Second cohomology group for trivial group action of direct product of Z4 and Z2 on Z2

Description of the group
We consider here the second cohomology group for trivial group action of the direct product of Z4 and Z2 on the cyclic group:Z2, i.e.,

$$\! H^2(G,A)$$

where $$G \cong \mathbb{Z}_4 \times \mathbb{Z}_2$$ and $$A \cong \mathbb{Z}_2$$.

The cohomology group is isomorphic to elementary abelian group:E8.

Computation of cohomology group
The cohomology group can be computed as an abstract group using the group cohomology of direct product of Z4 and Z2.

The general formula for $$H^2(G;A)$$ for this choice of $$G$$ is:

$$A/4A \oplus A/2A \oplus \operatorname{Ann}_A(2)$$

In this case, with $$A = \mathbb{Z}/2\mathbb{Z}$$, all three summands are $$\mathbb{Z}/2\mathbb{Z}$$, so overall we get $$(\mathbb{Z}/2\mathbb{Z})^3$$.

Elements
We list here the elements, grouped by similarity under the action of the automorphism groups on both sides.

In all cases, order of extension group is product of order of normal subgroup and quotient group, so the order is $$2 \times 8 = 16$$. Also, since these are central extensions (because the action is trivial), the nilpotency class of the extension group is at least 1 (the nilpotency class of the quotient) and at most 2.

The minimum size of generating set is at least 2 (the minimum size of generating set for the quotient) and at most 3 (the sum of the minimum size of generating set for the normal subgroup and the quotient). See minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group and minimum size of generating set of quotient group is at most minimum size of generating set of group.

The Schur multiplier $$H_2(G;\mathbb{Z})$$ is cyclic group:Z2, which is the base of the extension. Thus, some of these extensions are stem extensions, and the corresponding extension groups are Schur covering groups for $$G$$.

Under the action of the automorphism group of the acting group
Under the action of $$\operatorname{Aut}(G)$$, there are six orbits -- one for each of the cohomology class types listed above. In particular, there are four fixed points: direct product of Z4 and V4, direct product of Z4 and Z4, SmallGroup(16,3), and nontrivial semidirect product of Z4 and Z4. There are two orbits of size two: direct product of Z8 and Z2 and M16.

General background
We know from the general theory that there is a natural short exact sequence:

$$0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0$$

where the image of $$\operatorname{Ext}^1$$ is $$H^2_{sym}(G;A)$$, i.e., the group of cohomology classes represented by symmetric 2-cocycles. We also know, again from the general theory, that the short exact sequence above splits, i.e., $$H^2_{sym}(G;A)$$ has a complement inside $$H^2$$. However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

In this case
For this choice of $$G$$ and $$A$$, the subgroup $$H^2_{CP}(G;A)$$ of cyclicity-preserving cohomology classes is trivial, hence the sum $$H^2_{sym}(G;A) + H^2_{CP}(G;A)$$ is $$H^2_{sym}(G;A)$$ and is not the whole group $$H^2(G;A)$$. Thus, $$H^2_{CP}(G;A)$$ is not the desired complement.

However, the subgroup $$H^2_{IIP}(G,A)$$ (the IIP subgroup of second cohomology group for trivial group action) is a complement to $$H^2_{sym}(G,A)$$ in $$H^2(G,A)$$. This comprises the trivial extension and the extension that gives SmallGroup(16,3). Further, this complement is invariant under the action of $$\operatorname{Aut}(G) \times \operatorname{Aut}(A)$$. As an internal direct sum:

$$H^2(G,A) = H^2_{sym}(G,A) + H^2_{IIP}(G,A)$$

as an internal direct sum. A pictorial description of this would be as follows. Here, each column is a coset of $$H^2_{IIP}(G,A)$$ and each row is a coset of $$H^2_{sym}(G,A)$$. The top left entry is the identity element, hence the top row corresponds to abelian group extensions and the left column corresponds to $$H^2_{IIP}(G,A)$$.

The group $$\operatorname{Aut}(G) \times \operatorname{Aut}(A)$$ acts to permute the two right columns. Both rows remain intact. The two left columns also remain intact.

Construction of the cohomology group
The cohomology group can be constucted using the GAP functions CyclicGroup, DirectProduct, TwoCohomology, TrivialGModule, GF.

gap> G := DirectProduct(CyclicGroup(4),CyclicGroup(2));; gap> A := TrivialGModule(G,GF(2));; gap> T := TwoCohomology(G,A); rec( group := , module := rec( field := GF(2), isMTXModule := true, dimension := 1, generators := [ , ,  ] ), collector := rec( relators := [ [ [ 2, 1 ] ], [ [ 2, 1 ], 0 ], [ [ 3, 1 ], [ 3, 1 ], 0 ] ], orders := [ 2, 2, 2 ], wstack := [ [ 2, 1 ], [ 2, 1 ] ], estack := [ ], pstack := [ 3, 3 ], cstack := [ 1, 1 ], mstack := [ 0, 0 ], list := [ 1, 1, 0 ], module := [ , ,  ], mone := , mzero := , avoid := [ ], unavoidable := [ 1, 2, 3, 4, 5, 6 ] ),  cohom :=  -> ( GF(2)^3 )>,  presentation := rec( group := , relators := [ f1^2*f2^-1, f1^-1*f2*f1*f2^-1, f2^2, f1^-1*f3*f1*f3^-1, f2^-1*f3*f2*f3^-1, f3^2 ] ) )

Construction of extensions
The extensions can be constructed using the additional command Extensions.

gap> G := DirectProduct(CyclicGroup(4),CyclicGroup(2));; gap> A := TrivialGModule(G,GF(2));; gap> L := Extensions(G,A);; gap> List(L,IdGroup); [ [ 16, 10 ], [ 16, 2 ], [ 16, 3 ], [ 16, 4 ], [ 16, 5 ], [ 16, 5 ], [ 16, 6 ], [ 16, 6 ] ]

Under the action of the various automorphism groups
This uses additionally the GAP functions AutomorphismGroup, DirectProduct, CompatiblePairs, and ExtensionRepresentatives.

gap> G := DirectProduct(CyclicGroup(4),CyclicGroup(2));; gap> A := TrivialGModule(G,GF(2));; gap> A1 := AutomorphismGroup(G);; gap> A2 := GL(1,2);; gap> D := DirectProduct(A1,A2);; gap> P := CompatiblePairs(G,A,D);; gap> M := ExtensionRepresentatives(G,A,P);; gap> List(M,IdGroup); [ [ 16, 10 ], [ 16, 2 ], [ 16, 3 ], [ 16, 4 ], [ 16, 5 ], [ 16, 6 ] ]