Extensions for trivial outer action of Z2 on D8

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.

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 trivial.

Description in terms of cohomology groups

We have the induced outer action which is trivial:

$Q\to \mathrm{O}\mathrm{u}\mathrm{t}(N)$

Composing with the natural mapping $\mathrm{O}\mathrm{u}\mathrm{t}(N)\to \mathrm{A}\mathrm{u}\mathrm{t}(Z(N))$, we get a trivial map:

$Q\to \mathrm{A}\mathrm{u}\mathrm{t}(Z(N))$

Thus, the extensions for the trivial outer action of $Q$ on $N$ correspond to the elements of the second cohomology group for trivial group action:

$H^2(Q;Z(N))$

The correspondence is as follows: an element of $H^2(Q;Z(N))$ gives an extension with base $Z(N)$ and quotient $Q$. We take the central product of this extension group with $N$, identifying the common $Z(N)$.

See second cohomology group for trivial group action of Z2 on Z2, which is isomorphic to cyclic group:Z2.

Extensions

Cohomology class type	Number of cohomology classes	Corresponding group extension for $Q$ on $Z(N)$	Second part of GAP ID (order is 4)	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 semidirect product?
trivial	1	Klein four-group	2	direct product of D8 and Z2	11	Yes	No
nontrivial	1	cyclic group:Z4	1	central product of D8 and Z4	13	Yes	No