Extensions for nontrivial outer action of Z4 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 cyclic group:Z4.


We consider here the group extensions where the base normal subgroup N is dihedral group:D8, the quotient group Q is cyclic group:Z4, and the induced outer action of the quotient group on the normal subgroup is the unique nontrivial map.

More explicitly, note that \operatorname{Out}(N) is isomorphic to cyclic group:Z2, and thus there is a unique nontrivial map from Q to it.

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 Z4 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 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
1 SmallGroup(32,9) 9 Yes Yes 3 2 2
1 wreath product of Z4 and Z2 11 Yes No 3 2 2