# Extensions for trivial outer action of Z2 on D8

This article describes all the group extensions corresponding to a particular outer action with normal subgroup dihedral group:D8 and quotient group cyclic group:Z2.

We consider here the group extensions where the base normal subgroup is dihedral group:D8, the quotient group is cyclic group:Z2, 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 Z2 on Z2, which is isomorphic to cyclic group:Z2.

## Extensions

Cohomology class type | Number of cohomology classes | Corresponding group extension for on | Second part of GAP ID (order is 4) | Corresponding group extension for on (obtained by taking the central product with of the extension for on ) | Second part of GAP ID (order is 16) | 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 | Klein four-group | 2 | direct product of D8 and Z2 | 11 | Yes | No | 2 | 2 | 3 |

nontrivial | 1 | cyclic group:Z4 | 1 | central product of D8 and Z4 | 13 | Yes | No | 2 | 2 | 3 |