# Second cohomology group for trivial group action of Q8 on Z4

From Groupprops

This article gives information about the second cohomology group for trivial group action (i.e., the second cohomology group with trivial action) of the group quaternion group on cyclic group:Z4. The elements of this classify the group extensions with cyclic group:Z4 in the center and quaternion group the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.

The value of this cohomology group is Klein four-group.

Get more specific information about quaternion group |Get more specific information about cyclic group:Z4|View other constructions whose value is Klein four-group

## Description of the group

This article describes the second cohomology group for trivial group action of quaternion group on cyclic group:Z4, i.e., the group:

where is the quaternion group and is cyclic group:Z4.

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

## Computation of cohomology group

The cohomology group can be computed abstractly as the group cohomology of quaternion group.

## Elements

Cohomology class type | Number of cohomology classes | Corresponding group extension | Nilpotency class of group | Second part of GAP ID (order is 32) |
---|---|---|---|---|

trivial | 1 | direct product of Q8 and Z4 | 2 | 26 |

nontrivial | 3 | nontrivial semidirect product of Z4 and Z8 | 2 | 12 |