# Second cohomology group for trivial group action of Z4 on V4

From Groupprops

## Description of the group

We consider here the second cohomology group for trivial group action of cyclic group:Z4 on the Klein four-group, i.e.,

where and .

The cohomology group is isomorphic to the Klein four-group.

Note that since cyclic over central implies abelian, all the corresponding group extensions are abelian. Equivalently, all the 2-cocycles are symmetric 2-cocycles.

## Elements

We list here the elements, grouped by similarity under the action of the automorphism groups on both sides.

Cohomology class type | Number of cohomology classes | Representative 2-cocycle | Corresponding group extension | Second part of GAP ID (order is 16) |
---|---|---|---|---|

trivial | 1 | everywhere | direct product of Z4 and V4 | 10 |

nontrivial | 3 | direct product of Z8 and Z2 | 5 |