Extensions for nontrivial outer action of Z2 on Q8

This article describes all the group extensions corresponding to a particular outer action with normal subgroup quaternion group and quotient group cyclic group:Z2.

We consider here the group extensions where the base normal subgroup $N$ is dihedral group:D8, the quotient group $Q$ is cyclic group:Z2, and the induced outer action of the quotient group on the normal subgroup is a nontrivial map. There are three such possible maps, but they are all conjugate to each other, and hence there is essentially only one type of map.

More explicitly, note that $\operatorname{Out}(N)$ is isomorphic to symmetric group:S3, with three conjugate copies of cyclic group:Z2 in it (the three S2 in S3s), and thus there is a unique (up to conjugacy) nontrivial map from $Q$ to the group.

Description in terms of cohomology groups

We have the induced outer action which is nontrivial:

$Q \to \operatorname{Out}(N)$

Composing with the natural mapping $\operatorname{Out}(N) \to \operatorname{Aut}(Z(N))$, we get a trivial map:

$Q \to \operatorname{Aut}(Z(N))$

Thus, the number of extensions for the trivial outer action of $Q$ on $N$ equals the number of elements in the second cohomology group for trivial group action $H^2(Q;Z(N))$ for the trivial group action. More explicitly, $H^2(Q;Z(N))$ acts on the set of extensions (possibly with repetitions) in a manner that is equivalent to the regular group action. However, the extension set does not have a natural choice of extension corresponding to the identity element.

$H^2(Q;Z(N))$ is the second cohomology group for trivial group action of Z2 on Z2, and is isomorphic to cyclic group:Z2. The extension set is thus a set of size two with this group acting on it.

Extensions

Number of cohomology classes giving the extension	Corresponding group extension for $Q$ on $N$	Second part of GAP ID (order is 16)	Is the extension a semidirect product of $N$ by $Q$?	Is the base characteristic in the whole group?	Nilpotency class of extension group	Derived length of extension group	Minimum size of generating set of whole group
1	semidihedral group:SD16	8	Yes	No	3	2	2
1	generalized quaternion group:Q16	9	No	Yes	3	2	2