# Difference between revisions of "Second cohomology group for trivial group action of S4 on Z2"

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+ | Note that in ''all'' cases, the base of the group extension is characteristic in the whole group because it is ''precisely'' the [[center]] of the whole group. This is because the quotient group is a [[centerless group]] and hence cannot partake of any of the center. | ||

{| class="sortable" border="1" | {| class="sortable" border="1" | ||

− | ! Cohomology class type !! Number of cohomology classes !! Corresponding group extension !! Second part of GAP ID (order is 48) !! Derived length | + | ! Cohomology class type !! Number of cohomology classes !! Corresponding group extension !! Second part of GAP ID (order is 48) !! Base characteristic in whole group? !! Derived length |

|- | |- | ||

− | | trivial || 1 || [[direct product of S4 and Z2]] || 48 || 3 | + | | trivial || 1 || [[direct product of S4 and Z2]] || 48 || Yes || 3 |

|- | |- | ||

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

|- | |- | ||

− | | nontrivial || 1 || [[binary octahedral group]] || 28 || 4 | + | | nontrivial || 1 || [[binary octahedral group]] || 28 || Yes || 4 |

|- | |- | ||

− | | nontrivial || 1 || [[general linear group:GL(2,3)]] || 29 || 4 | + | | nontrivial || 1 || [[general linear group:GL(2,3)]] || 29 || Yes || 4 |

|- | |- | ||

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

|} | |} |

## Revision as of 01:15, 1 November 2011

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

Note that in *all* cases, the base of the group extension is characteristic in the whole group because it is *precisely* the center of the whole group. This is because the quotient group is a centerless group and hence cannot partake of any of the center.

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

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

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

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

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

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