# Extensions for nontrivial outer action of V4 on D8

This article describes all the group extensions corresponding to a particular outer action with normal subgroup dihedral group:D8 and quotient group Klein four-group.

We consider here the group extensions where the base normal subgroup $N$ is dihedral group:D8, the quotient group $Q$ is Klein four-group $V_4$, and the induced outer action of the quotient group on the normal subgroup is a nontrivial map to $\operatorname{Out}(N)$ which is isomorphic to $\mathbb{Z}_2$. There are three possibilities for the nontrivial map, each with kernel one of the copies of Z2 in V4. They are all equivalent under pre-composition by automorphisms of the Klein four-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 V4 on Z2, and is isomorphic to elementary abelian group:E8. The extension set is thus a set of size two with this group acting on it.

## Extensions

Note that there are two different and non-pseudo-congruent extensions both giving isomorphic overall extension groups (holomorph of Z8). See series-equivalent not implies automorphic.

Number of cohomology classes giving the extension Corresponding group extension for $Q$ on $N$ Second part of GAP ID (order is 32) 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 whole group Minimum size of generating set of whole group
? direct product of D16 and Z2 39 Yes No 3 2 3
? direct product of SD16 and Z2 40 No No 3 2 3
? central product of D16 and Z4 42 No No 3 2 3
? holomorph of Z8 43 Yes Yes 3 2 3
? holomorph of Z8 43 No No 3 2 3
? SmallGroup(32,44) 44 No Yes 3 2 3