# Groups of order 27

This article gives information about, and links to more details on, groups of order 27

See pages on algebraic structures of order 27| See pages on groups of a particular order

## Contents

## Statistics at a glance

Since is a prime power, and prime power order implies nilpotent, all the groups of order 27 are nilpotent groups.

Quantity | Value | Explanation |
---|---|---|

Total number of groups up to isomorphism | 5 | |

Number of abelian groups | 3 | equal to the number of unordered integer partitions of 3. See classification of finite abelian groups |

Number of groups of nilpotency class exactly two |
2 |

## The list

To learn more about how tocome upwith the list and prove that it is exhaustive (i.e., that these are precisely the isomorphism classes of groups of order 27), see classification of groups of prime-cube order

To understand these in a broader context, see

groups of order 3^n|groups of prime-cube order

Common name for group | Second part of GAP ID (GAP ID is (27,second part)) | Nilpotency class | Probability in cohomology tree probability distribution (as proper fraction) | Probability in cohomology tree probability distribution (as numerical value) |
---|---|---|---|---|

cyclic group:Z27 | 1 | 1 | 4/9 | 0.4444 |

direct product of Z9 and Z3 | 2 | 1 | 26/81 | 0.3210 |

prime-cube order group:U(3,3) | 3 | 2 | 2/81 | 0.0247 |

M27 (semidirect product of Z9 and Z3) | 4 | 2 | 16/81 | 0.1975 |

elementary abelian group:E27 | 5 | 1 | 1/81 | 0.0123 |

## Arithmetic functions

### Functions taking values between 0 and 3

Group | GAP ID (second part) | prime-base logarithm of exponent | nilpotency class | derived length | Frattini length | minimum size of generating set | subgroup rank | rank as p-group | normal rank | characteristic rank | prime-base logarithm of order of derived subgroup | prime-base logarithm of order of inner automorphism group |
---|---|---|---|---|---|---|---|---|---|---|---|---|

cyclic group:Z27 | 1 | 3 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |

direct product of Z9 and Z3 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 |

prime-cube order group:U(3,3) | 3 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 |

M27 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 2 |

elementary abelian group:E27 | 5 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 0 | 0 |

mean (with equal weight on all groups) | -- | 1.8 | 1.4 | 1.4 | 2 | 2 | 2 | 2 | 2 | 1.8 | 0.4 | 0.8 |

mean (weighting by cohomology tree probability distribution) | -- | 2.4074 | 1.2222 | 1.2222 | 2.4321 | 1.5679 | 1.5679 | 1.5679 | 1.5679 | 1.5432 | 0.2222 | 0.4444 |

Here is the same information, with rows and columns interchanged:

### Arithmetic functions values of a counting nature

### Numerical invariants

Group | Conjugacy class sizes | degrees of irreducible representations |
---|---|---|

cyclic group:Z27 | 1 (27 times) | 1 (27 times) |

direct product of Z9 and Z3 | 1 (27 times) | 1 (27 times) |

prime-cube order group:U(3,3) | 1 (3 times), 3 (8 times) | 1 (9 times), 3 (2 times) |

semidirect product of Z9 and Z3 | 1 (3 times), 3 (8 times) | 1 (9 times), 3 (2 times) |

elementary abelian group:E27 | 1 (27 times) | 1 (27 times) |

## Group properties

Property | cyclic group:Z27 | direct product of Z9 and Z3 | prime-cube order group:U(3,3) | semidirect product of Z9 and Z3 | elementary abelian group:E27 |
---|---|---|---|---|---|

cyclic group | Yes | No | No | No | No |

homocyclic group | Yes | No | No | No | Yes |

metacyclic group | Yes | Yes | Yes | Yes | No |

abelian group | Yes | Yes | No | No | Yes |

ambivalent group | No | No | No | No | No |

group of nilpotency class two | Yes | Yes | Yes | Yes | Yes |

group in which every element is automorphic to its inverse | Yes | Yes | Yes | No | Yes |

group in which any two elements generating the same cyclic subgroup are automorphic | Yes | Yes | Yes | No | Yes |