# Groups of prime-fourth order

From Groupprops

This article is about the groups of prime-fourth 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 16 for a summary of information on these groups.

Among the odd primes , the case is *slightly* different from the other primes.

## Statistics at a glance

### Group counts

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

Total number of groups | 14 | 15 | 15 | See classification of groups of order 16, classification of groups of prime-fourth order for odd prime |

Number of abelian groups | 5 | 5 | 5 | See classification of finite abelian groups and structure theorem for finitely generated abelian groups.. In this case, the number of unordered integer partitions of 4 equals 5. |

Number of groups of nilpotency class exactly two |
6 | 6 | 6 | |

Number of groups of nilpotency class exactly three |
3 | 4 | 4 |

## Particular cases

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

2 | 16 | 14 | groups of order 16 -- behaves quite differently from the others. |

3 | 81 | 15 | groups of order 81 -- behaves somewhat differently from the others. |

5 | 625 | 15 | groups of order 625 |

7 | 2401 | 15 | groups of order 2401 |

11 | 14641 | 15 | groups of order 14641 |