# Group cohomology of groups of order 27

From Groupprops

This article gives specific information, namely, group cohomology, about a family of groups, namely: groups of order 27.

View group cohomology of group families | View group cohomology of groups of a particular order |View other specific information about groups of order 27

With the exception of the zeroth homology group and cohomology group, *all* homology and cohomology groups over *all* possible abelian groups are 3-groups.

## Homology groups for trivial group action

FACTS TO CHECK AGAINST(homology group for trivial group action):

First homology group: first homology group for trivial group action equals tensor product with abelianization

Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier

General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

### Over the integers

The table below lists the first few homology groups with coefficients in the integers. We use to denote the cyclic group of order .

We use 0 to denote the trivial group.

Group | GAP ID 2nd part | Nilpotency class | (= abelianization) | (= Schur multiplier) | |||
---|---|---|---|---|---|---|---|

cyclic group:Z27 | 1 | 1 | 0 | 0 | |||

direct product of Z9 and Z3 | 2 | 1 | |||||

prime-cube order group:U(3,3) | 3 | 2 | ? | ? | ? | ||

M27 | 4 | 2 | 0 | ? | ? | ? | |

elementary abelian group:E27 | 5 | 1 |

## Second cohomology groups and extensions

### Schur multiplier and Schur covering groups

The Schur multiplier is defined as second cohomology group for trivial group action, , and also as the second homology group . A corresponding Schur covering group of is a group that arises as a stem extension with base normal subgroup the Schur multiplier and the quotient group is .

Group | GAP ID 2nd part | Nilpotency class | (= Schur multiplier) | Order of | Possibilities for Schur covering groups | Cohomology group information | Orders of Schur covering groups |
---|---|---|---|---|---|---|---|

cyclic group:Z27 | 1 | 1 | trivial group | 1 | cyclic group:Z27 | -- | 27 |

direct product of Z9 and Z3 | 2 | 1 | cyclic group:Z3 | 3 | SmallGroup(27,3), nontrivial semidirect product of Z9 and Z9, semidirect product of Z27 and Z3 | second cohomology group for trivial group action of direct product of Z9 and Z3 on Z3 | 81 |

prime-cube order group:U(3,3) | 3 | 2 | elementary abelian group:E9 | 9 | SmallGroup(243,3) | second cohomology group for trivial group action of U(3,3) on E9 | 243 |

M27 | 4 | 2 | trivial group | 1 | M27 | -- | 27 |

elementary abelian group:E27 | 5 | 1 | elementary abelian group:E27 | 27 | lots of them | second cohomology group for trivial group action of E27 on E27 | 729 |