Extensions with normal subgroup D8 and quotient group Z2

This article describes the full solution to the group extension problem for normal subgroup dihedral group:D8 and quotient group cyclic group:Z2: find all congruence classes of group extensions with normal subgroup identified with dihedral group:D8 and quotient group identified with cyclic group:Z2.

We follow the procedure outlined at the group extension problem page, so we set $$N$$ to be dihedral group:D8 and $$Q$$ to be cyclic group:Z2.

Finding a list of outer actions
We begin by trying to classify all possibilities for the induced outer action. This basically involves computing the group:

$$\operatorname{Hom}(Q,\operatorname{Out}(N))$$.

To begin with, we need to determine what $$\operatorname{Out}(N)$$ is as a group. Based on the endomorphism structure of dihedral group:D8, we know that $$\operatorname{Out}(N)$$ is isomorphic to cyclic group:Z2 (more detail: $$\operatorname{Aut}(N)$$ is isomorphic to dihedral group:D8 and the subgroup $$\operatorname{Inn}(N)$$ looks like one of the Klein four-subgroups of dihedral group:D8). Thus, we need to classify:

$$\operatorname{Hom}(\mathbb{Z}_2,\mathbb{Z}_2)$$

There are two possible homomorphisms: the trivial homomorphism (or zero homomorphism) and the nontrivial homomorphism, which would be an isomorphism. We will dub the corresponding outer actions the trivial outer action and nontrivial outer action for the rest of this page.

Finding all congruence classes for a given outer action
For both the trivial and nontrivial outer action, the induced action on the center is trivial. Thus, we are looking at the second cohomology group for trivial group action $$H^2(Q,Z(N))$$ in both cases, which becomes $$H^2(\mathbb{Z}_2,\mathbb{Z}_2)$$, i.e., second cohomology group for trivial group action of Z2 on Z2.

We thus have one copy each of this group corresponding to the trivial outer action and the nontrivial outer action, thus giving rise to four extensions. Full details in the table below: