# Groups of prime-cube order

This article is about the groups of prime-cube order for an odd prime number, i.e., the groups of order where is an odd prime. The special case is somewhat different -- see groups of order 8 for a summary of information on these groups.

Want to know how this list of groups is obtained? See classification of groups of prime-cube order

## Contents

## Statistics at a glance

Quantity | Value case | Value case odd | Explanation |
---|---|---|---|

Total number of groups | 5 | 5 | See classification of groups of prime-cube order |

Number of abelian groups | 3 | 3 | See classification of finite abelian groups and structure theorem for finitely generated abelian groups. In this case, the number of unordered integer partitions of 3 is 3, so that is the number of abelian groups. |

Number of groups of class exactly two |
2 | 2 | See classification of groups of prime-cube order |

## Particular cases

Prime | Information on groups of order | |
---|---|---|

2 | 8 | groups of order 8 |

3 | 27 | groups of order 27 |

5 | 125 | groups of order 125 |

7 | 343 | groups of order 343 |

## The list

The list below is valid for odd primes. The list is somewhat different for ; see groups of order 8.

Common name for group | Second part of GAP ID (GAP ID is (p^3, second part)) | Case | Nilpotency class | Probability in cohomology tree probability distribution |
---|---|---|---|---|

cyclic group of prime-cube order | 1 | cyclic group:Z27 | 1 | |

direct product of cyclic group of prime-square order and cyclic group of prime order | 2 | direct product of Z9 and Z3 | 1 | |

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

semidirect product of cyclic group of prime-square order and cyclic group of prime order (also denoted ) | 4 | semidirect product of Z9 and Z3, also denoted | 2 | |

elementary abelian group of prime-cube order | 5 | elementary abelian group:E27 | 1 |

## Presentations

`Further information: presentations for groups of prime-cube order`

Group | Second part of GAP ID (GAP ID is (p^3,2nd part) | Nilpotency class | Minimum size of generating set | Prime-base logarithm of exponent | full power-commutator presentation | ||||
---|---|---|---|---|---|---|---|---|---|

cyclic group of prime-cube order | 1 | 1 | 1 | 3 | 1 | 0 | 1 | 0 | [SHOW MORE] |

direct product of cyclic group of prime-square order and cyclic group of prime order | 2 | 1 | 2 | 2 | 0 | 1 | 0 | 0 | [SHOW MORE] |

prime-cube order group:U(3,p) | 3 | 2 | 2 | 1 | 0 | 0 | 0 | 1 | [SHOW MORE] |

semidirect product of cyclic group of prime-square order and cyclic group of prime order | 4 | 2 | 2 | 2 | 0 | 1 | 0 | 1 | [SHOW MORE] |

elementary abelian group of prime-cube order | 5 | 1 | 3 | 1 | 0 | 0 | 0 | 0 | [SHOW MORE] |

## Arithmetic functions

### Functions taking values between 0 and 3

These arithmetic function values are the same for *all* for the corresponding groups. For , the behavior for the abelian groups is exactly the same, but the two non-abelian groups behave a little differently.

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 | 1 | 3 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |

direct product of... | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 |

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

semidirect product of... | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 2 |

elementary abelian | 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) | -- |

Same, with rows and columns interchanged:

[SHOW MORE]### Arithmetic function values of a counting nature

Group | GAP ID (second part) | number of conjugacy classes | number of subgroups | number of conjugacy classes of subgroups | number of normal subgroups | number of automorphism classes of subgroups | number of characteristic subgroups |
---|---|---|---|---|---|---|---|

cyclic group of prime-cube order | 1 | 4 | 4 | 4 | 4 | 4 | |

direct product of ... | 2 | 6 | 4 | ||||

prime-cube order group:U(3,p) | 3 | 5 | 3 | ||||

semidirect product of ... | 4 | 6 | 4 | ||||

elementary abelian group of prime-cube order | 5 | 4 | 2 |

Same, with rows and columns interchanged:

Function | cyclic group of prime-cube order | direct product of ... | prime-cube order group:U(3,p) | semidirect product of ... | elementary abelian group of prime-cube order |
---|---|---|---|---|---|

number of conjugacy classes | |||||

number of subgroups | 4 | ||||

number of conjugacy classes of subgroups | 4 | ||||

number of normal subgroups | 4 | ||||

number of automorphism classes of subgroups | 4 | 6 | 5 | 6 | 4 |

number of characteristic subgroups | 4 | 4 | 3 | 4 | 2 |

## Group properties

Property | cyclic group of prime-cube order | direct product of ... | prime-cube order group:U(3,p) | semidirect product of ... | elementary abelian group of prime-cube order |
---|---|---|---|---|---|

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

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

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

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

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

metabelian group | Yes | Yes | Yes | Yes | Yes |

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

maximal class group | No | No | Yes | Yes | No |

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

rational group | No | No | No | No | No |

rational-representation group | No | No | No | No | No |

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 |

T-group | Yes | Yes | No | No | Yes |

C-group | No | No | No | No | Yes |

SC-group | No | No | No | No | Yes |

UL-equivalent group | Yes | Yes | Yes | Yes | Yes |

algebra group | No | Yes | Yes | No | Yes |

## Element structure

`Further information: element structure of groups of prime-cube order`

### Order statistics

Group | Second part of GAP ID | Number of elements of order | Number of elements of order | Number of elements of order | Number of elements of order |
---|---|---|---|---|---|

cyclic group of prime-cube order | 1 | 1 | |||

direct product of ... | 2 | 1 | 0 | ||

prime-cube order group:U(3,p) | 3 | 1 | 0 | 0 | |

semidirect product of ... | 4 | 1 | 0 | ||

elementary abelian group of prime-cube order | 5 | 1 | 0 | 0 |

Here are the *cumulative* order statistics, where the number of roots is the number of elements whose order divides .

Group | Second part of GAP ID | Number of elements of order | Number of roots | Number of roots | Number of roots |
---|---|---|---|---|---|

cyclic group of prime-cube order | 1 | 1 | |||

direct product of ... | 2 | 1 | |||

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

semidirect product of ... | 4 | 1 | |||

elementary abelian group of prime-cube order | 5 | 1 |

### Equivalence classes

Up to order statistics-equivalence, there are three equivalence classes. Moreover, these are the same as the eqiuvalence classes up to 1-isomorphism. A deeper explanation of this is that all the group in a given equivalence class actually have the same additive group of their respective Lazard Lie rings.

`Further information: order statistics-equivalent not implies 1-isomorphic, Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring`

Order statistics | Abelian group with those order statistics | Non-abelian group with those order statistics |
---|---|---|

cyclic group of prime-cube order | None | |

direct product of cyclic group of prime-square order and cyclic group of prime order | semidirect product of cyclic group of prime-square order and cyclic group of prime order | |

elementary abelian group of prime-cube order | prime-cube order group:U(3,p) |

## Subgroup structure

`Further information: subgroup structure of groups of prime-cube order`