Extensions for nontrivial outer action of V4 on D8

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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.



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