Dear Thomas,
thanks a lot for your help! In the mean time, Burkhard Külshammer and I
determined the isomorphism type by theoretical arguments. In case this
is of any interest, the Sylow 7-subgroup of the Monster is characterized
by the following data:
Size: 7^6,
Exponent: 7,
7-Rank: 3,
contains 7^{1+4}_+.
Best wishes,
Benjamin
Am 24.10.2012 09:25, schrieb Thomas Breuer:
Dear Benjamin, dear GAP Forum,
if the embedding into the Monster group need not be explicit
then the following works.
The current version of GAP's character table library contains
a few character tables of subgroups of the Monster group
that contain a Sylow 7 subgroup of the Monster group.
gap> m:= CharacterTable( "M" );
CharacterTable( "M" )
gap> Collected( Factors( Size( m ) ) );
[ [ 2, 46 ], [ 3, 20 ], [ 5, 9 ], [ 7, 6 ], [ 11, 2 ], [ 13, 3 ],
[ 17, 1 ], [ 19, 1 ], [ 23, 1 ], [ 29, 1 ], [ 31, 1 ], [ 41, 1 ],
[ 47, 1 ], [ 59, 1 ], [ 71, 1 ] ]
gap> src:= NamesOfFusionSources( m );;
gap> srctbls:= List( src, CharacterTable );;
gap> filt:= Filtered( srctbls, x -> Size( x ) mod 7^6 = 0 );
[ CharacterTable( "7^(2+1+2):GL2(7)" ),
CharacterTable( "7^(1+4):(3x2.S7)" ), CharacterTable( "7^1+4.2A7" ) ]
For some of these character tables, a construction of the corresponding
group is known.
gap> info:= List( filt, GroupInfoForCharacterTable );
[ [ [ "AtlasGroup", [ "7^(2+1+2):GL2(7)" ] ] ],
[ [ "AtlasGroup", [ "7^(1+4):(3x2.S7)" ] ] ], [ ] ]
gap> g:= GroupForGroupInfo( info[1][1] );
<permutation group of size 33882912 with 2 generators>
gap> syl:= SylowSubgroup( g, 7 );
<permutation group of size 117649 with 6 generators>
gap> pc:= Image( IsomorphismPcGroup( syl ) );
Group([ f1, f2, f3, f4, f5, f6 ])
I hope this helps.
All the best,
Thomas
On Tue, Oct 23, 2012 at 01:06:50PM +0200, Benjamin wrote:
Dear all,
how can I construct a Sylow 7-subgroup of the Monster group in GAP?
Best wishes,
Benjamin
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