Extensions for trivial 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 trivial.

Description in terms of cohomology groups

We have the induced outer action which is trivial:

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 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 Z4 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 8) Corresponding group extension for Q on N (obtained by taking the central product with N of the extension for Q on Z(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 semidirect product?
trivial 1 direct product of Z4 and Z2 2 direct product of D8 and Z4 25 Yes No
nontrivial 1 cyclic group:Z8 1 central product of D8 and Z8 38 No No