


A
generalized triangle group
is a group given by a presentation < a, b  a^{p} = b^{q} = R^{m} = 1 >
G = < a, b
 a^{2}, b^{3}, (abababab^{2}ab^{2}abab^{2}ab^{2})^{2}
>
and has order G = 2^{20} 3^{4} 5 . It has been named the Rosenberger monster. The following GAP commands, which were shown to me by R.F. Morse, create a solvable subgroup K in G of index 5. 

gap>
F := FreeGroup("a","b");; a:=F.1;; b:=F.2;; gap> R := [a^2, b^3, (a*b*a*b*a*b*a*b^2*a*b^2*a*b*a*b^2*a*b^2)^2];; gap> G := F/R;; gap> a:= G.1;; b:=G.2;; gap> K:=Subgroup(G, [a, (b^a)^b, (b^a)^(b^2)]);; gap> Index(G,K); 5 

The
Schur Multiplier H_{2}(G,Z) of the Rosenberger monster is a
quotient of H_{2}(K,Z) since K contains the Sylow 2subgroup
and Sylow 3subgroup of G, and the Sylow 5subgroup is cyclic and thus
has trivial Schur multiplier. The following commands first create a
pcgroup Kpc isomorphism to K, then construct three terms of a
resolution for K, and finally show that H_{1}(K,Z)=H_{2}(K,Z)
= Z_{3}+Z_{6} . 

gap>
K_iso_Kfp := IsomorphismFpGroup(K);; gap> Kfp:=Image(K_iso_Kfp);; gap> Kfp_iso_Kpc:= EpimorphismSolvableQuotient(Kfp,2^20*3^4);; gap> Kpc:=Image(Kfp_iso_Kpc);; gap> D:=DerivedSeries(Kpc)[3];; gap> NatHom:=NaturalHomomorphismByNormalSubgroup(Kpc,D);; gap> Q:=Image(NatHom);; gap> N:=NormalSubgroups(Q)[12];; gap> RQ:=ResolutionNormalSeries([Q,N,TrivialSubgroup(Q)],3);; gap> RD:=ResolutionNilpotentGroup(D,3);; gap> RKpc:=ResolutionExtension(NatHom,RD,RQ);; gap> TRKpc:=TensorWithIntegers(RKpc);; gap> Homology(TRKpc,1); [ 3, 6 ] gap> Homology(TRKpc,2); [ 3, 6 ] 

The following commands construct the group D=[[G,G],[G,G]] and show that H_{2}(G/D,Z)=Z_{6} and D=[G,D] .  
gap>
D:=DerivedSubgroup(DerivedSubgroup(G));; gap> GroupHomology(G/D,2); [ 2, 3 ] gap> Index(G,D)=Index(G,CommutatorSubgroup(G,D)); true 

The
five term exact sequence in integral homology (see this page for details) arising
from the normal subgroup D in G yields a surjection H_{2}(G,Z)
→ H_{2}(G/D,Z) . We conclude that the Schur multiplier of the
Rosenberger monster is either H_{2}(G,Z) = Z_{3}+Z_{6}
or
H_{2}(G,Z) = Z_{6 }._{.}
We can complete the calculation of H_{2}(G,Z) using the following basic result.
This result follows from the fact that H_{*}(G,Z) is finite and is the homology of a chain complex of free abelian groups with 1 generator in dimension 0, x generators in dimension 1 and r generators in dimension 2. We now have the following.


