Second cohomology group for trivial group action of D8 on Z2

Definition
This article is about the second cohomology group for trivial group action where the acting group is dihedral group:D8 (the dihedral group of order 8 and degree 4) and the base group is cyclic group:Z2 (the cyclic group of order 2). In other words, we are interested in:

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

where $$G \cong D_8$$ and $$A \cong \mathbb{Z}_2$$.

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

Computation of cohomology group
The cohomology group can be computed as an abstract group using group cohomology of dihedral group:D8.

Summary
Note that all these extensions are central extensions with the base normal subgroup isomorphic to cyclic group:Z2 and the quotient group isomorphic to dihedral group:D8. Due to the fact that order of extension group is product of order of normal subgroup and quotient group, the order of each of the extension groups is $$2 \times 8 = 16$$.

Some, but not all, of the extensions are stem extensions. Since cyclic group:Z2 is in fact also the Schur multiplier of dihedral group:D8, the stem extensions here are precisely the Schur covering groups of dihedral group:D8.

The minimum size of generating set of the extension group is at least equal to 2 (which is the minimum size of generating set of the quotient group) and at most equal to 3 (which is the sum of the minimum size of generating set of the normal subgroup and the quotient group). 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 nilpotency class is at least 2 and at most 3 in all cases. It is at least 2 because the quotient dihedral group:D8 has nilpotency class two. It is at most 3 because the sum of the nilpotency class of the normal subgroup and quotient group is 3, and the extension is a central extension. The derived length is always exactly 2 because nilpotency class 2 or 3 forces derived length exactly 2, using derived length is logarithmically bounded by nilpotency class.

Explicit description and relation with power-commutator presentation
Consider an extension group $$E$$ with central subgroup isomorphic to $$A$$ (cyclic group:Z2) and quotient group $$G$$ isomorphic to dihedral group:D8. Denote by $$\overline{a_1}, \overline{a_2}, \overline{a_3}$$ elements of $$G$$ such that $$\overline{a_1}^2 = \overline{a_3}$$, $$\overline{a_2}$$ has order two, and $$[\overline{a_1},\overline{a_2}] = \overline{a_3}$$. Explicitly, we are considering $$G$$ as given by the presentation $$\langle \overline{a_1}, \overline{a_2}, \overline{a_3} \mid \overline{a_1}^2 = \overline{a_3}, \overline{a_2}^2 = e, \overline{a_3}^2 = e, [\overline{a_1},\overline{a_2}] = \overline{a_3}, [\overline{a_1},\overline{a_3}] = e, [\overline{a_2},\overline{a_3}] = e \rangle$$.

We pick elements $$a_1,a_2,a_3$$ of $$E$$ that live in the cosets corresponding to $$\overline{a_1}, \overline{a_2}, \overline{a_3}$$, with the additional condition that $$a_3 = a_1^2$$. Next, we pick $$a_4$$ as the non-identity element of $$A$$.

We can now write a power-commutator presentation of $$E$$ in terms of $$a_1,a_2,a_3,a_4$$. With the usual notation, we already have $$\beta(1,3) = \beta(1,2,3) = 1$$, and all the other coefficients involving 1-3 are zero. This leaves us to choose the values $$\beta(1,4), \beta(2,4), \beta(3,4), \beta(1,2,4), \beta(1,3,4), \beta(2,3,4)$$. However, we note that $$\beta(1,4) = \beta(1,3,4) = 0$$ because in our specific choice, $$a_3$$ equals $$a_1^2$$. Thus, we in fact have only four values to choose: $$\beta(2,4), \beta(3,4), \beta(1,2,4), \beta(2,3,4)$$.

Further, it turns out that of these four values, we must have $$\beta(3,4) = \beta(2,3,4)$$ under these conditions. To see this, note that if $$[a_3,a_2] = [a_1^2,a_2]$$. Because $$a_1$$ commutes with $$[a_1,a_2]$$ this can be rewritten as $$[a_1,a_2]^2$$. Note that $$[a_1,a_2] = a_3$$ or $$[a_1,a_2] = a_3a_4$$, so its square is $$a_3^2$$. The upshot is that $$[a_3,a_2] = a_3^2$$. Inverting, we get $$[a_2,a_3] = a_3^2$$.

The upshot of all this is that we can freely vary the parameters $$\beta(2,4), \beta(3,4), \beta(1,2,4)$$ and the remaining parameters are fixed. This gives $$2^3 = 8$$ possibilities. Moreover, the coordinate-wise addition of these corresponds to addition in the cohomology group.

Note that the positions of dihedral group:D16 and semidihedral group:SD16 are sensitive to the commutator convention.

Under the action of the automorphism group of the acting group
By pre-composition, the automorphism group of dihedral group:D8, which is itself isomorphic to dihedral group:D8, acts on the second cohomology group. This action is transitive on all the extensions in each cohomology class type. In particular, the four fixed points are the cohomology classes corresponding to the extension groups direct product of D8 and Z2, nontrivial semidirect product of Z4 and Z4, dihedral group:D16 and generalized quaternion group:Q16, and the two orbits of size two correspond to the extensions SmallGroup(16,3) and semidihedral group:SD16.

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

$$0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(H_2(G;\mathbb{Z}),A) \to 0$$

Here:


 * The group $$\operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A)$$ corresponds to the subgroup $$H^2_{\operatorname{sym}}(G;A)$$ of cohomology classes represented by symmetric 2-cocycles. $$G^{\operatorname{ab}}$$ is the abelianization of $$G$$ and its image comprises those extensions where the restricted extension of the derived subgroup $$[G,G]$$ on $$A$$ is trivial and the corresponding extension of the quotient group is abelian.
 * The group $$\operatorname{Hom}(H_2(G;\mathbb{Z}),A)$$ is the second cohomology group up to isoclinism: Explicitly, $$H_2(G;\mathbb{Z})$$ is the Schur multiplier of $$G$$.

We also know, again from the general theory, that the short exact sequence above splits, i.e., the image of $$\operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A)$$ in $$H^2(G;A)$$ has a complement inside $$H^2(G;A)$$. 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 $$A$$, $$G^{\operatorname{ab}}$$ is isomorphic to the Klein four-group. The corresponding group $$\operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A)$$ is also a Klein four-group.

The Schur multiplier $$H_2(G;\mathbb{Z})$$ is cyclic group:Z2, hence $$\operatorname{Hom}(H_2(G;\mathbb{Z}),A)$$ is also isomorphic to cyclic group:Z2.

The image of $$\operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A)$$ in $$H^2(G;A)$$ comprises the four non-stem extensions. It has two cosets in the whole second cohomology group. To split the short exact sequence in an automorphism-invariant fashion, we could pick as our complement either dihedral group:D16 or generalized quaternion group:Q16. The two possibilities are shown below:

Using dihedral group:D16 as the choice of complement, where the rows represent the cosets of the image of $$\operatorname{Ext}^1$$, and the columns represent the cosets of the complement:

Using generalized quaternion group:Q16 as the choice of complement, where the rows represent the cosets of the image of $$\operatorname{Ext}^1$$, and the columns represent the cosets of the complement:

More on the mapping from $$\operatorname{Ext}^1$$
Below is an explicit description of the mapping from $$\operatorname{Ext}^1(G^{\operatorname{ab}},A)$$ to $$H^2(G;A)$$, in terms of the original extension for Klein four-group (the abelianization of dihedral group:D8) on top of cyclic group:Z2 and what the new extension looks like for dihedral group:D8 on top of cyclic group:Z2.

Note that in $$\operatorname{Ext}^1$$, all the three nontrivial elements are in the same orbit under the natural action of the automorphism group of the Klein four-group. But they split into two orbits when we consider those automorphisms of the Klein four-group that are induced by automorphisms of the dihedral group:D8, which is why they give different outputs in $$H^2(G;A)$$.

Homomorphisms on $$A$$
The unique injective homomorphism $$A = \mathbb{Z}_2$$ to $$\mathbb{Z}_4$$ induces a homomorphism:

$$\! H^2(G;A) \to H^2(G;\mathbb{Z}_4)$$

The group on the right is also isomorphic to elementary abelian group:E8 (see second cohomology group for trivial group action of D8 on Z4). However, the induced map above is not an isomorphism. Rather, it has kernel of order four precisely the image of $$\operatorname{Ext}^1(G^{\operatorname{ab}},A)$$ in $$H^2(G;A)$$ (see the direct sum decomposition section) and its image is a subgroup of order two in $$H^2(G;\mathbb{Z}_4)$$.

In terms of extensions, the map is interpreted as follows: it involves taking the central product of a given extension with cyclic group:Z4, identifying the base cyclic group:Z2 in the original extension with the $$\mathbb{Z}_2$$ in $$\mathbb{Z}_4$$.

The map is given in the table below:

The unique surjective map from $$\mathbb{Z}_4$$ to $$A = \mathbb{Z}_2$$ induces a homomorphism:

$$H^2(G;\mathbb{Z}_4) \to H^2(G;A)$$

The kernel of this map is the image of the preceding map and the image of this map is the kernel of the preceding map. The map is given in the table below:

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

gap> G := DihedralGroup(8);; 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 := [ [ 0 ], [ [ 2, 1, 3, 1 ], [ 3, 1 ] ], [ [ 3, 1 ], [ 3, 1 ], 0 ] ], orders := [ 2, 2, 2 ], wstack := [ [ 1, 1 ], [ 2, 1, 3, 1 ] ], estack := [ ], pstack := [ 3, 5 ], cstack := [ 1, 1 ], mstack := [ 0, 0 ], list := [ 0, 0, 0 ], module := [ , ,  ], mone := , mzero := , avoid := [ ], unavoidable := [ 1, 2, 3, 4, 5, 6 ] ), cohom :=  -> ( GF(2)^3 )>, presentation := rec( group :=  ,     relators := [ f1^2, f1^-1*f2*f1*f3^-1*f2^-1, f2^2*f3^-1, 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 := DihedralGroup(8);; gap> A := TrivialGModule(G,GF(2));; gap> L := Extensions(G,A);; gap> List(L,IdGroup); [ [ 16, 11 ], [ 16, 8 ], [ 16, 3 ], [ 16, 7 ], [ 16, 4 ], [ 16, 8 ], [ 16, 3 ], [ 16, 9 ] ]

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

gap> G := ElementaryAbelianGroup(4);; 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, 11 ], [ 16, 8 ], [ 16, 3 ], [ 16, 7 ], [ 16, 4 ], [ 16, 9 ] ]