[GAP Forum] Schur multiplier of GL(3,4)?

[Derek Holt]

Dera Hung, Dear Forum,

Yes, you can do it with cohomolo. cohomolo works only with permutation groups,
and you have to do the calculation one prime at at a time.

gap> LoadPackage("cohomolo");
true
gap> G:=GL(3,4);;
gap> P:=Image(IsomorphismPermGroup(G));
<permutation group of size 181440 with 2 generators>
gap> C:=CHR(P,2);; SchurMultiplier(C);
[  ]
gap> C:=CHR(P,3);; SchurMultiplier(C);
[  ]
gap> C:=CHR(P,5);; SchurMultiplier(C);
The Sylow p-subgroup of the group is cyclic - so the multiplier is trivial.
[  ]
gap> C:=CHR(P,7);; SchurMultiplier(C);        
The Sylow p-subgroup of the group is cyclic - so the multiplier is trivial.
[  ]

So the Schur multiplier of GL(3,4) is trivial.

(SL(3,4) on the other hand has multiplier 4 X 4, but the outer automorphism
of SL(3,4) of order 3 induces a fixed-point-free action on this group, and
hence kills it in GL(3,4).)

Derek.

> Dear Forum,
> 
> I am trying to compute the Schur multiplier of GL(3,4) and here is what I got:
> 
> >G:=GeneralLinearGroup(3,4);
> [GL(3,4)
> >AbelianInvariantsMultiplier(G);
> [Error, the coset enumeration has defined more than 256000 cosets
> 
> I just wonder if we can do it by using the ''cohomolo'' package and  working under UNIX/LINUX systems? 
> 
> Thank you very much for your help,
> Hung.
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