Second cohomology group for trivial group action of E8 on Z4

Description of the group
We consider here the second cohomology group for trivial group action of elementary abelian group:E8 on the cyclic group:Z4, i.e.,

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

where $$G \cong E_8 \cong \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$$ and $$A \cong \mathbb{Z}_4$$.

The cohomology group is isomorphic to elementary abelian group:E64.

Under the action of the automorphism group of the acting group
The acting group $$G$$, which is isomorphic to elementary abelian group:E8, has a huge automorphism group, general linear group:GL(3,2) of order $$168$$. Each cohomology class type in the table above is one orbit. In particular, the trivial group extension is the only fixed point, there are two orbits of size $$7$$, and there is one orbit each of size $$21$$ and $$28$$.

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
In terms of the general background, one way of putting this is that the skew map:

$$H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A)$$

has a section (i.e., a reverse map):

$$\operatorname{Hom}(\bigwedge^2G,A) \to H^2(G;A)$$

whose image is $$H^2_{CP}(G;A)$$ of cohomology classes represented by cyclicity-preserving 2-cocycles (see cyclicity-preserving subgroup of second cohomology group for trivial group action). Thus, the natural short exact sequence splits, and we get an internal direct sum decomposition:

$$H^2(G;A) = H^2_{sym}(G;A) + H^2_{CP}(G;A)$$

A pictorial description of this is as follows. Here, each column is a coset of $$H^2_{CP}(G,A)$$ 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 cyclicity-preserving 2-cocycles.

Note that apart from the choice of first row and first column representing the subgroups, the ordering of rows and columns has no significance, since the automorphism group acts transitively on all the rows other than the first row and all the columns other than the first column. Also, note that the $$7 \times 7$$ matrix that we get on deleting the first row and first column has the configuration of a Fano plane -- such a configuration is dictated by the various symmetry considerations.

Generalized Baer Lie rings
The direct sum decomposition (discussed in the preceding section):

$$H^2(G;A) = H^2_{sym}(G;A) + H^2_{CP}(G;A)$$

gives rise to some examples of the cocycle halving generalization of Baer correspondence. For any group extension arising as an element of $$H^2(G;A)$$, the additive group of its Lie ring arises as the group extension corresponding to the projection onto $$H^2_{sym}(G;A)$$, and the Lie bracket coincides with the group commutator.

In the description (in the preceding section), the additive group of the Lie ring of a given group is the unique abelian group in the column corresponding to that group.

Thus, we have the following correspondences emerging: