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#############################################################################
##
#W idgrp3.g GAP group library Hans Ulrich Besche
## Bettina Eick, Eamonn O'Brien
##
## This file contains the identification routines for groups of order
## 2^n * p with 3 <= n <= 8 and p an odd prime
##
Revision.idgrp3_g :=
"@(#)$Id: idgrp3.g,v 1.9 2000/01/23 13:58:54 gap Exp $";
#############################################################################
##
## tell GAP about the component
##
DeclareComponent("id3","2.1");
#############################################################################
##
#F ID_AVAILABLE_FUNCS[ 3 ]
##
ID_AVAILABLE_FUNCS[ 3 ] := function( size )
local p;
p := FactorsInt( size );
if Length( p ) > 9 or p[ Length( p ) - 1 ] <> 2 or size = 512 then
return fail;
fi;
return rec( func := 13,
lib := 3,
n := Length( p ) - 1,
p := p[ Length( p ) ] );
end;
#############################################################################
##
#F ID_GROUP_FUNCS[ 13 ]( G, inforec )
##
## standard lookup in the case 2^n * p
##
ID_GROUP_FUNCS[ 13 ] := function( G, inforec )
local size, sid, num, pos, cid, n, typ, c, sp, i, j, k, tmp, listInt,
rank, id, g, gens, fgens, sgens, inv, coeff, p, root1, root2,
pcgs, pelm, rels, C;
listInt := function( int )
local r, i;
i := 1;
r := [ ];
while int > 0 do
if int mod 2 = 1 then
Add( r, i );
fi;
i := i + 1;
int := QuoInt( int, 2 );
od;
return r;
end;
if IsNilpotent( G ) then
return IdGroup( SylowSubgroup( G, 2 ) )[ 2 ];
fi;
size := Size( G );
if ( not IsNormal( G, SylowSubgroup( G, 2 ) ) ) and
( not IsNormal( G, SylowSubgroup( G, inforec.p ) ) ) then
if size = 768 and not IsBound( ID_GROUP_TREE.next[ 768 ] ) then
ID_GROUP_TREE.next[ 768 ] := rec( fp:= [ 1, 2 ], next:= [ ] );
fi;
return ID_GROUP_FUNCS[ 8 ]( G, inforec, [ size, 2 ] );
fi;
sid := IdGroup( SylowSubgroup( G, 2 ) )[ 2 ];
num := NUMBER_SMALL_GROUPS_FUNCS[ 11 ]( size, SMALL_AVAILABLE( size ) );
if IsNormal( G, SylowSubgroup( G, 2 ) ) then
num := num.pos[ Position( num.types, "p-autos" ) ];
if not IsBound( SMALL_GROUP_LIB[ size ] ) then
SMALL_GROUP_LIB[ size ] := rec();
fi;
if not IsBound( SMALL_GROUP_LIB[ size ].pnil ) then
ReadSmallLib( "sml", 3, size, [ ] );
fi;
pos := Position( SMALL_GROUP_LIB[ size ].pnil.2syl, sid );
if ( not IsBound( SMALL_GROUP_LIB[ size ].pnil.2syl[ pos+1 ] ) )
or ( SMALL_GROUP_LIB[ size ].pnil.2syl[ pos + 1 ] <> sid ) then
return num + pos;
fi;
if size = 768 and not IsBound( ID_GROUP_TREE.next[ 768 ] ) then
ID_GROUP_TREE.next[ 768 ] := rec( fp:= [ 1, 2 ], next:= [ ] );
fi;
return ID_GROUP_FUNCS[ 8 ]( G, inforec, [ size, 1, sid ] );
fi;
# p-sylow-subgroup is normal
cid := IdGroup( Centralizer( SylowSubgroup( G, 2 ),
SylowSubgroup( G, inforec.p ) ) );
n := inforec.n;
typ := Log( 2 ^ n / cid[ 1 ], 2 );
cid := cid[ 2 ];
num := num.pos[ Position( num.types, typ ) ];
if typ = 1 and n in [ 7, 8 ] then
if n = 7 then
c := [ 0, 4514, 13729 ];
else
c := [ 0, 3465, 8461, 14319, 20208, 25076 , 30536,
36802, 48659, 61838, 75468, 84609, 94875, 106350,
115482, 124305, 135946, 146216, 158916, 171446, 185870,
199622, 211598, 223026, 236294, 250682, 263271, 271778,
286530, 307239, 328434, 351278, 379078, 402238, 431410,
461222, 485658, 515470, 544547, 568726, 596286, 625438,
656042, 686670, 716722, 746080, 771660, 799616, 830416,
861172, 891634, 922178, 951604, 976054, 988277, 1006917,
1026007 ];
fi;
sp := QuoInt( sid - 1, 1000 ) * 1000 + 1;
num := num + c[ QuoInt( sid + 999, 1000 ) ];
else
sp := 1;
fi;
if n < 8 then
if not IsBound( SMALL_GROUP_LIB[ n ] ) then
ReadSmallLib( "nor", 3, 2, [ n ] );
fi;
else
if not IsBound( SMALL_GROUP_LIB[ 8 ] ) then
SMALL_GROUP_LIB[ 8 ] := [ ];
fi;
if not IsBound( SMALL_GROUP_LIB[ 8 ][ typ ] ) then
c := [ 1, 2, 3, 3, 3, 3, 3, 3 ];
ReadSmallLib( "nor", 3, 2, [ 8, c[ typ ] ] );
fi;
fi;
if IsRecord( SMALL_GROUP_LIB[ n ][ typ ] ) then
tmp := [ ];
i := 0;
for j in [ 1 .. Length( SMALL_GROUP_LIB[ n ][ typ ].pos ) ] do
if SMALL_GROUP_LIB[ n ][ typ ].pos[ j ] > 0 then
i := i + 1;
tmp[ SMALL_GROUP_LIB[ n ][ typ ].pos[ j ] ] :=
SMALL_GROUP_LIB[ n ][ typ ].val[ i ];
else
for k in [ SMALL_GROUP_LIB[ n ][ typ ].pos[ j - 1] + 1 ..
-SMALL_GROUP_LIB[ n ][ typ ].pos[ j ] ] do
i := i + 1;
tmp[ k ] := SMALL_GROUP_LIB[ n ][ typ ].val[ i ];
od;
fi;
od;
SMALL_GROUP_LIB[ n ][ typ ] := tmp;
fi;
for i in [ sp .. sid ] do
if typ = 1 and ( not IsBound( SMALL_GROUP_LIB[n][typ][ i ] ) ) then
SMALL_GROUP_LIB[n][typ][ i ] := SMALL_GROUP_LIB[n][typ][ i-1 ];
fi;
if IsBound( SMALL_GROUP_LIB[ n ][ typ ][ i ] ) then
if IsInt( SMALL_GROUP_LIB[ n ][ typ ][ i ] ) then
if SMALL_GROUP_LIB[ n ][ typ ][ i ] < 0 then
if IsInt( SMALL_GROUP_LIB[ n ][ typ ][
-SMALL_GROUP_LIB[ n ][ typ ][ i ] ] ) then
SMALL_GROUP_LIB[ n ][ typ ][
-SMALL_GROUP_LIB[ n ][ typ ][ i ] ] :=
listInt( SMALL_GROUP_LIB[ n ][ typ ][
-SMALL_GROUP_LIB[ n ][ typ ][ i ] ]);
fi;
SMALL_GROUP_LIB[ n ][ typ ][ i ] := SMALL_GROUP_LIB[ n ]
[ typ ][ -SMALL_GROUP_LIB[ n ][ typ ][ i ] ];
else
SMALL_GROUP_LIB[ n ][ typ ][ i ] := listInt(
SMALL_GROUP_LIB[ n ][ typ ][ i ] );
fi;
fi;
num := num + Length( SMALL_GROUP_LIB[ n ][ typ ][ i ] );
fi;
od;
# num is the last group of the correct n, typ, sid
if Length( SMALL_GROUP_LIB[ n ][ typ ][ sid ] ) = 1 then
return num;
fi;
# look if there will be a conflict
c := [ ,, [],
[ 1 ],
[ 1, 1 ],
[ 1, 1, 1 ],
[ 6, 1, 1, 1 ],
[ 300, 20, 1, 1, 1 ] ];
if IsBound( c[ n ][ typ ] ) then
c := c[ n ][ typ ];
if c > 1 then
i := ID_GROUP_FUNCS[ 8 ]( G, inforec,
[ n, typ, sid mod c, sid, cid ], true );
else
i := ID_GROUP_FUNCS[ 8 ]( G, inforec,
[ n, typ, sid, cid ], true );
fi;
p := [ 3, 5, 17, 17, 97 ];
p := p[ typ ];
if i <> fail then
if p <> inforec.p then
# construct the group of size 2^n * p (<>inforec.p) with the
# analogue operation of the 2-sylowsubgroup on the p-sylowsub
g := FreeGroup( n + 1 );
gens := GeneratorsOfGroup( g );
pcgs := Pcgs( SylowSubgroup( G, 2 ) );
pelm := First( GeneratorsOfGroup( SylowSubgroup( G,
inforec.p )), x -> Order( x ) = inforec.p );
rels := RelatorsCode( CodePcgs( pcgs ), 2^n, gens{[1 ..n ]});
Add( rels, gens[ n + 1 ] ^ p );
root1 := PrimitiveRootMod( p ) ^ ((p-1)/2^typ) mod p;
root2 := PrimitiveRootMod( inforec.p ) ^
((inforec.p-1)/2^typ) mod inforec.p;
root2 := List( [ 0 .. 2 ^ typ - 1 ], x-> pelm ^ (root2^x) );
for j in [ 1 .. n ] do
Add( rels, gens[ n+1 ] ^ gens[ j ] / gens[ n+1 ] ^
( root1 ^ ( ( Position( root2, pelm^pcgs[j] ) - 1 )
mod p ) ) );
od;
G := PcGroupFpGroup( g / rels );
if Size( G ) < 1000 and n <> 8 then
return i.next[ Position( i.fp, IdGroup( G )[ 2 ] ) ];
fi;
fi;
if c > 1 then
return ID_GROUP_FUNCS[ 8 ]( G, inforec,
[ n, typ, sid mod c, sid, cid ] );
else
return ID_GROUP_FUNCS[ 8 ]( G, inforec, [ n, typ, sid,cid] );
fi;
fi;
fi;
# there is no conflict. Analyse the ids of the centralizer of the
# p-sylowsubgroup in the 2-sylowsubgroup
c := [ ,, [ 1, 4 ],
[ 1, 9, 13 ],
[ 1, 20, 44, 50 ],
[ 1, 54, 191, 259, 266 ],
[ 1, 163, 996, 2149, 2318, 2327 ],
[ 1, 541, 6731, 26972, 55625, 56081, 56091 ] ];
c := c[ n ];
rank := 1;
repeat
rank := rank + 1;
until c[ rank ] >= sid;
g := SmallGroup( 2^n, sid );
gens := GeneratorsOfGroup( g );
if typ = 1 then
fgens := Reversed( gens{[ rank + 1 .. n ]} );
for j in [ 1 .. Length( SMALL_GROUP_LIB[ n ][ typ ][ sid ] ) ] do
coeff := CoefficientsMultiadic( 2 + 0 * [ 1 .. rank ],
SMALL_GROUP_LIB[ n ][ typ ][ sid ][ j ] );
sgens := ShallowCopy( fgens );
Unbind( inv );
for i in Reversed( [ 1 .. rank ] ) do
if coeff[ i ] = 0 then
Add( sgens, gens[ i ] );
else
if IsBound( inv ) then
Add( sgens, gens[ i ] * inv );
else
inv := gens[ i ];
fi;
fi;
od;
if cid = IdGroup( GroupByGenerators( Reversed( sgens ) ) )[2] then
return num - Length( SMALL_GROUP_LIB[n][typ][ sid ] ) + j;
fi;
od;
Error( "fatal Error in function ID_GROUP_FUNCS[ 13 ]" );
fi;
# typ in [ 2 ..
C := CyclicGroup( 2 ^ typ );
c := GeneratorsOfGroup( C )[ 1 ];
gens := gens{[ 1 .. rank ]};
for j in [ 1 .. Length( SMALL_GROUP_LIB[ n ][ typ ][ sid ] ) ] do
if SMALL_GROUP_LIB[ n ][ typ ][ sid ][ j ] > 0 then
coeff := CoefficientsMultiadic( 2 ^ typ + 0 * [ 1 .. rank ],
SMALL_GROUP_LIB[ n ][ typ ][ sid ][ j ] );
if IdGroup( Kernel( GroupHomomorphismByImages( g, C, gens,
List( coeff, x-> c ^ x ) ) ) ) = [ 2 ^ ( n - typ ), cid ] then
return num - Length( SMALL_GROUP_LIB[n][typ][ sid ] ) + j;
fi;
fi;
od;
Error( "fatal Error in function ID_GROUP_FUNCS[ 13 ]" );
end;
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