# Second cohomology group for trivial group action of S4 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:S4 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and symmetric group:S4 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 symmetric group:S4 |Get more specific information about cyclic group:Z2|View other constructions whose value is Klein four-group

## Description of the group

This article describes the second cohomology group for trivial group action

where is symmetric group:S4 (the symmetric group on a set of size four) and is cyclic group:Z2. itself has order 24 and has order 2.

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

## Computation of the group

See group cohomology of symmetric group:S4#Cohomology groups for trivial group action. It is clear that we have:

In the case that is cyclic of order two, both the groups and are cyclic of order two, so their direct sum is the Klein four-group.

## Elements

Cohomology class type | Number of cohomology classes | Corresponding group extension | Second part of GAP ID (order is 48) | Derived length |
---|---|---|---|---|

trivial | 1 | direct product of S4 and Z2 | 48 | 3 |

nontrivial (but symmetric on commuting pairs? may be) | 1 | special linear group:SL(2,Z4) | 30 | 3 |

nontrivial | 1 | binary octahedral group | 28 | 4 |

nontrivial | 1 | general linear group:GL(2,3) | 29 | 4 |

Total (--) | 4 | -- | -- | -- |