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The first step in attempting to calculate the cohomology of a group G is to decide on how best to represent the group. In some cases G will admit a decomposition into subgroups and quotient groups where the various component groups need to be represented differently. We try to illustrate this with a simple example.
Consider the tessellation of the plane R2 by congruent equilateral triangles, and let G be the group of isometries of the plane which preserve this tessellation.
The group G is generated by two translations S,T together with two reflections X,Y. By using the embedding R2 → R3 (u,v) → (u,v,1) these affine transformations can be represented as 3×3 matrices involving the square root of 3. The following commands construct G.

gap> x:=Indeterminate(Rationals);;
gap> p:=x^2-3;;
gap> K:=AlgebraicExtension(Rationals,p);;
gap> one:=One(K);;
gap> rt3:=RootOfDefiningPolynomial(K);;
gap> reflectionX:=[[-1,0,0],[0,1,0],[0,0,1]]*one;;
gap> reflectionY:=[[1/2,rt3/2,0],[rt3/2,-1/2,0],[0,0,1]]*one;;
gap> translationS:=[[1,0,2],[0,1,0],[0,0,1]]*one;;
gap> translationT:=[[1,0,1],[0,1,rt3],[0,0,1]]*one;;
gap> G:=Group([reflectionX,reflectionY,translationS,translationT]);;
gap> P:=Group([reflectionX,reflectionY]);;
gap> N:=Group([translationS,translationT]);;

The linear isometries in G form a finite group P (called the point group). The translations S and T generate a free abelian group N. The general theory of space groups tells us that N is normal in G and that the quotient G/N is isomorphic to P. The groups N and P are created above. The following command shows that P is the symmetric group of degree 3.
gap> StructureDescription(P); "S3"
We can construct a ZP-resolution as follows.
gap> RP:=ResolutionFiniteGroup(P,4);;
Using the fact that N is free abelian of rank 2 we construct a ZN-resolution as follows.
gap> RN:=ResolutionAbelianGroup([0,0],4);; gap> fpN:=RN.group;; gap> fpNhomN:=GroupHomomorphismByImages(fpN,N, GeneratorsOfGroup(fpN),[translationS,translationT]);; gap> RN.group:=N;; gap> RN.elts:=List(RN.elts,x->Image(fpNhomN,x));;
We construct a homomorphism G→P, together with a section P→G, as follows.
gap> ################################ gap> GhomPfn:=function(MM) > local M,i,j; > M:=[]; > for i in [1..3] do M[i]:=[]; > for j in [1..3] do > M[i][j]:=MM[i][j]; > od; > od; > M[1][3]:=0*one; > M[2][3]:=0*one; > return M; > end;; gap> ################################ gap> GhomP:=GroupHomomorphismByFunction(G,P,GhomPfn); gap> PmapG:=function(MM); return MM; end;
We can now combine the ZP-resolution and ZN-resolution into a free ZG-resolution as follows.
gap> RG:=ResolutionExtension(GhomP,RN,RP,"Don't Test Finiteness", PmapG);;
The following commands suggest that H1(G,Z)=Z2 and H2(G,Z)=Z2.
gap> TRG:=TensorWithIntegers(RG);; gap> Homology(TRG,1); [ 2 ] gap> Homology(TRG,2) [ 2 ]
However, there is a problem that needs looking into: the boundary map in RG doesn't square to zero in higher dimension!! So the reliability of the above homology calculations is also in question. I suspect that the problem lies with the contracting homotopy in the function ResolutionDirectProduct() . Things work with mod 2 coefficients, so maybe it's a problem with signs.

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