),
[ 110, -2, 5, 2, 2, 0, 0, 1, 0,
-2*E(9)^2+E(9)^3-E(9)^4-E(9)^5+E(9)^6-2*E(9)^7,
E(9)^2+E(9)^3-E(9)^4-E(9)^5+E(9)^6+E(9)^7,
E(9)^2+E(9)^3+2*E(9)^4+2*E(9)^5+E(9)^6+E(9)^7, -2, -2, -1, 0,
0, E(17)^3+E(17)^5+E(17)^6+E(17)^7+E(17)^10+E(17)^11+E(17)^12
+E(17)^14,
E(17)+E(17)^2+E(17)^4+E(17)^8+E(17)^9+E(17)^13+E(17)^15+E(17)^16
] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">9A:= Position( OrdersClassRepresentatives( tmod19 ), 9 );</span>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">deg110[1][ 9A ];</span>
-2*E(9)^2+E(9)^3-E(9)^4-E(9)^5+E(9)^6-2*E(9)^7
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "-1+2y9+&4" ) = deg110[1][ 9A ];</span>
true
</pre></div>
<p>It turns out that four of the twelve possible class fusions are not compatible with the <span class="SimpleMath">\(19\)</span>-modular tables.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">smod19:= s mod 19;</span>
BrauerTable( "L2(19)", 19 )
<span class="GAPprompt">gap></span> <span class="GAPinput">compatible:= [];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for map in sfust do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> comp:= CompositionMaps( InverseMap( GetFusionMap( tmod19, t ) ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> CompositionMaps( map, GetFusionMap( smod19, s ) ) );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> rest:= List( Irr( tmod19 ), phi -> phi{ comp } );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if not fail in Decomposition( Irr( smod19 ), rest, "nonnegative" ) then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Add( compatible, map );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">compatible;</span>
[ [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 20, 21 ],
[ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 21, 20 ],
[ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 20, 21 ],
[ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 21, 20 ],
[ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 20, 21 ],
[ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 21, 20 ],
[ 1, 2, 4, 7, 6, 12, 10, 11, 14, 13, 20, 21 ],
[ 1, 2, 4, 7, 6, 12, 10, 11, 14, 13, 21, 20 ] ]
</pre></div>
<p>Moreover, the subgroups of those table automorphisms of the ordinary tables that leave the set of compatible fusions invariant make two orbits on this set. Indeed, the two orbits belong to essentially different decompositions of the restriction of <span class="SimpleMath">\(\varphi_3\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">reps:= RepresentativesFusions( s, compatible, t );</span>
[ [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 20, 21 ],
[ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 20, 21 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">compatiblemod19:= List( reps, map -> CompositionMaps(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> InverseMap( GetFusionMap( tmod19, t ) ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> CompositionMaps( map, GetFusionMap( smod19, s ) ) ) );</span>
[ [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14 ],
[ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">rest:= List( compatiblemod19, map -> Irr( tmod19 )[3]{ map } );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">dec:= Decomposition( Irr( smod19 ), rest, "nonnegative" );</span>
[ [ 0, 0, 1, 2, 1, 2, 2, 1, 0, 1 ], [ 0, 2, 0, 2, 0, 1, 2, 0, 2, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( Irr( smod19 ), phi -> phi[1] );</span>
[ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 ]
</pre></div>
<p>In order to decide which class fusion is correct, we take the matrix representation of <span class="SimpleMath">\(J_3\)</span> that affords <span class="SimpleMath">\(\varphi_3\)</span>, restrict it to <span class="SimpleMath">\(L_2(19)\)</span>, which is the second maximal subgroup of <span class="SimpleMath">\(J_3\)</span>, and compute the composition factors. For that, we use a representation from the <strong class="pkg">Atlas</strong> of Group Representations <a href="chapBib_mj.html#biBAGRv3">[WWT+]</a>, and access it via the <strong class="pkg">GAP</strong> package <strong class="pkg">AtlasRep</strong> (<a href="chapBib_mj.html#biBAtlasRep">[WPN+22]</a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "atlasrep", false );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">prog:= AtlasProgram( "J3", "maxes", 2 );</span>
rec( groupname := "J3", identifier := [ "J3", "J3G1-max2W1", 1 ],
program := <straight line program>, size := 3420,
standardization := 1, subgroupname := "L2(19)", version := "1" )
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= OneAtlasGeneratingSetInfo( "J3", Characteristic, 19,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Dimension, 110 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= AtlasGenerators( gens );</span>
rec( charactername := "110a", constituents := [ 3 ],
contents := "core", dim := 110,
generators := [ < immutable compressed matrix 110x110 over GF(19) >,
< immutable compressed matrix 110x110 over GF(19) > ],
groupname := "J3", id := "",
identifier := [ "J3", [ "J3G1-f19r110B0.m1", "J3G1-f19r110B0.m2" ],
1, 19 ], repname := "J3G1-f19r110B0", repnr := 35,
ring := GF(19), size := 50232960, standardization := 1,
type := "matff" )
<span class="GAPprompt">gap></span> <span class="GAPinput">restgens:= ResultOfStraightLineProgram( prog.program, gens.generators );</span>
[ < immutable compressed matrix 110x110 over GF(19) >,
< immutable compressed matrix 110x110 over GF(19) > ]
<span class="GAPprompt">gap></span> <span class="GAPinput">module:= GModuleByMats( restgens, GF( 19 ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">facts:= SMTX.CollectedFactors( module );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( facts );</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">List( facts, x -> x[1].dimension );</span>
[ 5, 7, 9, 11, 13, 15, 19 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( facts, x -> x[2] );</span>
[ 1, 2, 1, 2, 2, 1, 1 ]
</pre></div>
<p>This means that there are seven pairwise nonisomorphic composition factors, the smallest one of dimension five. In other words, the first of the two maps is the correct one. Let us check whether this map equals the one that is stored on the library table.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( s, t ) = reps[1];</span>
true
</pre></div>
<p><em>Remark:</em></p>
<p>In May 2015, the <span class="SimpleMath">\(19\)</span>-modular character table of <span class="SimpleMath">\(J_3\)</span> has been corrected, by swapping the two classes of element order <span class="SimpleMath">\(17\)</span>. This affects the above computations only in one place, where the values of the character <code class="code">deg110</code> are shown.</p>
<p><a id="X8225D9FA80A7D20F" name="X8225D9FA80A7D20F"></a></p>
<h4>9.7 <span class="Heading">Fusions Determined by Information about the Groups</span></h4>
<p>In the examples in this section, character theoretic arguments do not suffice for determining the class fusions. So we use computations with the groups in question or information about these groups beyond the character table, and perhaps additionally character theoretic arguments.</p>
<p>The group representations are taken from the <strong class="pkg">Atlas</strong> of Group Representations <a href="chapBib_mj.html#biBAGRv3">[WWT+]</a> and are accessed via the <strong class="pkg">GAP</strong> package <strong class="pkg">AtlasRep</strong> (<a href="chapBib_mj.html#biBAtlasRep">[WPN+22]</a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "atlasrep", false );</span>
true
</pre></div>
<p><a id="X7AE2962E82B4C814" name="X7AE2962E82B4C814"></a></p>
<h5>9.7-1 <span class="Heading"><span class="SimpleMath">\(U_3(3).2 \rightarrow Fi_{24}^{\prime}\)</span> (November 2002)</span></h5>
<p>The group <span class="SimpleMath">\(G = Fi_{24}^{\prime}\)</span> contains a maximal subgroup <span class="SimpleMath">\(H\)</span> of type <span class="SimpleMath">\(U_3(3).2\)</span>. From the character tables of <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(H\)</span>, one gets a lot of essentially different possibilities (and additionally this takes quite some time). We use the description of <span class="SimpleMath">\(H\)</span> as the normalizer in <span class="SimpleMath">\(G\)</span> of a <span class="SimpleMath">\(U_3(3)\)</span> type subgroup containing elements in the classes <code class="code">2B</code>, <code class="code">3D</code>, <code class="code">3E</code>, <code class="code">4C</code>, <code class="code">4C</code>, <code class="code">6J</code>, <code class="code">7B</code>, <code class="code">8C</code>, and <code class="code">12M</code> (see <a href="chapBib_mj.html#biBBN95">[BN95]</a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "F3+" );</span>
CharacterTable( "F3+" )
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "U3(3).2" );</span>
CharacterTable( "U3(3).2" )
<span class="GAPprompt">gap></span> <span class="GAPinput">tnames:= ClassNames( t, "ATLAS" );</span>
[ "1A", "2A", "2B", "3A", "3B", "3C", "3D", "3E", "4A", "4B", "4C",
"5A", "6A", "6B", "6C", "6D", "6E", "6F", "6G", "6H", "6I", "6J",
"6K", "7A", "7B", "8A", "8B", "8C", "9A", "9B", "9C", "9D", "9E",
"9F", "10A", "10B", "11A", "12A", "12B", "12C", "12D", "12E",
"12F", "12G", "12H", "12I", "12J", "12K", "12L", "12M", "13A",
"14A", "14B", "15A", "15B", "15C", "16A", "17A", "18A", "18B",
"18C", "18D", "18E", "18F", "18G", "18H", "20A", "20B", "21A",
"21B", "21C", "21D", "22A", "23A", "23B", "24A", "24B", "24C",
"24D", "24E", "24F", "24G", "26A", "27A", "27B", "27C", "28A",
"29A", "29B", "30A", "30B", "33A", "33B", "35A", "36A", "36B",
"36C", "36D", "39A", "39B", "39C", "39D", "42A", "42B", "42C",
"45A", "45B", "60A" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( s );</span>
[ 1, 2, 3, 3, 4, 4, 6, 7, 8, 12, 2, 4, 6, 8, 12, 12 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= List( [ "1A", "2B", "3D", "3E", "4C", "4C", "6J", "7B", "8C",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "12M" ], x -> Position( tnames, x ) );</span>
[ 1, 3, 7, 8, 11, 11, 22, 25, 28, 50 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t, rec( fusionmap:= sfust ) );</span>
[ [ 1, 3, 7, 8, 11, 11, 22, 25, 28, 50, 3, 9, 23, 28, 43, 43 ],
[ 1, 3, 7, 8, 11, 11, 22, 25, 28, 50, 3, 11, 23, 28, 50, 50 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( s );</span>
[ 1, 2, 3, 3, 4, 4, 6, 7, 8, 12, 2, 4, 6, 8, 12, 12 ]
</pre></div>
<p>So we still have two possibilities, which differ on the outer classes of element order <span class="SimpleMath">\(4\)</span> and <span class="SimpleMath">\(12\)</span>.</p>
<p>Our idea is to take a subgroup <span class="SimpleMath">\(U\)</span> of <span class="SimpleMath">\(H\)</span> that contains such elements, and to compute the possible class fusions of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(G\)</span>, via the factorization through a suitable maximal subgroup <span class="SimpleMath">\(M\)</span> of <span class="SimpleMath">\(G\)</span>.</p>
<p>We take <span class="SimpleMath">\(U = N_H(\langle g \rangle)\)</span> where <span class="SimpleMath">\(g\)</span> is an element in the first class of order three elements of <span class="SimpleMath">\(H\)</span>; this is a maximal subgroup of <span class="SimpleMath">\(H\)</span>, of order <span class="SimpleMath">\(216\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Maxes( s );</span>
[ "U3(3)", "3^(1+2):SD16", "L3(2).2", "2^(1+4).S3", "4^2:D12" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SizesCentralizers( s );</span>
[ 12096, 192, 216, 18, 96, 32, 24, 7, 8, 12, 48, 48, 6, 8, 12, 12 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( Maxes( s )[2] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufuss:= GetFusionMap( u, s );</span>
[ 1, 2, 11, 3, 4, 5, 12, 7, 13, 9, 9, 15, 16, 10 ]
</pre></div>
<p>Candidates for <span class="SimpleMath">\(M\)</span> are those subgroups of <span class="SimpleMath">\(G\)</span> that contain elements in the class <code class="code">3D</code> of <span class="SimpleMath">\(G\)</span> whose centralizer is the full <code class="code">3D</code> centralizer in <span class="SimpleMath">\(G\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">3Dcentralizer:= SizesCentralizers( t )[7];</span>
153055008
<span class="GAPprompt">gap></span> <span class="GAPinput">cand:= [];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for name in Maxes( t ) do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> m:= CharacterTable( name );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> mfust:= GetFusionMap( m, t ); </span>
<span class="GAPprompt">></span> <span class="GAPinput"> if ForAny( [ 1 .. Length( mfust ) ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> i -> mfust[i] = 7 and SizesCentralizers( m )[i] = 3Dcentralizer ) </span>
<span class="GAPprompt">></span> <span class="GAPinput"> then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Add( cand, m );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cand;</span>
[ CharacterTable( "3^7.O7(3)" ),
CharacterTable( "3^2.3^4.3^8.(A5x2A4).2" ) ]
</pre></div>
<p>For these two groups <span class="SimpleMath">\(M\)</span>, we show that the possible class fusions from <span class="SimpleMath">\(U\)</span> to <span class="SimpleMath">\(G\)</span> via <span class="SimpleMath">\(M\)</span> factorize through <span class="SimpleMath">\(H\)</span> only if the second possible class fusion from <span class="SimpleMath">\(H\)</span> to <span class="SimpleMath">\(G\)</span> is chosen.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">possufust:= List( sfust, x -> CompositionMaps( x, ufuss ) );</span>
[ [ 1, 3, 3, 7, 8, 11, 9, 22, 23, 28, 28, 43, 43, 50 ],
[ 1, 3, 3, 7, 8, 11, 11, 22, 23, 28, 28, 50, 50, 50 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= cand[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusm:= PossibleClassFusions( u, m );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( ufusm );</span>
242
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= List( ufusm, x -> CompositionMaps( GetFusionMap( m, t ), x ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( possufust, comp );</span>
[ [ 1, 3, 3, 7, 8, 11, 11, 22, 23, 28, 28, 50, 50, 50 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= cand[2];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusm:= PossibleClassFusions( u, m );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( ufusm ); </span>
256
<span class="GAPprompt">gap></span> <span class="GAPinput">comp:= List( ufusm, x -> CompositionMaps( GetFusionMap( m, t ), x ) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( possufust, comp );</span>
[ [ 1, 3, 3, 7, 8, 11, 11, 22, 23, 28, 28, 50, 50, 50 ] ]
</pre></div>
<p>Finally, we check that the correct fusion is stored in the <strong class="pkg">GAP</strong> Character Table Library.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( s, t ) = sfust[2];</span>
true
</pre></div>
<p><a id="X83061094871EE241" name="X83061094871EE241"></a></p>
<h5>9.7-2 <span class="Heading"><span class="SimpleMath">\(L_2(13).2 \rightarrow Fi_{24}^{\prime}\)</span> (September 2002)</span></h5>
<p>The class fusion of maximal subgroups <span class="SimpleMath">\(U\)</span> of type <span class="SimpleMath">\(L_2(13).2\)</span> in <span class="SimpleMath">\(G = Fi_{24}^{\prime}\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "F3+" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( "L2(13).2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= PossibleClassFusions( u, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( u, fus, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( repr );</span>
3
</pre></div>
<p>In <a href="chapBib_mj.html#biBLW91">[LW91, p. 155]</a>, it is stated that <span class="SimpleMath">\(U^{\prime}\)</span> contains elements in the classes <code class="code">2B</code>, <code class="code">3D</code>, and <code class="code">7B</code> of <span class="SimpleMath">\(G\)</span>. (Note that the two conjugacy classes of groups isomorphic to <span class="SimpleMath">\(U\)</span> have the same class fusion because the outer automorphism of <span class="SimpleMath">\(G\)</span> fixes the relevant classes.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( repr, x -> t.2b in x and t.3d in x and t.7b in x );</span>
[ [ 1, 3, 7, 22, 25, 25, 25, 51, 3, 9, 43, 43, 53, 53, 53 ],
[ 1, 3, 7, 22, 25, 25, 25, 51, 3, 11, 50, 50, 53, 53, 53 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassNames( t ){ [ 43, 50 ] };</span>
[ "12f", "12m" ]
</pre></div>
<p>So we have to decide whether <span class="SimpleMath">\(U\)</span> contains elements in the class <code class="code">12F</code> or in <code class="code">12M</code> of <span class="SimpleMath">\(G\)</span>.</p>
<p>The order <span class="SimpleMath">\(12\)</span> elements in question lie inside subgroups of type <span class="SimpleMath">\(13 : 12\)</span> in <span class="SimpleMath">\(U\)</span>. These subgroups are clearly contained in the Sylow <span class="SimpleMath">\(13\)</span> normalizers of <span class="SimpleMath">\(G\)</span>, which are contained in maximal subgroups of type <span class="SimpleMath">\((3^2:2 \times G_2(3)).2\)</span> in <span class="SimpleMath">\(G\)</span>; the class fusion of the latter groups is unique up to table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos:= Position( OrdersClassRepresentatives( t ), 13 );</span>
51
<span class="GAPprompt">gap></span> <span class="GAPinput">SizesCentralizers( t )[ pos ];</span>
234
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassOrbit( t, pos );</span>
[ 51 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">cand:= [];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for name in Maxes( t ) do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> m:= CharacterTable( name );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> pos:= Position( OrdersClassRepresentatives( m ), 13 );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if pos <> fail and </span>
<span class="GAPprompt">></span> <span class="GAPinput"> SizesCentralizers( m )[ pos ] = 234 </span>
<span class="GAPprompt">></span> <span class="GAPinput"> and ClassOrbit( m, pos ) = [ pos ] then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Add( cand, m );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cand;</span>
[ CharacterTable( "(3^2:2xG2(3)).2" ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= cand[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t );;</span>
</pre></div>
<p>As no <span class="SimpleMath">\(13:12\)</span> type subgroup is contained in the derived subgroup of <span class="SimpleMath">\((3^2:2 \times G_2(3)).2\)</span>, we look at the elements of order <span class="SimpleMath">\(12\)</span> in the outer half.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">der:= ClassPositionsOfDerivedSubgroup( s );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">outer:= Difference( [ 1 .. NrConjugacyClasses( s ) ], der );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">imgs:= Set( Flat( List( sfust, x -> x{ outer } ) ) );</span>
[ 2, 3, 10, 11, 15, 17, 18, 19, 21, 22, 26, 44, 45, 49, 50, 52, 62,
83, 87, 98 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">t.12f in imgs;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">t.12m in imgs;</span>
true
</pre></div>
<p>So <span class="SimpleMath">\(L_2(13).2 \setminus L_2(13)\)</span> does not contain <code class="code">12F</code> elements of <span class="SimpleMath">\(G\)</span>, i. e., we have determined the class fusion of <span class="SimpleMath">\(U\)</span> in <span class="SimpleMath">\(G\)</span>.</p>
<p>Finally, we check whether the correct fusion is stored in the <strong class="pkg">GAP</strong> Character Table Library.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( u, t ) = filt[2];</span>
true
</pre></div>
<p><a id="X7E9C203C7C4D709D" name="X7E9C203C7C4D709D"></a></p>
<h5>9.7-3 <span class="Heading"><span class="SimpleMath">\(M_{11} \rightarrow B\)</span> (April 2009)</span></h5>
<p>The sporadic simple group <span class="SimpleMath">\(B\)</span> contains a maximal subgroup <span class="SimpleMath">\(M\)</span> of the type <span class="SimpleMath">\(M_{11}\)</span> whose class fusion is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CharacterTable( "B" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m11:= CharacterTable( "M11" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m11fusb:= PossibleClassFusions( m11, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( m11fusb );</span>
31
<span class="GAPprompt">gap></span> <span class="GAPinput">CompositionMaps( ClassNames( b, "ATLAS" ), Parametrized( m11fusb ) );</span>
[ "1A", [ "2B", "2D" ], [ "3A", "3B" ],
[ "4B", "4E", "4G", "4H", "4J" ], [ "5A", "5B" ],
[ "6C", "6E", "6H", "6I", "6J" ],
[ "8B", "8E", "8G", "8J", "8K", "8L", "8M", "8N" ],
[ "8B", "8E", "8G", "8J", "8K", "8L", "8M", "8N" ], "11A", "11A" ]
</pre></div>
<p>According to <a href="chapBib_mj.html#biBWil93a">[Wil93a, Thm. 12.1]</a>, <span class="SimpleMath">\(M\)</span> contains no <code class="code">5A</code> elements of <span class="SimpleMath">\(B\)</span>. By the proof of <a href="chapBib_mj.html#biBWil99">[Wil99, Prop. 4.1]</a>, the involutions in any <span class="SimpleMath">\(S_5\)</span> type subgroup <span class="SimpleMath">\(U\)</span> of <span class="SimpleMath">\(M\)</span> lie in the class <code class="code">2C</code> or <code class="code">2D</code> of <span class="SimpleMath">\(B\)</span>, and since the possible class fusions of <span class="SimpleMath">\(M\)</span> computed above admit only involutions in the class <code class="code">2B</code> or <code class="code">2D</code>, all involutions of <span class="SimpleMath">\(U\)</span> lie in the class <code class="code">2D</code>. Again by the proof of <a href="chapBib_mj.html#biBWil99">[Wil99, Prop. 4.1]</a>, <span class="SimpleMath">\(U\)</span> is contained in a maximal subgroup of type <span class="SimpleMath">\(Th\)</span> in <span class="SimpleMath">\(B\)</span>.</p>
<p>Now we use the embedding of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(B\)</span> via <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(Th\)</span> for determining the class fusion of <span class="SimpleMath">\(M\)</span> into <span class="SimpleMath">\(B\)</span>. The class fusion of the embedding of <span class="SimpleMath">\(U\)</span> via <span class="SimpleMath">\(Th\)</span> is uniquely determined.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">th:= CharacterTable( "Th" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s5:= CharacterTable( "S5" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s5fusth:= PossibleClassFusions( s5, th );</span>
[ [ 1, 2, 4, 8, 2, 7, 11 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">thfusb:= PossibleClassFusions( th, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s5fusb:= Set( thfusb, x -> CompositionMaps( x, s5fusth[1] ) );</span>
[ [ 1, 5, 7, 19, 5, 17, 29 ] ]
</pre></div>
<p>Also the class fusion of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(M\)</span> is unique, and this determines the class fusion of <span class="SimpleMath">\(M\)</span> into <span class="SimpleMath">\(B\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s5fusm11:= PossibleClassFusions( s5, m11 );</span>
[ [ 1, 2, 3, 5, 2, 4, 6 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">m11fusb:= Filtered( m11fusb,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map -> CompositionMaps( map, s5fusm11[1] ) = s5fusb[1] );</span>
[ [ 1, 5, 7, 17, 19, 29, 45, 45, 54, 54 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CompositionMaps( ClassNames( b, "ATLAS" ), m11fusb[1] );</span>
[ "1A", "2D", "3B", "4J", "5B", "6J", "8N", "8N", "11A", "11A" ]
</pre></div>
<p>(Using the information that the <span class="SimpleMath">\(M_{10}\)</span> type subgroups of <span class="SimpleMath">\(M\)</span> are also contained in <span class="SimpleMath">\(Th\)</span> type subgroups would not have helped us, since these subgroups do not contain elements of order <span class="SimpleMath">\(6\)</span>, and two possibilities would have remained.)</p>
<p><a id="X85821D748716DC7E" name="X85821D748716DC7E"></a></p>
<h5>9.7-4 <span class="Heading"><span class="SimpleMath">\(L_2(11):2 \rightarrow B\)</span> (April 2009)</span></h5>
<p>The sporadic simple group <span class="SimpleMath">\(B\)</span> contains a maximal subgroup <span class="SimpleMath">\(L\)</span> of the type <span class="SimpleMath">\(L_2(11):2\)</span> whose class fusion is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CharacterTable( "B" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= CharacterTable( "L2(11).2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">lfusb:= PossibleClassFusions( l, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( lfusb );</span>
16
<span class="GAPprompt">gap></span> <span class="GAPinput">CompositionMaps( ClassNames( b, "ATLAS" ), Parametrized( lfusb ) );</span>
[ "1A", [ "2B", "2D" ], [ "3A", "3B" ], [ "5A", "5B" ],
[ "5A", "5B" ], [ "6C", "6H", "6I", "6J" ], "11A", [ "2C", "2D" ],
[ "4D", "4E", "4F", "4G", "4H", "4J" ], [ "10C", "10E", "10F" ],
[ "10C", "10E", "10F" ],
[ "12E", "12F", "12H", "12I", "12J", "12L", "12N", "12P", "12Q",
"12R", "12S" ],
[ "12E", "12F", "12H", "12I", "12J", "12L", "12N", "12P", "12Q",
"12R", "12S" ] ]
</pre></div>
<p>According to <a href="chapBib_mj.html#biBWil93a">[Wil93a, Thm. 12.1]</a>, <span class="SimpleMath">\(L\)</span> contains no <code class="code">5A</code> elements of <span class="SimpleMath">\(B\)</span>. By the proof of <a href="chapBib_mj.html#biBWil99">[Wil99, Prop. 4.1]</a>, <span class="SimpleMath">\(B\)</span> contains exactly one class of <span class="SimpleMath">\(L_2(11)\)</span> type subgroups with this property. Hence the subgroup <span class="SimpleMath">\(U\)</span> of index two in <span class="SimpleMath">\(L\)</span> is contained in a maximal subgroup <span class="SimpleMath">\(M\)</span> of type <span class="SimpleMath">\(M_{11}\)</span> in <span class="SimpleMath">\(B\)</span>, whose class fusion was determined in Section <a href="chap9_mj.html#X7E9C203C7C4D709D"><span class="RefLink">9.7-3</span></a>.</p>
<p>In the same way as we proceeded in Section <a href="chap9_mj.html#X7E9C203C7C4D709D"><span class="RefLink">9.7-3</span></a>, we use the embedding of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(B\)</span> via <span class="SimpleMath">\(L\)</span> and <span class="SimpleMath">\(M\)</span> for determining the class fusion of <span class="SimpleMath">\(L\)</span> into <span class="SimpleMath">\(B\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= CharacterTable( "M11" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( "L2(11)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusm:= PossibleClassFusions( u, m );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mfusb:= GetFusionMap( m, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusb:= Set( ufusm, x -> CompositionMaps( mfusb, x ) );</span>
[ [ 1, 5, 7, 19, 19, 29, 54, 54 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusl:= PossibleClassFusions( u, l );</span>
[ [ 1, 2, 3, 4, 5, 6, 7, 7 ], [ 1, 2, 3, 5, 4, 6, 7, 7 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">lfusb:= Filtered( lfusb, </span>
<span class="GAPprompt">></span> <span class="GAPinput"> map2 -> ForAny( ufusl, </span>
<span class="GAPprompt">></span> <span class="GAPinput"> map1 -> CompositionMaps( map2, map1 ) in ufusb ) );</span>
[ [ 1, 5, 7, 19, 19, 29, 54, 5, 15, 53, 53, 73, 73 ] ]
</pre></div>
<p><a id="X828D81487F57D612" name="X828D81487F57D612"></a></p>
<h5>9.7-5 <span class="Heading"><span class="SimpleMath">\(L_3(3) \rightarrow B\)</span> (April 2009)</span></h5>
<p>The sporadic simple group <span class="SimpleMath">\(B\)</span> contains a maximal subgroup <span class="SimpleMath">\(T\)</span> of the type <span class="SimpleMath">\(L_3(3)\)</span> whose class fusion is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CharacterTable( "B" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "L3(3)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tfusb:= PossibleClassFusions( t, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( tfusb );</span>
36
</pre></div>
<p>According to <a href="chapBib_mj.html#biBWil99">[Wil99, Section 9]</a>, <span class="SimpleMath">\(T\)</span> contains a subgroup <span class="SimpleMath">\(U\)</span> of the type <span class="SimpleMath">\(3^2:2S_4\)</span> that is contained also in a maximal subgroup <span class="SimpleMath">\(M\)</span> of the type <span class="SimpleMath">\(3^2.3^3.3^6.(S_4 \times 2S_4)\)</span>. So we throw away the possible fusions from <span class="SimpleMath">\(T\)</span> to <span class="SimpleMath">\(B\)</span> that are not compatible with the compositions of the embeddings of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(B\)</span> via <span class="SimpleMath">\(T\)</span> and <span class="SimpleMath">\(M\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= CharacterTable( "3^2.3^3.3^6.(S4x2S4)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= PSL(3,3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mx:= MaximalSubgroupClassReps( g );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= First( mx, x -> Size( x ) = 432 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( u );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusm:= PossibleClassFusions( u, m );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufust:= PossibleClassFusions( u, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mfusb:= GetFusionMap( m, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusb:= Set( ufusm, map -> CompositionMaps( mfusb, map ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tfusb:= Filtered( tfusb, map -> ForAny( ufust,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map2 -> CompositionMaps( map, map2 ) in ufusb ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tfusb;</span>
[ [ 1, 5, 6, 7, 12, 27, 41, 41, 75, 75, 75, 75 ],
[ 1, 5, 7, 6, 12, 28, 41, 41, 75, 75, 75, 75 ],
[ 1, 5, 7, 7, 12, 28, 41, 41, 75, 75, 75, 75 ],
[ 1, 5, 7, 7, 12, 29, 41, 41, 75, 75, 75, 75 ],
[ 1, 5, 7, 7, 17, 29, 45, 45, 75, 75, 75, 75 ] ]
</pre></div>
<p>Now we use that <span class="SimpleMath">\(T\)</span> does not contain <code class="code">4E</code> elements of <span class="SimpleMath">\(B\)</span> (again see <a href="chapBib_mj.html#biBWil99">[Wil99, Section 9]</a>). Thus the last of the five candidates is the correct class fusion.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassNames( b, "ATLAS" ){ [ 12, 17 ] };</span>
[ "4E", "4J" ]
</pre></div>
<p>We check that this map is stored on the library table.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( t, b ) = tfusb[5];</span>
true
</pre></div>
<p><a id="X7B4E13337D66020F" name="X7B4E13337D66020F"></a></p>
<h5>9.7-6 <span class="Heading"><span class="SimpleMath">\(L_2(17).2 \rightarrow B\)</span> (March 2004)</span></h5>
<p>The sporadic simple group <span class="SimpleMath">\(B\)</span> contains a maximal subgroup <span class="SimpleMath">\(U\)</span> of the type <span class="SimpleMath">\(L_2(17).2\)</span> whose class fusion is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CharacterTable( "B" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( "L2(17).2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusb:= PossibleClassFusions( u, b );</span>
[ [ 1, 5, 7, 15, 42, 42, 47, 47, 47, 91, 4, 30, 89, 89, 89, 89, 97,
97, 97 ],
[ 1, 5, 7, 15, 44, 44, 46, 46, 46, 91, 5, 29, 90, 90, 90, 90, 96,
96, 96 ],
[ 1, 5, 7, 15, 44, 44, 47, 47, 47, 91, 5, 29, 90, 90, 90, 90, 95,
95, 95 ] ]
</pre></div>
<p>According to <a href="chapBib_mj.html#biBWil99">[Wil99, Prop. 11.1]</a>, <span class="SimpleMath">\(U\)</span> contains elements in the classes <code class="code">8M</code> and <code class="code">9A</code> of <span class="SimpleMath">\(B\)</span>. This determines the fusion map.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">names:= ClassNames( b, "ATLAS" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos:= List( [ "8M", "9A" ], x -> Position( names, x ) );</span>
[ 44, 46 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusb:= Filtered( ufusb, map -> IsSubset( map, pos ) );</span>
[ [ 1, 5, 7, 15, 44, 44, 46, 46, 46, 91, 5, 29, 90, 90, 90, 90, 96,
96, 96 ] ]
</pre></div>
<p>We check that this map is stored on the library table.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( u, b ) = ufusb[1];</span>
true
</pre></div>
<p><a id="X8528432A84851F7B" name="X8528432A84851F7B"></a></p>
<h5>9.7-7 <span class="Heading"><span class="SimpleMath">\(L_2(49).2_3 \rightarrow B\)</span> (June 2006)</span></h5>
<p>The sporadic simple group <span class="SimpleMath">\(B\)</span> contains a class of maximal subgroups of the type <span class="SimpleMath">\(L_2(49).2_3\)</span> (a non-split extension of <span class="SimpleMath">\(L_2(49)\)</span>, see <a href="chapBib_mj.html#biBWilson93">[Wil93b, Theorem 2]</a>). Let <span class="SimpleMath">\(U\)</span> be such a subgroup. The class fusion of <span class="SimpleMath">\(U\)</span> in <span class="SimpleMath">\(B\)</span> is not determined by the character tables of <span class="SimpleMath">\(U\)</span> and <span class="SimpleMath">\(B\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( "L2(49).2_3" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CharacterTable( "B" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusb:= PossibleClassFusions( u, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( u, ufusb, b ) );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">ufusb;</span>
[ [ 1, 5, 7, 15, 19, 28, 31, 42, 42, 71, 125, 125, 128, 128, 128,
128, 128, 15, 71, 71, 89, 89, 89, 89 ],
[ 1, 5, 7, 15, 19, 28, 31, 42, 42, 71, 125, 125, 128, 128, 128,
128, 128, 17, 72, 72, 89, 89, 89, 89 ] ]
</pre></div>
<p>We show that the fusion is determined by the embeddings of the Sylow <span class="SimpleMath">\(7\)</span> normalizer <span class="SimpleMath">\(N\)</span>, say, of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(U\)</span> and into the Sylow <span class="SimpleMath">\(7\)</span> normalizer of <span class="SimpleMath">\(B\)</span>. (Note that the fusion of the latter group into <span class="SimpleMath">\(B\)</span> has been determined in Section <a href="chap9_mj.html#X7F5186E28201B027"><span class="RefLink">9.3-1</span></a>.)</p>
<p>For that, we compute the character table of <span class="SimpleMath">\(N\)</span> from a representation of <span class="SimpleMath">\(U\)</span>. Note that <span class="SimpleMath">\(U\)</span> is a non-split extension of the simple group <span class="SimpleMath">\(L_2(49)\)</span> by the product of a diagonal automorphism and a field automorphism. In <a href="chapBib_mj.html#biBWilson93">[Wil93b]</a>, the structure of <span class="SimpleMath">\(N\)</span> is described as <span class="SimpleMath">\(7^2:(3 \times Q_{16})\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= SL( 2, 49 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( g );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= GF(49);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mats:= List( gens, x -> IdentityMat( 4, f ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [ 1 .. Length( gens ) ] do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> mats[i]{ [ 1, 2 ] }{ [ 1, 2 ] }:= gens[i];</span>
<span class="GAPprompt">></span> <span class="GAPinput"> mats[i]{ [ 3, 4 ] }{ [ 3, 4 ] }:= List( gens[i],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x -> List( x, y -> y^7 ) );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fieldaut:= PermutationMat( (1,3)(2,4), 4, f );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">diagaut:= IdentityMat( 4, f );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">diagaut[1][1]:= Z(49);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">diagaut[3][3]:= Z(49)^7;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( Concatenation( mats, [ fieldaut * diagaut ] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">v:= [ 1, 0, 0, 0 ] * Z(7)^0;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">orb:= Orbit( g, v, OnLines );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">act:= Action( g, orb, OnLines );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:= Normalizer( act, SylowSubgroup( act, 7 ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ntbl:= CharacterTable( n );;</span>
</pre></div>
<p>Now we compute the possible class fusions of <span class="SimpleMath">\(N\)</span> into <span class="SimpleMath">\(B\)</span>, via the Sylow <span class="SimpleMath">\(7\)</span> normalizer in <span class="SimpleMath">\(B\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">bn7:= CharacterTable( "BN7" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">nfusbn7:= PossibleClassFusions( ntbl, bn7 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( ntbl, nfusbn7, bn7 ) );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">nfusb:= SetOfComposedClassFusions( PossibleClassFusions( bn7, b ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> nfusbn7 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( ntbl, nfusb, b ) );</span>
5
</pre></div>
<p>Although there are several possibilities, this information is enough to exclude one of the possible fusions of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(B\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">nfusu:= PossibleClassFusions( ntbl, u );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( nfusu );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">filt:= Filtered( ufusb,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x -> ForAny( nfusu, y -> CompositionMaps( x, y ) in nfusb ) );</span>
[ [ 1, 5, 7, 15, 19, 28, 31, 42, 42, 71, 125, 125, 128, 128, 128,
128, 128, 17, 72, 72, 89, 89, 89, 89 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassNames( b, "ATLAS" ){ filt[1] };</span>
[ "1A", "2D", "3B", "4H", "5B", "6I", "7A", "8K", "8K", "12Q", "24L",
"24L", "25A", "25A", "25A", "25A", "25A", "4J", "12R", "12R",
"16G", "16G", "16G", "16G" ]
</pre></div>
<p>So the class fusion of <span class="SimpleMath">\(U\)</span> into <span class="SimpleMath">\(B\)</span> can be described by the property that the elements of order four inside and outside the simple subgroup <span class="SimpleMath">\(L_2(49)\)</span> are not conjugate in <span class="SimpleMath">\(B\)</span>.</p>
<p>We check that the correct map is stored on the library table.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( u, b ) in filt;</span>
true
</pre></div>
<p>Let us confirm that the two groups of the types <span class="SimpleMath">\(L_2(49).2_1\)</span> and <span class="SimpleMath">\(L_2(49).2_2\)</span> cannot occur as subgroups of <span class="SimpleMath">\(B\)</span>. First we show that <span class="SimpleMath">\(L_2(49).2_1\)</span> is isomorphic with PGL<span class="SimpleMath">\((2,49)\)</span>, an extension of <span class="SimpleMath">\(L_2(49)\)</span> by a diagonal automorphism, and <span class="SimpleMath">\(L_2(49).2_2\)</span> is an extension by a field automorphism.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClasses( u ); NrConjugacyClasses( act );</span>
24
24
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( "L2(49).2_1" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( Concatenation( mats, [ diagaut ] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">orb:= Orbit( g, v, OnLines );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">act:= Action( g, orb, OnLines );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(act );</span>
117600
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClasses( u ); NrConjugacyClasses( act );</span>
51
51
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= CharacterTable( "L2(49).2_2" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( Concatenation( mats, [ fieldaut ] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">orb:= Orbit( g, v, OnLines );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">act:= Action( g, orb, OnLines );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClasses( u ); NrConjugacyClasses( act );</span>
27
27
</pre></div>
<p>The group <span class="SimpleMath">\(L_2(49).2_1\)</span> can be excluded because no class fusion into <span class="SimpleMath">\(B\)</span> is possible.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PossibleClassFusions( CharacterTable( "L2(49).2_1" ), b );</span>
[ ]
</pre></div>
<p>For <span class="SimpleMath">\(L_2(49).2_2\)</span>, it is not that easy. We would get several possible class fusions into <span class="SimpleMath">\(B\)</span>. However, the Sylow <span class="SimpleMath">\(7\)</span> normalizer of <span class="SimpleMath">\(L_2(49).2_2\)</span> does not admit a class fusion into the Sylow <span class="SimpleMath">\(7\)</span> normalizer of <span class="SimpleMath">\(B\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:= Normalizer( act, SylowSubgroup( act, 7 ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( PossibleClassFusions( CharacterTable( n ), bn7 ) );</span>
0
</pre></div>
<p><a id="X7EAD52AA7A28D956" name="X7EAD52AA7A28D956"></a></p>
<h5>9.7-8 <span class="Heading"><span class="SimpleMath">\(2^3.L_3(2) \rightarrow G_2(5)\)</span> (January 2004)</span></h5>
<p>The Chevalley group <span class="SimpleMath">\(G = G_2(5)\)</span> contains a maximal subgroup <span class="SimpleMath">\(U\)</span> of the type <span class="SimpleMath">\(2^3.L_3(2)\)</span> whose class fusion is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "G2(5)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "2^3.L3(2)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sfust:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativesFusions( s, sfust, t );</span>
[ [ 1, 2, 2, 5, 6, 4, 13, 16, 17, 15, 15 ],
[ 1, 2, 2, 5, 6, 4, 14, 16, 17, 15, 15 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( s );</span>
[ 1, 2, 2, 4, 4, 3, 6, 8, 8, 7, 7 ]
</pre></div>
<p>So the question is whether <span class="SimpleMath">\(U\)</span> contains elements in the class <code class="code">6B</code> or <code class="code">6C</code> of <span class="SimpleMath">\(G\)</span> (position <span class="SimpleMath">\(13\)</span> or <span class="SimpleMath">\(14\)</span> in the <strong class="pkg">Atlas</strong> table). We use a permutation representation of <span class="SimpleMath">\(G\)</span>, restrict it to <span class="SimpleMath">\(U\)</span>, and compute the centralizer in <span class="SimpleMath">\(G\)</span> of a suitable element of order <span class="SimpleMath">\(6\)</span> in <span class="SimpleMath">\(U\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= AtlasGroup( "G2(5)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= AtlasSubgroup( "G2(5)", 7 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( u );</span>
1344
<span class="GAPprompt">gap></span> <span class="GAPinput">repeat</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x:= Random( u );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> until Order( x ) = 6;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">siz:= Size( Centralizer( g, x ) );</span>
36
<span class="GAPprompt">gap></span> <span class="GAPinput">Filtered( [ 1 .. NrConjugacyClasses( t ) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> i -> SizesCentralizers( t )[i] = siz );</span>
[ 14 ]
</pre></div>
<p>So <span class="SimpleMath">\(U\)</span> contains <code class="code">6C</code> elements in <span class="SimpleMath">\(G_2(5)\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( s, t ) in Filtered( sfust, map -> 14 in map ); </span>
true
</pre></div>
<p><a id="X79617107849A6CEA" name="X79617107849A6CEA"></a></p>
<h5>9.7-9 <span class="Heading"><span class="SimpleMath">\(5^{{1+4}}.2^{{1+4}}.A_5.4 \rightarrow B\)</span> (April 2009)</span></h5>
<p>The sporadic simple group <span class="SimpleMath">\(B\)</span> contains a maximal subgroup <span class="SimpleMath">\(M\)</span> of the type <span class="SimpleMath">\(5^{{1+4}}.2^{{1+4}}.A_5.4\)</span> whose class fusion is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CharacterTable( "B" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= CharacterTable( "5^(1+4).2^(1+4).A5.4" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mfusb:= PossibleClassFusions( m, b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( mfusb );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">repres:= RepresentativesFusions( m, mfusb, b );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( repres );</span>
2
</pre></div>
<p>The restriction of the unique irreducible character of degree <span class="SimpleMath">\(4\,371\)</span> distinguishes the two possibilities,</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">char:= Filtered( Irr( b ), x -> x[1] = 4371 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( char );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">rest:= List( repres, map -> char[1]{ map } );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">scprs:= MatScalarProducts( m, Irr( m ), rest );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">constit:= List( scprs,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x -> Filtered( [1 .. Length(x) ], i -> x[i] <> 0 ) );</span>
[ [ 2, 27, 60, 63, 73, 74, 75, 79, 82 ],
[ 2, 27, 60, 63, 70, 72, 75, 79, 84 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( constit, x -> List( Irr( m ){ x }, Degree ) );</span>
[ [ 1, 6, 384, 480, 400, 400, 500, 1000, 1200 ],
[ 1, 6, 384, 480, 100, 300, 500, 1000, 1600 ] ]
</pre></div>
<p>The database <a href="chapBib_mj.html#biBAGRv3">[WWT+]</a> contains the <span class="SimpleMath">\(3\)</span>-modular reduction of the irreducible representation of degree <span class="SimpleMath">\(4\,371\)</span> and also a straight line program for restricting this representation to <span class="SimpleMath">\(M\)</span>. We access these data via the <strong class="pkg">GAP</strong> package <strong class="pkg">AtlasRep</strong> (see <a href="chapBib_mj.html#biBAtlasRep">[WPN+22]</a>), and compute the composition factors of the natural module of this restriction.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= AtlasSubgroup( "B", Dimension, 4371, Ring, GF(3), 21 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">module:= GModuleByMats( GeneratorsOfGroup( g ), GF(3) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">dec:= MTX.CompositionFactors( module );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SortedList( List( dec, x -> x.dimension ) );</span>
[ 1, 6, 100, 384, 400, 400, 400, 480, 1000, 1200 ]
</pre></div>
<p>We see that exactly one ordinary constituent does not stay irreducible upon restriction to characteristic <span class="SimpleMath">\(3\)</span>. Thus the first of the two possible class fusions is the correct one.</p>
<p><a id="X85C48EEB7B711C09" name="X85C48EEB7B711C09"></a></p>
<h5>9.7-10 <span class="Heading">The fusion from the character table of <span class="SimpleMath">\(7^2:2L_2(7).2\)</span>
into the table of marks (January 2004)</span></h5>
<p>It can happen that the class fusion from the ordinary character table of a group <span class="SimpleMath">\(G\)</span> into the table of marks of <span class="SimpleMath">\(G\)</span> is not unique up to table automorphisms of the character table of <span class="SimpleMath">\(G\)</span>.</p>
<p>As an example, consider <span class="SimpleMath">\(G = 7^2:2L_2(7).2\)</span>, a maximal subgroup in the sporadic simple group <span class="SimpleMath">\(He\)</span>.</p>
<p><span class="SimpleMath">\(G\)</span> contains four classes of cyclic subgroups of order <span class="SimpleMath">\(7\)</span>. One contains the elements in the normal subgroup of type <span class="SimpleMath">\(7^2\)</span>, and the other three are preimages of the order <span class="SimpleMath">\(7\)</span> elements in the factor group <span class="SimpleMath">\(L_2(7)\)</span>. The conjugacy classes of nonidentity elements in the latter three classes split into two Galois conjugates each, which are permuted cyclicly by the table automorphisms of the character table of <span class="SimpleMath">\(G\)</span>, but on which the stabilizer of one class acts trivially. This means that determining one of the three classes determines also the other two.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "7^2:2psl(2,7)" );</span>
CharacterTable( "7^2:2psl(2,7)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">tom:= TableOfMarks( tbl );</span>
TableOfMarks( "7^2:2L2(7)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= PossibleFusionsCharTableTom( tbl, tom );</span>
[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 10, 9, 16, 7, 10, 9, 8, 16 ],
[ 1, 6, 2, 4, 3, 5, 13, 13, 7, 9, 8, 10, 16, 7, 8, 10, 9, 16 ],
[ 1, 6, 2, 4, 3, 5, 13, 13, 7, 10, 9, 8, 16, 7, 9, 8, 10, 16 ],
[ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 9, 10, 16, 7, 9, 10, 8, 16 ],
[ 1, 6, 2, 4, 3, 5, 13, 13, 7, 10, 8, 9, 16, 7, 8, 9, 10, 16 ],
[ 1, 6, 2, 4, 3, 5, 13, 13, 7, 9, 10, 8, 16, 7, 10, 8, 9, 16 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">reps:= RepresentativesFusions( tbl, fus, Group(()) ); </span>
[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 9, 10, 16, 7, 9, 10, 8, 16 ],
[ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 10, 9, 16, 7, 10, 9, 8, 16 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AutomorphismsOfTable( tbl );</span>
Group([ (9,14)(10,17)(11,15)(12,16)(13,18), (7,8), (10,11,12)
(15,16,17) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( tbl );</span>
[ 1, 7, 2, 4, 3, 6, 8, 8, 7, 7, 7, 7, 14, 7, 7, 7, 7, 14 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">perms1:= PermCharsTom( reps[1], tom );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">perms2:= PermCharsTom( reps[2], tom );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">perms1 = perms2; </span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( perms1 ) = Set( perms2 );</span>
true
</pre></div>
<p>The table of marks of <span class="SimpleMath">\(G\)</span> does not distinguish the three classes of cyclic subgroups, there are permutations of rows and columns that act as an <span class="SimpleMath">\(S_3\)</span> on them.</p>
<p>Note that an <span class="SimpleMath">\(S_3\)</span> acts on the classes in question in the <em>rational</em> character table. So it is due to the irrationalities in the character table that it contains more information.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tbl );</span>
7^2:2psl(2,7)
2 4 . 4 3 1 1 3 3 1 . . . 1 1 . . .
3 1 . 1 . 1 1 . . . . . . . . . . .
7 3 3 1 . . . . . 2 2 2 2 1 2 2 2 2
1a 7a 2a 4a 3a 6a 8a 8b 7b 7c 7d 7e 14a 7f 7g 7h 7i
2P 1a 7a 1a 2a 3a 3a 4a 4a 7b 7c 7d 7e 7b 7f 7g 7h 7i
3P 1a 7a 2a 4a 1a 2a 8b 8a 7f 7i 7g 7h 14b 7b 7d 7e 7c
5P 1a 7a 2a 4a 3a 6a 8b 8a 7f 7i 7g 7h 14b 7b 7d 7e 7c
7P 1a 1a 2a 4a 3a 6a 8a 8b 1a 1a 1a 1a 2a 1a 1a 1a 1a
11P 1a 7a 2a 4a 3a 6a 8b 8a 7b 7c 7d 7e 14a 7f 7g 7h 7i
13P 1a 7a 2a 4a 3a 6a 8b 8a 7f 7i 7g 7h 14b 7b 7d 7e 7c
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 3 3 3 -1 . . 1 1 B B B B B /B /B /B /B
X.3 3 3 3 -1 . . 1 1 /B /B /B /B /B B B B B
X.4 6 6 6 2 . . . . -1 -1 -1 -1 -1 -1 -1 -1 -1
X.5 7 7 7 -1 1 1 -1 -1 . . . . . . . . .
X.6 8 8 8 . -1 -1 . . 1 1 1 1 1 1 1 1 1
X.7 4 4 -4 . 1 -1 . . -B -B -B -B B -/B -/B -/B -/B
X.8 4 4 -4 . 1 -1 . . -/B -/B -/B -/B /B -B -B -B -B
X.9 6 6 -6 . . . A -A -1 -1 -1 -1 1 -1 -1 -1 -1
X.10 6 6 -6 . . . -A A -1 -1 -1 -1 1 -1 -1 -1 -1
X.11 8 8 -8 . -1 1 . . 1 1 1 1 -1 1 1 1 1
X.12 48 -1 . . . . . . 6 -1 -1 -1 . 6 -1 -1 -1
X.13 48 -1 . . . . . . C -1 /C /D . /C C D -1
X.14 48 -1 . . . . . . C /C /D -1 . /C D -1 C
X.15 48 -1 . . . . . . /C D -1 C . C -1 /C /D
X.16 48 -1 . . . . . . C /D -1 /C . /C -1 C D
X.17 48 -1 . . . . . . /C C D -1 . C /D -1 /C
X.18 48 -1 . . . . . . /C -1 C D . C /C /D -1
2 1
3 .
7 1
14b
2P 7f
3P 14a
5P 14a
7P 2a
11P 14b
13P 14a
X.1 1
X.2 /B
X.3 B
X.4 -1
X.5 .
X.6 1
X.7 /B
X.8 B
X.9 1
X.10 1
X.11 -1
X.12 .
X.13 .
X.14 .
X.15 .
X.16 .
X.17 .
X.18 .
A = E(8)-E(8)^3
= Sqrt(2) = r2
B = E(7)+E(7)^2+E(7)^4
= (-1+Sqrt(-7))/2 = b7
C = 2*E(7)+2*E(7)^2+2*E(7)^4
= -1+Sqrt(-7) = 2b7
D = -3*E(7)-3*E(7)^2-2*E(7)^3-3*E(7)^4-2*E(7)^5-2*E(7)^6
= (5-Sqrt(-7))/2 = 2-b7
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:= MatTom( tom );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mataut:= MatrixAutomorphisms( mat );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( mataut, "\n" );</span>
Group( [ (11,12)(23,24)(27,28)(46,47)(53,54)(56,57),
( 9,10)(20,21)(31,32)(38,39), ( 8, 9)(20,22)(31,33)(38,40) ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativesFusions( Group( () ), reps, mataut );</span>
[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 9, 10, 16, 7, 9, 10, 8, 16 ] ]
</pre></div>
<p>We could say that thus the fusion is unique up to table automorphisms and automorphisms of the table of marks. But since a group is associated with the table of marks, we compute the character table from the group, and decide which class fusion is correct.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= UnderlyingGroup( tom );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tg:= CharacterTable( g );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tgfustom:= FusionCharTableTom( tg, tom );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">trans:= TransformingPermutationsCharacterTables( tg, tbl );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tblfustom:= Permuted( tgfustom, trans.columns );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">orbits:= List( reps, map -> OrbitFusions( AutomorphismsOfTable( tbl ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map, Group( () ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PositionProperty( orbits, orb -> tblfustom in orb );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">PositionProperty( orbits, orb -> FusionToTom( tbl ).map in orb );</span>
2
</pre></div>
<p>So we see that the second one of the possibilities above is the right one.</p>
<p><a id="X7B1C689C7EFD07CB" name="X7B1C689C7EFD07CB"></a></p>
<h5>9.7-11 <span class="Heading"><span class="SimpleMath">\(3 \times U_4(2) \rightarrow 3_1.U_4(3)\)</span> (March 2010)</span></h5>
<p>According to the <strong class="pkg">Atlas</strong> (see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 52]</a>), the simple group <span class="SimpleMath">\(U_4(3)\)</span> contains two classes of maximal subgroups of the type <span class="SimpleMath">\(U_4(2)\)</span>. The class fusion of <span class="SimpleMath">\(U_4(2)\)</span> into <span class="SimpleMath">\(U_4(3)\)</span> is unique up to table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u42:= CharacterTable( "U4(2)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u43:= CharacterTable( "U4(3)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u42fusu43:= PossibleClassFusions( u42, u43 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( u42fusu43 );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( u42, u42fusu43, u43 ) );</span>
1
</pre></div>
<p>More precisely, take the outer automorphism group of <span class="SimpleMath">\(U_4(3)\)</span>, which is a dihedral group of order eight, and consider the subgroup generated by its central involution (this automorphism is denoted by <span class="SimpleMath">\(2_1\)</span> in the <strong class="pkg">Atlas</strong>) and another involution called <span class="SimpleMath">\(2_3\)</span> in the <strong class="pkg">Atlas</strong>. This subgroup is a Klein four group that induces a permutation group on the classes of <span class="SimpleMath">\(U_4(3)\)</span> and thus acts on the four possible class fusions of <span class="SimpleMath">\(U_4(2)\)</span> into <span class="SimpleMath">\(U_4(3)\)</span>. In fact, this action is transitive.</p>
<p>The automorphism <span class="SimpleMath">\(2_1\)</span> swaps each pair of mutually inverse classes of order nine, that is, <code class="code">9A</code> is swapped with <code class="code">9B</code> and <code class="code">9C</code> is swapped with <code class="code">9D</code>. All <span class="SimpleMath">\(U_4(2)\)</span> type subgroups of <span class="SimpleMath">\(U_4(3)\)</span> are invariant under this automorphism, they extend to subgroups of the type <span class="SimpleMath">\(U_4(2).2\)</span> in <span class="SimpleMath">\(U_4(3).2_1\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u43_21:= CharacterTable( "U4(3).2_1" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus1:= GetFusionMap( u43, u43_21 );</span>
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 17,
18 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">act1:= Filtered( InverseMap( fus1 ), IsList );</span>
[ [ 16, 17 ], [ 18, 19 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CompositionMaps( ClassNames( u43, "Atlas" ), act1 );</span>
[ [ "9A", "9B" ], [ "9C", "9D" ] ]
</pre></div>
<p>The automorphism <span class="SimpleMath">\(2_3\)</span> swaps <code class="code">6B</code> with <code class="code">6C</code>, <code class="code">9A</code> with <code class="code">9C</code>, and <code class="code">9B</code> with <code class="code">9D</code>. The two classes of <span class="SimpleMath">\(U_4(2)\)</span> type subgroups of <span class="SimpleMath">\(U_4(3)\)</span> are swapped by this automorphism.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u43_23:= CharacterTable( "U4(3).2_3" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus3:= GetFusionMap( u43, u43_23 );</span>
[ 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 13, 14,
15 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">act3:= Filtered( InverseMap( fus3 ), IsList );</span>
[ [ 4, 5 ], [ 11, 12 ], [ 13, 14 ], [ 16, 18 ], [ 17, 19 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CompositionMaps( ClassNames( u43, "Atlas" ), act3 );</span>
[ [ "3B", "3C" ], [ "6B", "6C" ], [ "7A", "7B" ], [ "9A", "9C" ],
[ "9B", "9D" ] ]
</pre></div>
<p>The <strong class="pkg">Atlas</strong> states that the permutation character induced by the first class of <span class="SimpleMath">\(U_4(2)\)</span> type subgroups is <code class="code">1a+35a+90a</code>, which means that the subgroups in this class contain <code class="code">9A</code> and <code class="code">9B</code> elements. Then the permutation character induced by the second class of <span class="SimpleMath">\(U_4(2)\)</span> type subgroups is <code class="code">1a+35b+90a</code>, and the subgroups in this class contain <code class="code">9C</code> and <code class="code">9D</code> elements.</p>
<p>So we choose appropriate fusions for the two classes of maximal <span class="SimpleMath">\(U_4(2)\)</span> type subgroups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">firstfus:= First( u42fusu43, x -> IsSubset( x, [ 16, 17 ] ) );</span>
[ 1, 2, 2, 3, 3, 5, 4, 7, 8, 9, 10, 10, 12, 12, 11, 12, 16, 17, 20,
20 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">secondfus:= First( u42fusu43, x -> IsSubset( x, [ 18, 19 ] ) );</span>
[ 1, 2, 2, 3, 3, 4, 5, 7, 8, 9, 10, 10, 11, 11, 12, 11, 18, 19, 20,
20 ]
</pre></div>
<p>Let us now consider the central extension <span class="SimpleMath">\(3_1.U_4(3)\)</span>. Since the Schur multiplier of <span class="SimpleMath">\(U_4(2)\)</span> has order two, the <span class="SimpleMath">\(U_4(2)\)</span> type subgroups of <span class="SimpleMath">\(U_4(3)\)</span> lift to groups of the structure <span class="SimpleMath">\(3 \times U_4(2)\)</span> in <span class="SimpleMath">\(3_1.U_4(3)\)</span>. There are eight possible class fusions from <span class="SimpleMath">\(3 \times U_4(2)\)</span> to <span class="SimpleMath">\(3_1.U_4(3)\)</span>, in two orbits of length four under the action of table automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">3u42:= CharacterTable( "Cyclic", 3 ) * u42;</span>
CharacterTable( "C3xU4(2)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">3u43:= CharacterTable( "3_1.U4(3)" );</span>
CharacterTable( "3_1.U4(3)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">3u42fus3u43:= PossibleClassFusions( 3u42, 3u43 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( 3u42fus3u43 );</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( RepresentativesFusions( 3u42, 3u42fus3u43, 3u43 ) );</span>
2
</pre></div>
<p>More precisely, each of the four fusions from <span class="SimpleMath">\(U_4(2)\)</span> to <span class="SimpleMath">\(U_4(3)\)</span> has exactly two lifts. The four lifts of those fusions from <span class="SimpleMath">\(U_4(2)\)</span> to <span class="SimpleMath">\(U_4(3)\)</span> with <code class="code">9A</code> and <code class="code">9B</code> in their image form one orbit under the action of table automorphisms. The other orbit consists of the lifts of those fusions with <code class="code">9C</code> and <code class="code">9D</code> in their image.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">inducedmaps:= List( 3u42fus3u43, map -> CompositionMaps(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GetFusionMap( 3u43, u43 ), CompositionMaps( map,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> InverseMap( GetFusionMap( 3u42, u42 ) ) ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( inducedmaps, map -> Position( u42fusu43, map ) );</span>
[ 1, 1, 2, 2, 4, 4, 3, 3 ]
</pre></div>
<p>This solves the ambiguity: Fusions from each of the two orbits occur, and we can assign them to the two classes of subgroups by the choice of the fusions from <span class="SimpleMath">\(U_4(2)\)</span> to <span class="SimpleMath">\(U_4(3)\)</span>.</p>
<p>The reason for the asymmetry is that the automorphism <span class="SimpleMath">\(2_3\)</span> of <span class="SimpleMath">\(U_4(3)\)</span> does not lift to <span class="SimpleMath">\(3_1.U_4(3)\)</span>. Note that each of the classes <code class="code">9A</code>, <code class="code">9B</code> of <span class="SimpleMath">\(U_4(3)\)</span> has three preimages in <span class="SimpleMath">\(3_1.U_4(3)\)</span>, whereas each of the classes <code class="code">9C</code>, <code class="code">9D</code> has only one preimage.</p>
<p>In fact the two classes of <span class="SimpleMath">\(3 \times U_4(2)\)</span> type subgroups of <span class="SimpleMath">\(3_1.U_4(3)\)</span> behave differently. For example, inducing the irreducible characters of a <span class="SimpleMath">\(3 \times U_4(2)\)</span> type subgroup in the first class of maximal subgroups of <span class="SimpleMath">\(3_1.U_4(3)\)</span> yields no irreducible character, whereas the two irreducible characters of degree <span class="SimpleMath">\(630\)</span> are obtained by inducing the irreducible characters of a subgroup in the second class.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">rep:= RepresentativesFusions( 3u42, 3u42fus3u43, 3u43 );</span>
[ [ 1, 4, 4, 7, 7, 10, 13, 15, 18, 21, 24, 24, 27, 27, 30, 27, 48,
49, 50, 50, 2, 5, 5, 8, 8, 11, 13, 16, 19, 22, 25, 25, 28, 28,
31, 28, 48, 49, 51, 51, 3, 6, 6, 9, 9, 12, 13, 17, 20, 23, 26,
26, 29, 29, 32, 29, 48, 49, 52, 52 ],
[ 1, 4, 4, 8, 9, 13, 10, 15, 18, 21, 25, 26, 31, 32, 27, 30, 46,
44, 51, 52, 2, 5, 5, 9, 7, 13, 11, 16, 19, 22, 26, 24, 32, 30,
28, 31, 47, 42, 52, 50, 3, 6, 6, 7, 8, 13, 12, 17, 20, 23, 24,
25, 30, 31, 29, 32, 45, 43, 50, 51 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">irr:= Irr( 3u42 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ind:= InducedClassFunctionsByFusionMap( 3u42, 3u43, irr, rep[1] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( ind, Irr( 3u43 ) );</span>
[ Character( CharacterTable( "3_1.U4(3)" ),
[ 630, 630*E(3)^2, 630*E(3), 6, 6*E(3)^2, 6*E(3), 9, 9*E(3)^2,
9*E(3), -9, -9*E(3)^2, -9*E(3), 0, 0, 2, 2*E(3)^2, 2*E(3), -2,
-2*E(3)^2, -2*E(3), 0, 0, 0, -3, -3*E(3)^2, -3*E(3), 3,
3*E(3)^2, 3*E(3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, -1, -E(3)^2, -E(3) ] ),
Character( CharacterTable( "3_1.U4(3)" ),
[ 630, 630*E(3), 630*E(3)^2, 6, 6*E(3), 6*E(3)^2, 9, 9*E(3),
9*E(3)^2, -9, -9*E(3), -9*E(3)^2, 0, 0, 2, 2*E(3), 2*E(3)^2,
-2, -2*E(3), -2*E(3)^2, 0, 0, 0, -3, -3*E(3), -3*E(3)^2, 3,
3*E(3), 3*E(3)^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, -1, -E(3), -E(3)^2 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ind:= InducedClassFunctionsByFusionMap( 3u42, 3u43, irr, rep[2] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( ind, Irr( 3u43 ) );</span>
[ ]
</pre></div>
<p>For <span class="SimpleMath">\(6_1.U_4(3)\)</span> and <span class="SimpleMath">\(12_1.U_4(3)\)</span>, one gets the same phenomenon: We have two orbits of class fusions, one corresponding to each of the two classes of subgroups of the type <span class="SimpleMath">\(3 \times 4 Y 2.U_4(2)\)</span>. We get <span class="SimpleMath">\(10\)</span> irreducible induced characters from a subgroup in the second class (four faithful ones, four with kernel of order two, and the two abovementioned degree <span class="SimpleMath">\(630\)</span> characters with kernel of order four) and no irreducible character from a subgroup in the first class.</p>
<p><a id="X7A94F78C792122D5" name="X7A94F78C792122D5"></a></p>
<h5>9.7-12 <span class="Heading"><span class="SimpleMath">\(2.3^4.2^3.S_4 \rightarrow 2.A12\)</span> (September 2011)</span></h5>
<p>The double cover <span class="SimpleMath">\(G\)</span> of the alternating group <span class="SimpleMath">\(A_{12}\)</span> contains a maximal subgroup <span class="SimpleMath">\(M\)</span> of the type <span class="SimpleMath">\(2.3^4.2^3.S_4\)</span> whose class fusion is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">2a12:= CharacterTable( "2.A12" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mtbl:= CharacterTable( "2.3^4.2^3.S4" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mtblfus2a12:= PossibleClassFusions( mtbl, 2a12 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( mtblfus2a12 );</span>
32
<span class="GAPprompt">gap></span> <span class="GAPinput">repres:= RepresentativesFusions( mtbl, mtblfus2a12, 2a12 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( repres );</span>
2
</pre></div>
<p>We decide the question which of the essentially different two possible class fusion is the right one, by explicitly constructing <span class="SimpleMath">\(M\)</span> as a subgroup of <span class="SimpleMath">\(G\)</span>.</p>
<p>For that, let <span class="SimpleMath">\(\pi\)</span> denote the natural epimorphism from <span class="SimpleMath">\(G\)</span> to <span class="SimpleMath">\(A_{12}\)</span>, and note that <span class="SimpleMath">\(\pi(M)\)</span> can be described as the intersection of a <span class="SimpleMath">\(S_3 \wp S_4\)</span> type subgroup of <span class="SimpleMath">\(S_{12}\)</span> with <span class="SimpleMath">\(A_{12}\)</span>. Further note that the generators for <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(A_{12}\)</span> provided by <a href="chapBib_mj.html#biBAGRv3">[WWT+]</a> are compatible in the sense that <span class="SimpleMath">\(\pi\)</span> can be defined by mapping the generators of <span class="SimpleMath">\(G\)</span> to those of <span class="SimpleMath">\(A_{12}\)</span>.</p>
<p>We need <span class="SimpleMath">\(\pi\)</span> only for computing one preimage of each given element. Therefore, we represent <span class="SimpleMath">\(\pi\)</span> implicitly by two epimorphisms from a free group to <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(A_{12}\)</span>, respectively, in order to avoid that <strong class="pkg">GAP</strong>precomputes a lot of unnecessary information for <span class="SimpleMath">\(G\)</span>. This way, computing a preimage of an element of <span class="SimpleMath">\(A_{12}\)</span> under <span class="SimpleMath">\(\pi\)</span> is cheap. However, computing the preimage of a subgroup of <span class="SimpleMath">\(A_{12}\)</span> would be very expensive. So we construct the subgroup of <span class="SimpleMath">\(G\)</span> that is generated by preimages of a set of generators of <span class="SimpleMath">\(\pi(M)\)</span>; later we see that this subgroup is in fact equal to <span class="SimpleMath">\(M\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= AtlasGroup( "A12" );</span>
Group([ (1,2,3), (2,3,4,5,6,7,8,9,10,11,12) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">2g:= AtlasGroup( "2.A12" );</span>
<matrix group of size 479001600 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= FreeGroup( 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pi1:= GroupHomomorphismByImagesNC( f, 2g, GeneratorsOfGroup( f ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GeneratorsOfGroup( 2g ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pi2:= GroupHomomorphismByImagesNC( f, g, GeneratorsOfGroup( f ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GeneratorsOfGroup( g ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">w:= WreathProduct( SymmetricGroup( 3 ), SymmetricGroup(4) );</span>
<permutation group of size 31104 with 10 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints( w );</span>
12
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= Intersection( w, g ); Size( s );</span>
<permutation group with 8 generators>
15552
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= SubgroupNC( 2g, List( SmallGeneratingSet( s ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x -> ImagesRepresentative( pi1,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PreImagesRepresentative( pi2, x ) ) ) );;</span>
</pre></div>
<p>Now we compute the character table of <span class="SimpleMath">\(M\)</span>, using a faithful permutation representation of <span class="SimpleMath">\(M\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">iso:= IsomorphismPermGroup( m );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( Image( iso ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( t );</span>
31104
<span class="GAPprompt">gap></span> <span class="GAPinput">trans:= TransformingPermutationsCharacterTables( mtbl, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRecord( trans );</span>
true
</pre></div>
<p>Now let us see where the two fusion candidates differ.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">para:= Parametrized( repres );</span>
[ 1, 2, 6, 10, 8, 12, 7, 11, 9, 13, 5, 5, 17, 17, 17, 17, 3, 4, 24,
22, 27, 25, 12, 10, 13, 11, 28, 29, 35, 37, 39, 36, 38, 40, 5, 23,
28, 29, 26, 14, 14, 16, 16, 33, 34, [ 33, 34 ], [ 33, 34 ], 49, 49,
48, 48 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PositionsProperty( para, IsList );</span>
[ 46, 47 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( repres, map -> map{ [ 44 .. 47 ] } );</span>
[ [ 33, 34, 33, 34 ], [ 33, 34, 34, 33 ] ]
</pre></div>
<p>So the question is whether the elements in class 44 are conjugate in <span class="SimpleMath">\(G\)</span> to the elements in class 46 or in class 47. In order to answer this question, we compute preimages of the relevant class representatives in the matrix group <span class="SimpleMath">\(M\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">positions:= OnTuples( [ 44 .. 47 ], trans.columns );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">classreps:= List( ConjugacyClasses( t ){ positions },</span>
<span class="GAPprompt">></span> <span class="GAPinput"> c -> PreImagesRepresentative( iso, Representative( c ) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">traces:= List( classreps, TraceMat );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( traces, x -> Position( traces, x ) );</span>
[ 1, 2, 2, 1 ]
</pre></div>
<p>We are lucky, already the traces of the elements allow us to decide which pairs of elements are <span class="SimpleMath">\(G\)</span>-conjugate; there is no need for an explicit (and expensive) conjugacy test in the matrix group <span class="SimpleMath">\(G\)</span>.</p>
<p>Finally, we check whether the stored fusion is correct.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">good:= First( repres,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> map -> map{ [ 44 .. 47 ] } = [ 33, 34, 34, 33 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( mtbl, 2a12 ) = good;</span>
true
</pre></div>
<p><a id="X7E2AF30C7E8F89F9" name="X7E2AF30C7E8F89F9"></a></p>
<h5>9.7-13 <span class="Heading"><span class="SimpleMath">\(127:7 \rightarrow L_7(2)\)</span> (January 2012)</span></h5>
<p>The simple group <span class="SimpleMath">\(G = L_7(2)\)</span> contains a maximal subgroup <span class="SimpleMath">\(M\)</span> of the type <span class="SimpleMath">\(127:7\)</span> (the normalizer of an extension field type subgroup GL<span class="SimpleMath">\((1,2^7)\)</span>) whose class fusion is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "L7(2)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "127:7" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( s, fus, t );</span>
[ [ 1, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111,
112, 113, 114, 115, 117, 116, 76, 76, 77, 76, 77, 77 ],
[ 1, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111,
112, 113, 114, 115, 117, 116, 83, 83, 83, 83, 83, 83 ] ]
</pre></div>
<p>The two fusion candidates differ only for elements of order <span class="SimpleMath">\(7\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">diff:= Filtered( [ 1 .. Length( repr[1] ) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> i -> repr[1][i] <> repr[2][i] );</span>
[ 20, 21, 22, 23, 24, 25 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( s ){ diff };</span>
[ 7, 7, 7, 7, 7, 7 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( repr, l -> l{ diff } );</span>
[ [ 76, 76, 77, 76, 77, 77 ], [ 83, 83, 83, 83, 83, 83 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SizesCentralizers( t ){ [ 76, 77, 83 ] };</span>
[ 3528, 3528, 49 ]
</pre></div>
<p>We can decide which candidate is the correct one if we know the centralizer order in G of the elements of order <span class="SimpleMath">\(7\)</span> in <span class="SimpleMath">\(M\)</span>. So we compute this centralizer order.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Image( IsomorphismPermGroup( GL(7,2) ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repeat x:= Random( g ); until Order(x) = 127;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:= Normalizer( g, SubgroupNC( g, [ x ] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( n ) / 127;</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">repeat x:= Random( n ); until Order( x ) = 7;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c:= Centralizer( g, x );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( c );</span>
49
</pre></div>
<p>We see that the second candidate is the fusion from <span class="SimpleMath">\(M\)</span> into <span class="SimpleMath">\(G\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( s, t ) = repr[2];</span>
true
</pre></div>
<p><a id="X7E7B2AD67ACD27AE" name="X7E7B2AD67ACD27AE"></a></p>
<h5>9.7-14 <span class="Heading"><span class="SimpleMath">\(L_2(59) \rightarrow M\)</span> (May 2009)</span></h5>
<p>The sporadic simple Monster group <span class="SimpleMath">\(M\)</span> contains a maximal subgroup <span class="SimpleMath">\(G\)</span> of the type <span class="SimpleMath">\(L_2(59)\)</span>, see <a href="chapBib_mj.html#biBHW04">[HW04]</a>. The class fusion of <span class="SimpleMath">\(G\)</span> into <span class="SimpleMath">\(M\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "M" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "L2(59)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( s, fus, t );</span>
[ [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
97, 98, 52, 32, 52, 14, 12, 98, 52, 32, 5, 98, 12, 98, 52, 3 ],
[ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
97, 100, 50, 30, 50, 15, 11, 100, 50, 30, 4, 100, 11, 100, 50,
3 ],
[ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
97, 101, 51, 30, 51, 14, 11, 101, 51, 30, 5, 101, 11, 101, 51,
3 ],
[ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
97, 102, 53, 32, 53, 18, 12, 102, 53, 32, 6, 102, 12, 102, 53,
3 ],
[ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52,
3 ] ]
</pre></div>
<p>The candidates differ on the classes of element order <span class="SimpleMath">\(30\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ord:= OrdersClassRepresentatives( s );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ord30:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = 30 );</span>
[ 18, 24, 28, 30 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( repr, x -> x{ ord30 } );</span>
[ [ 98, 98, 98, 98 ], [ 100, 100, 100, 100 ], [ 101, 101, 101, 101 ],
[ 102, 102, 102, 102 ], [ 104, 104, 104, 104 ] ]
</pre></div>
<p>According to <a href="chapBib_mj.html#biBHW04">[HW04]</a>, <span class="SimpleMath">\(G\)</span> contains elements in the class <code class="code">30G</code> of <span class="SimpleMath">\(M\)</span>. This determines the class fusion up to Galois automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos:= Position( ClassNames( t, "Atlas" ), "30G" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">good:= Filtered( fus, map -> pos in map );</span>
[ [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52,
3 ],
[ 1, 153, 152, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52,
3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( s, good, t );</span>
[ [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52,
3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( s, t ) = repr[1];</span>
true
</pre></div>
<p><a id="X8409DA2E83A41ABE" name="X8409DA2E83A41ABE"></a></p>
<h5>9.7-15 <span class="Heading"><span class="SimpleMath">\(L_2(71) \rightarrow M\)</span> (May 2009)</span></h5>
<p>The sporadic simple Monster group <span class="SimpleMath">\(M\)</span> contains a maximal subgroup <span class="SimpleMath">\(G\)</span> of the type <span class="SimpleMath">\(L_2(71)\)</span>, see <a href="chapBib_mj.html#biBHW08">[HW08]</a>. The class fusion of <span class="SimpleMath">\(G\)</span> into <span class="SimpleMath">\(M\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "M" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "L2(71)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( s, fus, t );</span>
[ [ 1, 169, 170, 112, 112, 112, 112, 19, 112, 11, 112, 112, 19, 112,
112, 112, 11, 19, 112, 112, 114, 60, 36, 27, 114, 17, 114, 27,
7, 60, 114, 5, 114, 60, 36, 27, 114, 3 ],
[ 1, 169, 170, 112, 112, 112, 112, 19, 112, 11, 112, 112, 19, 112,
112, 112, 11, 19, 112, 112, 115, 61, 36, 28, 115, 17, 115, 28,
7, 61, 115, 5, 115, 61, 36, 28, 115, 3 ],
[ 1, 169, 170, 112, 112, 112, 112, 19, 112, 11, 112, 112, 19, 112,
112, 112, 11, 19, 112, 112, 117, 61, 43, 28, 117, 17, 117, 28,
9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ],
[ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113,
113, 113, 12, 20, 113, 113, 114, 60, 36, 27, 114, 17, 114, 27,
7, 60, 114, 5, 114, 60, 36, 27, 114, 3 ],
[ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113,
113, 113, 12, 20, 113, 113, 115, 61, 36, 28, 115, 17, 115, 28,
7, 61, 115, 5, 115, 61, 36, 28, 115, 3 ],
[ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113,
113, 113, 12, 20, 113, 113, 117, 61, 43, 28, 117, 17, 117, 28,
9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ] ]
</pre></div>
<p>The candidates differ on the classes of the element orders <span class="SimpleMath">\(7\)</span> and <span class="SimpleMath">\(36\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ord:= OrdersClassRepresentatives( s );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ord36:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = 36 );</span>
[ 21, 25, 27, 31, 33, 37 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( repr, x -> x{ ord36 } );</span>
[ [ 114, 114, 114, 114, 114, 114 ], [ 115, 115, 115, 115, 115, 115 ],
[ 117, 117, 117, 117, 117, 117 ], [ 114, 114, 114, 114, 114, 114 ],
[ 115, 115, 115, 115, 115, 115 ], [ 117, 117, 117, 117, 117, 117 ] ]
</pre></div>
<p>According to <a href="chapBib_mj.html#biBNW02">[NW02, Table 3]</a>, <span class="SimpleMath">\(G\)</span> contains elements in the classes <code class="code">7B</code> and <code class="code">36D</code> of <span class="SimpleMath">\(M\)</span>. This determines the class fusion up to Galois automorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos1:= Position( ClassNames( t, "Atlas" ), "7B" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos2:= Position( ClassNames( t, "Atlas" ), "36D" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos:= [ pos1, pos2 ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">good:= Filtered( fus, map -> IsSubset( map, pos ) );</span>
[ [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113,
113, 113, 12, 20, 113, 113, 117, 61, 43, 28, 117, 17, 117, 28,
9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ],
[ 1, 170, 169, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113,
113, 113, 12, 20, 113, 113, 117, 61, 43, 28, 117, 17, 117, 28,
9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( s, good, t );</span>
[ [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113,
113, 113, 12, 20, 113, 113, 117, 61, 43, 28, 117, 17, 117, 28,
9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( s, t ) = repr[1];</span>
true
</pre></div>
<p><a id="X78B3B1BE7A2CA4D1" name="X78B3B1BE7A2CA4D1"></a></p>
<h5>9.7-16 <span class="Heading"><span class="SimpleMath">\(L_2(41) \rightarrow M\)</span> (April 2012)</span></h5>
<p>The sporadic simple Monster group <span class="SimpleMath">\(M\)</span> contains a maximal subgroup <span class="SimpleMath">\(G\)</span> of the type <span class="SimpleMath">\(L_2(41)\)</span>, see <a href="chapBib_mj.html#biBNW12">[NW13]</a>. The class fusion of <span class="SimpleMath">\(G\)</span> into <span class="SimpleMath">\(M\)</span> is ambiguous.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "M" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= CharacterTable( "L2(41)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= PossibleClassFusions( s, t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= RepresentativesFusions( s, fus, t );</span>
[ [ 1, 127, 127, 64, 30, 64, 11, 7, 30, 64, 11, 64, 3, 70, 70, 19,
70, 70, 19, 4, 70, 19, 70 ],
[ 1, 127, 127, 64, 30, 64, 11, 7, 30, 64, 11, 64, 3, 72, 72, 19,
72, 72, 19, 6, 72, 19, 72 ],
[ 1, 127, 127, 64, 30, 64, 11, 7, 30, 64, 11, 64, 3, 73, 73, 20,
73, 73, 20, 5, 73, 20, 73 ],
[ 1, 127, 127, 66, 33, 66, 12, 7, 33, 66, 12, 66, 3, 72, 72, 19,
72, 72, 19, 6, 72, 19, 72 ],
[ 1, 127, 127, 66, 33, 66, 12, 7, 33, 66, 12, 66, 3, 73, 73, 20,
73, 73, 20, 5, 73, 20, 73 ],
[ 1, 127, 127, 67, 30, 67, 11, 10, 30, 67, 11, 67, 3, 72, 72, 19,
72, 72, 19, 6, 72, 19, 72 ],
[ 1, 127, 127, 67, 30, 67, 11, 10, 30, 67, 11, 67, 3, 73, 73, 20,
73, 73, 20, 5, 73, 20, 73 ],
[ 1, 127, 127, 68, 32, 68, 12, 10, 32, 68, 12, 68, 3, 72, 72, 19,
72, 72, 19, 6, 72, 19, 72 ],
[ 1, 127, 127, 68, 32, 68, 12, 10, 32, 68, 12, 68, 3, 73, 73, 20,
73, 73, 20, 5, 73, 20, 73 ],
[ 1, 127, 127, 69, 33, 69, 12, 9, 33, 69, 12, 69, 3, 72, 72, 19,
72, 72, 19, 6, 72, 19, 72 ],
[ 1, 127, 127, 69, 33, 69, 12, 9, 33, 69, 12, 69, 3, 73, 73, 20,
73, 73, 20, 5, 73, 20, 73 ] ]
</pre></div>
<p>The candidates differ on the classes of the element orders <span class="SimpleMath">\(3\)</span>–<span class="SimpleMath">\(8\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ambig:= Parametrized( repr );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ambigpos:= PositionsProperty( ambig, IsList );</span>
[ 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( OrdersClassRepresentatives( t ){ ambigpos } );</span>
[ 3, 4, 5, 6, 7, 8 ]
</pre></div>
<p>According to <a href="chapBib_mj.html#biBNW12">[NW13, Theorem 3]</a>, <span class="SimpleMath">\(G\)</span> contains elements in the classes <code class="code">3B</code> and <code class="code">4C</code> of <span class="SimpleMath">\(M\)</span>. This determines the class fusion uniquely.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos1:= Position( ClassNames( t, "Atlas" ), "3B" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos2:= Position( ClassNames( t, "Atlas" ), "4C" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos:= [ pos1, pos2 ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">good:= Filtered( fus, map -> IsSubset( map, pos ) );</span>
[ [ 1, 127, 127, 69, 33, 69, 12, 9, 33, 69, 12, 69, 3, 73, 73, 20,
73, 73, 20, 5, 73, 20, 73 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GetFusionMap( s, t ) = good[1];</span>
true
</pre></div>
<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap8_mj.html">[Previous Chapter]</a> <a href="chap10_mj.html">[Next Chapter]</a> </div>
<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chap3_mj.html">3</a> <a href="chap4_mj.html">4</a> <a href="chap5_mj.html">5</a> <a href="chap6_mj.html">6</a> <a href="chap7_mj.html">7</a> <a href="chap8_mj.html">8</a> <a href="chap9_mj.html">9</a> <a href="chap10_mj.html">10</a> <a href="chap11_mj.html">11</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>
<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
|
