# Extensions for trivial 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 trivial.

## Description in terms of cohomology groups

We have the induced outer action which is trivial:

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

Thus, the extensions for the trivial outer action of on correspond to the elements of the second cohomology group for trivial group action:

The correspondence is as follows: an element of gives an extension with base and quotient . We take the central product of this extension group with , identifying the common .

See second cohomology group for trivial group action of V4 on Z2, which is isomorphic to elementary abelian group:E8.

## Extensions

The table below builds on the knowledge of second cohomology group for trivial group action of V4 on Z2.

Cohomology class type | Number of cohomology classes | Corresponding group extension for on | Second part of GAP ID (order is 8) | Corresponding group extension for on (obtaining by taking a central product of the extension of on by , with identified) | Second part of GAP ID (order is 32) | Is the extension a semidirect product of by ? | Is the base characteristic in the semidirect product? | Nilpotency class of whole group | Derived length of whole group | Minimum size of generating set of whole group |
---|---|---|---|---|---|---|---|---|---|---|

trivial | 1 | elementary abelian group:E8 | 5 | direct product of D8 and V4 | 46 | Yes | No | 2 | 2 | 4 |

symmetric and nontrivial | 3 | direct product of Z4 and Z2 | 2 | direct product of SmallGroup(16,13) and Z2 | 48 | Yes | No | 2 | 2 | 4 |

non-symmetric | 3 | dihedral group:D8 | 3 | inner holomorph of D8 | 49 | Yes | No | 2 | 2 | 4 |

non-symmetric | 1 | quaternion group | 4 | central product of D8 and Q8 | 50 | No | No | 2 | 2 | 4 |

Total (--) | 8 | -- | -- | -- | -- | -- | -- | -- | -- | -- |