# 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 is dihedral group:D8, the quotient group is Klein four-group , and the induced outer action of the quotient group on the normal subgroup is a nontrivial map to which is isomorphic to . 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:

Composing with the natural mapping , we get a trivial map:

Thus, the number of extensions for the trivial outer action of on equals the number of elements in the second cohomology group for trivial group action for the trivial group action. More explicitly, 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.

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

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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 on | Second part of GAP ID (order is 32) | Is the extension a semidirect product of by ? | 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 |