# Second cohomology group for trivial group action of S3 on Z2

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 symmetric group:S3 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and symmetric group:S3 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.

Get more specific information about symmetric group:S3 |Get more specific information about cyclic group:Z2

## Description of the group

This article describes the second cohomology group for trivial group action:

where is symmetric group:S3 (i.e., the symmetric group on a set of size three) and is cyclic group:Z2.

The cohomology group is isomorphic to cyclic group:Z2.

## Computation of cohomology group

The cohomology group can be computed directly from group cohomology of symmetric group:S3#Cohomology groups for trivial group action.

## Elements

Cohomology class type | Number of cohomology classes | Corresponding group extension | Second part of GAP ID (order is 12) |
---|---|---|---|

trivial | 1 | dihedral group:D12 (same as direct product of S3 and Z2) | 4 |

nontrivial | 1 | dicyclic group:Dic12 | 1 |