Extensions with normal subgroup D8 and quotient group Z2

This page describes all possible group extensions for normal subgroup isomorphic to dihedral group:D8 and quotient group isomorphic to cyclic group:Z2, thus completely solving the group extension problem for this particular choice of normal subgroup and quotient group.

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:

Outer action, i.e., homomorphism from $Q$ to $\operatorname{Out}(N)$ Corresponding action of $Q$ on $Z(N)$, i.e., homomorphism from $Q$ to $\operatorname{Aut}(Z(N))$ Corresponding second cohomology group Isomorphism class of second cohomology group Do there exist extensions for this outer action? (If so, the second cohomology group has a regular group action on this extension set) Total number of congruence classes of extensions (zero or size of second cohomology group) List of extensions More information Is there a natural choice of basepoint for the extension?
trivial outer action, i.e., the trivial homomorphism trivial action second cohomology group for trivial group action of Z2 on Z2 cyclic group:Z2 Yes 2 direct product of D8 and Z2 and central product of D8 and Z4 extensions for trivial outer action of Z2 on D8 For the trivial action, we can take the direct product as the natural choice of basepoint. Note, however, that both extensions are semidirect products.
nontrivial outer action, i.e., an isomorphism trivial action second cohomology group for trivial group action of Z2 on Z2 cyclic group:Z2 Yes 2 dihedral group:D16 and semidihedral group:SD16 extensions for nontrivial outer action of Z2 on D8 No natural choice, both are semidirect products.
Total (2 rows) -- -- -- -- 4 -- -- --