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|
#############################################################################
##
#W tmstdrd.tom GAP table of marks library Thomas Merkwitz
#W Thomas Breuer
##
#H @(#)$Id: tmstdrd.tom,v 4.2 1999/11/19 17:23:11 gap Exp $
##
#Y Copyright (C) 1999, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
## This file contains the descriptions of standard generators of tables of
## marks in the {\GAP} library.
##
## Each entry in the list `LIBTOMKNOWN.STDGEN' is a list of length $4$.
## \beginlist
## \item{-}
## At the first position the *identifier* value of the table of marks in
## the {\GAP} library is stored.
## \item{-}
## The value at the second position is one of the strings `\"Y\"',
## `\"N\"', the former meaning that the standard generators coincide
## with those defined in Rob Wilson's {\ATLAS} of Group Representations
## (see~`http://www.mat.bham.ac.uk/atlas').
## More precisely, this was true at least in March~1998.
## \item{-}
## The value at position three is a string of names of the standard
## generators, for example `\"a, b\"' means that a pair of generators
## with names `a' and `b' is dealt with.
## \item{-}
## Finally, the value at position four is a string that describes how
## standard generators for the group $G$ in question are defined,
## in terms of their names.
## The following terms occur, separated by commas.
## $|g|=q$ means that the element $g$ has order $q$,
## $|C(g)|=n$ means that the centralizer of $g$ in $G$ has order $n$,
## $|x|=q, x^m=g$ means that the $m$-th power of of any element of order
## $q$ in $G$ is conjugate to $g$.
## Additionally, if $G$ is a cyclic downward extension of another group
## $H$ then the definition of standard generators of $G$ may include
## that they are preimages of standard generators of $H$;
## This is then mentioned in the string at position three.
## \endlist
##
Revision.tmstdrd_tom :=
"@(#)$Id: tmstdrd.tom,v 4.2 1999/11/19 17:23:11 gap Exp $";
LIBTOMKNOWN.STDGEN := [
["A5", "Y", "a, b",
"|a|=2, |b|=3, |ab|=5" ],
["S5", "Y", "c, d",
"|z|=6, z^3=c, |d|=4, |cd|=5" ],
["2.A5", "Y", "preimages e, f of b, a",
"|e|=3, |ef|=5" ],
["2.S5", "N", "g, h",
"|g|=2, |C(g)|=12, |h|=5, |gh|=8" ],
["2.S5'", "N", "i, j",
"|i|=10, |j|=12, |ij|=8, |ijj|=6" ],
["L2(7)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=7" ],
["L2(7).2", "N", "c, d",
"|c|=2, |C(c)|=12, |d|=3, |cd|=8, |cdcdd|=7" ],
["A6", "Y", "a, b",
"|a|=2, |b|=4, |ab|=5" ],
["S6", "Y", "c, d",
"|z|=6, z^3=c, |d|=5, |cd|=6, |cdd|=6" ],
["A6.2_2", "N", "e, f",
"|z|=10, z^5=e, |f|=3, |ef|=10" ],
["A6.2_3", "N", "g, h",
"|g|=3, |h|=8, |gh|=4, |ghh|=4" ],
["A6.2^2", "N", "i, j",
"|z|=6, z^3=i, |j|=10, |ij|=4" ],
["2.A6", "N", "preimages k, l of a, b",
"|kl|=10" ],
["2.S6", "N", "preimages m, n of c, d",
"|m|=2, |n|=5" ],
["L2(8)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=7" ],
["L2(8).3", "Y", "c, d",
"|c|=2, |d|=3, |cd|=9, |(cd)^3(dc)^2ddcdd|=7" ],
["L2(11)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=11" ],
["L2(11).2", "N", "c, d",
"|z|=10, z^5=c, |d|=3, |cd|=10, |cdcdd|=11" ],
["L2(13)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=13" ],
["L2(13).2", "N", "c, d",
"|z|=14, z^7=c, |d|=3, |cd|=14, |cdcdd|=6" ],
["L2(17)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=17" ],
["A7", "Y", "a, b",
"|z|=6, z^2=a, |b|=5, |ab|=7" ],
["S7", "N", "c, d",
"|z|=10, z^5=c, |d|=7, |cd|=6" ],
["2.A7", "Y", "preimages e, f of a, b",
"|e|=3, |f|=5, |ef|=7" ],
["2.S7", "N", "preimages g, h of c, d",
"|h|=7" ],
["L2(19)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=19" ],
["L2(16)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=15" ],
["L2(16).2", "N", "c, d",
"|c|=3, |d|=4, |cd|=10, |cd^3|=6" ],
["L2(16).4", "N", "e, f",
"|e|=3, |f|=8, |ef|=12, |effeefef|=2" ],
["L3(3)", "Y", "a, b",
"|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4" ],
["L3(3).2", "Y", "c, d",
"|c|=2, |C(c)|=48, |z|=12, z^3=d, |cd|=13, |(cd)^3d|=12" ],
["U3(3)", "Y", "a, b",
"|a|=2, |b|=6, |ab|=7" ],
["U3(3).2", "Y", "c, d",
"|y|=6, y^3=c, |z|=12, z^3=d, |cd|=7" ],
["L2(23)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=23" ],
["L2(25)", "N", "a, b",
"|a|=2, |b|=3, |ab|=13, |ababb|=13" ],
["L2(25).2_2", "N", "c, d",
"|c|=4, |C(c)|=8, |d|=12, |cd|=4, |cdcdd|=3,\
|cdc^3d^-1|=10" ],
["M11", "Y", "a, b",
"|a|=2, |b|=4, |ab|=11, |(ab)^4(ba)^2bbabb|=4" ],
["L2(27)", "N", "a, b",
"|a|=2, |b|=3, |ab|=14" ],
["L2(29)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=29" ],
["L2(31)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=31" ],
["A8", "Y", "a, b",
"|z|=15, z^5=a, |b|=7, |ab|=6, |abb|=15" ],
["S8", "Y", "c, d",
"|z|=10, z^5=c, |d|=7, |cd|=8" ],
["2.A8", "N", "e, f",
"|z|=15, z^5=e, |f|=7, |ef|=6, |ef^3|=30" ],
["2.S8", "N", "g, h",
"|g|=2, |C(g)|=96, |h|=15, |hghhgh^-1g|=20" ],
["L3(4)", "Y", "a, b",
"|a|=2, |b|=4, |ab|=7, |abb|=5" ],
["L3(4).2_1", "Y", "c, d",
"|z|=6, z^3=c, |d|=4, |C(d)|=16, |cd|=7" ],
["L3(4).2_2", "Y", "e, f",
"|z|=14, z^7=e, |f|=5, |ef|=14, |eff|=8" ],
["L3(4).2_3", "N", "g, h",
"|z|=8, z^4=g, |h|=10, |gh|=6, |ghh|=3" ],
["L3(4).3", "N", "i, j",
"|i|=2, |j|=21, |ij|=21, |ij^3|=7" ],
["L3(4).2^2", "N", "k, l",
"|z|=10, z^5=k, |l|=6, |C(l)|=36, |kl|=8, |kllklkll|=14" ],
["L3(4).6", "Y", "m, n",
"|m|=2, |C(m)|=432, |n|=6, |C(n)|=36, |mn|=21, |mnmnn|=4" ],
["L3(4).3.2_2", "N", "p, q",
"|z|=14, z^7=p, |q|=21, |pq|=6, |pqqpqpq^3|=4" ],
["L3(4).3.2_3", "N", "r, s",
"|z|=10, z^5=r, |s|=21, |rs|=8, |rss|=8,\
|srssrs^-1r|=8" ],
["L3(4).D12", "N", "t, u",
"|z|=14, z^7=t, |u|=12, |tu|=8, |tuu|=14" ],
["3.L3(4)", "Y", "preimages A, B of a, b",
"|A|=2, |B|=4" ],
["3.L3(4).2_3", "N", "preimages C, D of g, h",
"|C|=2" ],
["2^2.L3(4)", "N", "E, F",
"|z|=12, z^6=E, |F|=10, |EF|=6, |EFF|=3" ],
["2^2.L3(4).3", "N", "G, H",
"|z|=15, z^5=G, |H|=7, |GH|=15, |GGHGHH|=6" ],
["2^2.L3(4).2_2", "N", "I, J",
"|I|=3, |J|=28, |IJ|=8, |IJJ|=6, |IJ^3|=4" ],
["2^2.L3(4).3.2_2", "N", "K, L",
"|z|=15, z^5=K, |L|=28, |KL|=8, |KL^3|=28, |KL^4|=6,\
|KLKKLL|=4" ],
["L2(37)", "N", "a, b",
"|a|=2, |b|=3, |ab|=31" ],
["U4(2)", "N", "a, b",
"|z|=12, z^6=a, |b|=9, |ab|=5" ],
["U4(2).2", "N", "c, d",
"|z|=10, z^5=c, |d|=9, |cd|=10" ],
["2.U4(2)", "N", "preimages e, f of a, b",
"|f|=18, |ef|=5" ],
["2.U4(2).2", "N", "preimages g, h of c, d",
"|h|=9" ],
["Sz(8)", "Y", "a, b",
"|a|=2, |b|=4, |abb|=7, |(ab)^2bbabb|=7" ],
["L2(32)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=31, |ababb|=11" ],
["L2(41)", "N", "a, b",
"|a|=2, |b|=3, |ab|=41" ],
["L2(43)", "N", "a, b",
"|a|=2, |b|=3, |ab|=43" ],
["L2(47)", "N", "a, b",
"|a|=2, |b|=3, |ab|=47" ],
["L2(49)", "N", "a, b",
"|a|=2, |b|=3, |ab|=47, |ababb|=8" ],
["U3(4)", "Y", "a, b",
"|a|=2, |b|=3, |ab|=13" ],
["U3(4).2", "N", "c, d",
"|z|=6, z^3=c, |d|=5, |C(d)|=50, |cd|=6, |cdd|=6" ],
["L2(53)", "N", "a, b",
"|a|=2, |b|=3, |ab|=53" ],
["M12", "Y", "a, b",
"|z|=4, z^2=a, |b|=3, |C(b)|=36, |ab|=11" ],
["M12.2", "Y", "c, d",
"|c|=2, |C(c)|=240, |d|=3, |C(d)|=108, |cd|=12" ],
["L2(59)", "N", "a, b",
"|a|=2, |b|=3, |ab|=59" ],
["L2(61)", "N", "a, b",
"|a|=2, |b|=3, |ab|=61" ],
["U3(5)", "Y", "a, b",
"|a|=3, |z|=10, z^2=b, |ab|=7" ],
["U3(5).2", "Y", "c, d",
"|c|=2, |C(c)|=240, |z|=8, z^2=d, |cd|=10,\
|(cd)^2ddcdd|=2" ],
#["U3(5).3", "N", "e, f",
# "|e|=2, |f|=6, |C(f)|=24, |C(f^2)|=480, |ef|=21,\
# |ef^3|=3" ],
["U3(5).S3", "N", "g, h",
"|z|=24, z^8=g, |h|=12, |C(h)|=12, |gh|=6, |ghh|=30" ],
["L2(67)", "N", "a, b",
"|a|=2, |b|=3, |ab|=67" ],
["J1", "Y", "a, b",
"|a|=2, |b|=3, |ab|=7, |ababb|=19" ],
["L2(71)", "N", "a, b",
"|a|=2, |b|=3, |ab|=71" ],
["A9", "Y", "a, b",
"|z|=15, z^5=a, |b|=7, |ab|=9" ],
["S9", "N", "c, d",
"|z|=14, z^7=c, |d|=8, |cd|=9" ],
["L2(73)", "N", "a, b",
"|a|=2, |b|=3, |ab|=73" ],
["L2(79)", "N", "a, b",
"|a|=2, |b|=3, |ab|=79" ],
["L2(64)", "N", "a, b",
"|a|=2, |b|=3, |ab|=63, |ababb|=13" ],
["L2(81)", "N", "a, b",
"|a|=2, |b|=3, |ab|=10, |ababb|=10" ],
["L2(83)", "N", "a, b",
"|a|=2, |b|=3, |ab|=83" ],
["L2(89)", "N", "a, b",
"|a|=2, |b|=3, |ab|=89" ],
["L3(5)", "Y", "a, b",
"|z|=10, z^2=a, |b|=3, |ab|=20" ],
["M22", "Y", "a, b",
"|a|=2, |z|=8, z^2=b, |ab|=11, |ababb|=11" ],
["M22.2", "Y", "c, d",
"|z|=14, z^7=c, |d|=4, |C(d)|=96, |cd|=11" ],
["L2(97)", "N", "a, b",
"|a|=2, |b|=3, |ab|=97" ],
["L2(101)", "N", "a, b",
"|a|=2, |b|=3, |ab|=101" ],
["L2(103)", "N", "a, b",
"|a|=2, |b|=3, |ab|=103" ],
["J2", "Y", "a, b",
"|z|=10, z^5=a, |b|=3, |C(b)|=36, |ab|=7" ],
["J2.2", "Y", "c, d",
"|y|=14, y^2=c, |z|=15, z^3=d, |cd|=14" ],
["L2(107)", "N", "a, b",
"|a|=2, |b|=3, |ab|=107" ],
["L2(109)", "N", "a, b",
"|a|=2, |b|=3, |ab|=109" ],
["L2(113)", "N", "a, b",
"|a|=2, |b|=3, |ab|=113" ],
["L2(121)", "N", "a, b",
"|a|=2, |b|=3, |ab|=61, |ababb|=12" ],
["L2(125)", "N", "a, b",
"|a|=2, |b|=3, |ab|=63, |ababb|=7" ],
["S4(4)", "Y", "a, b",
"|a|=2, |C(a)|=3840, |b|=5, |C(b)|=15, |ab|=17,\
|ababb|=15" ],
["S4(4).2", "Y", "c, d",
"|c|=2, |C(c)|=1440, |d|=4, |C(d)|=96, |cd|=17, |cdd|=4" ],
["S6(2)", "Y", "a, b",
"|z|=10, z^5=a, |b|=7, |ab|=9" ],
["2.S6(2)", "Y", "preimages c, d of a, b",
"|d|=7, |cd|=9" ],
["A10", "Y", "a, b",
"|z|=15, z^5=a, |b|=9, |ab|=8, |abb|=12" ],
["S10", "Y", "c, d",
"|z|=14, z^7=c, |d|=9, |cd|=10" ],
["L3(7)", "N", "a, b",
"|a|=2, |b|=7, |C(b)|=49, |ab|=4, |abb|=8" ],
["U4(3)", "Y", "a, b",
"|a|=2, |b|=6, |C(b)|=72, |ab|=7, |(ab)^3(ba)^2bb|=5" ],
["U4(3).2_1", "N", "c, d",
"|z|=8, z^4=c, |d|=10, |cd|=14, |cd^3|=10, |cd^4|=7,\
|cddcd^3cd|=5" ],
["U4(3).2_3", "N", "e, f",
"|z|=12, z^6=e, |f|=6, |C(f)|=18, |ef|=24,\
|efef^3efeffef|=6" ],
#["U4(3).(2^2)_133", "N", "a, b",
# "|a|=4, |C(a)|=2304, |ab|=6, |abb|=14,\
# |abbababbab^3ab|=10" ],
["G2(3)", "Y", "a, b",
"|a|=2, |z|=9, z^3=b, |ab|=13" ],
["G2(3).2", "N", "c, d",
"|z|=14, z^2=c, |d|=13, |cd|=18, |cdd|=12,\
|cd^3|=18, |cdcddcddcd^3|=6" ],
["S4(5)", "Y", "a, b",
"|a|=2, |C(a)|=480, |b|=3, |C(b)|=360, |ab|=13,\
|ababb|=12" ],
["U3(8)", "N", "a, b",
"|a|=2, |b|=19, |ab|=3" ],
["U3(7)", "N", "a, b",
"|a|=2, |b|=3, |ab|=56, |ababb|=8" ],
["L4(3)", "Y", "a, b",
"|y|=10, y^5=a, |z|=8, z^2=b, |ab|=13, |abb|=8" ],
["L5(2)", "Y", "a, b",
"|z|=14, z^7=a, |b|=5, |ab|=21" ],
["M23", "Y", "a, b",
"|a|=2, |b|=4, |ab|=23, |(ab)^4(ba)^2bbabb|=8" ],
["U5(2)", "Y", "a, b",
"|z|=8, z^4=a, |b|=5, |ab|=11" ],
["L3(8)", "N", "a, b",
"|a|=3, |z|=14, z^2=b, |ab|=73, |ab^3|=7, |aab|=73" ],
["2F4(2)'", "Y", "a, b",
"|z|=10, z^5=a, |b|=3, |ab|=13" ],
["A11", "Y", "a, b",
"|z|=21, z^7=a, |b|=9, |ab|=11" ],
["S11", "Y", "c, d",
"|z|=18, z^9=c, |d|=10, |C(d)|=10, |cd|=11" ],
["Sz(32)", "Y", "a, b",
"|a|=2, |b|=4, |ab|=5, |ababb|=25,\
|(ab)^3(bab)^2bbabb|=25" ],
["L3(9)", "N", "a, b",
"|a|=2, |b|=5, |ab|=80, |abb|=16" ],
["U3(9)", "N", "a, b",
"|a|=2, |b|=5, |C(b)|=100, |ab|=73, |abb|=15,\
|ab^3ab^3ab|=30" ],
["HS", "Y", "a, b",
"|y|=20, y^10=a, |z|=20, z^4=b, |ab|=11" ],
["J3", "Y", "a, b",
"|a|=2, |z|=6, z^2=b, |ab|=19, |ababb|=19" ],
["J3.2", "Y", "a, b",
"|y|=18, y^9=a, |z|=24, z^8=b, |ab|=24, |ababb|=9" ],
["U3(11)", "N", "a, b",
"|a|=3, |z|=44, z^4=b, |ab|=37, |baa|=20" ],
#["O8+(2)", "Y", "a, b",
# "|a|=2, |C(a)|=3072, |b|=5, |C(b)|=300, |ab|=12,\
# |C(ab)|=24, |abb|=15, |(ab)^4(ba)^2bbabb|=8" ],
["O8-(2)", "Y", "a, b",
"|a|=2, |C(a)|=3072, |z|=9, z^3=b, |ab|=17,\
|ababb|=17, |(ab)^3b|=30" ],
["3D4(2)", "Y", "a, b",
"|z|=8, z^4=a, |b|=9, |ab|=13, |abb|=8" ],
["L3(11)", "N", "a, b",
"|a|=3, |z|=55, z^5=b, |ab|=7" ],
["A12", "Y", "a, b",
"|z|=21, z^7=a, |b|=11, |ab|=10, |abb|=35" ],
["M24", "Y", "a, b",
"|y|=10, y^5=a, |z|=15, z^5=b, |ab|=23,\
|(ab)^3(bab)^2bbabb|=4" ],
["G2(4)", "N", "a, b",
"|z|=8, z^4=a, |b|=5, |ab|=13, |abbababb|=8" ],
["McL", "Y", "a, b",
"|y|=10, y^5=a, |z|=15, z^3=b, |ab|=11, |(ab)^4(ba)^2bbabb|=7" ],
["McL.2", "Y", "c, d",
"|z|=22, z^11=c, |d|=3, |C(d)|=1944, |cd|=22,\
|(cd)^4(dc)^2ddcdd|=24" ],
["HE", "Y", "a, b",
"|z|=10, z^5=a, |b|=7, |C(b)|=1029, |ab|=17" ],
["Co3", "Y", "a, b",
"|y|=9, y^3=a, |z|=20, z^5=b, |ab|=14" ],
];
#############################################################################
##
#E
|
