# Classification of groups of prime-fourth order

From Groupprops

## Statement

Let be a prime number. Then, the groups of order can be classified as follows, with slightly different classifications for the cases , , and . The case is different even in so far as the *number* of possible groups is concerned. The cases and have minor differences with each other.

Group property | Number of isomorphism classes of groups case , so order | Number of isomorphism classes of groups case odd prime |
---|---|---|

abelian group | 5 | 5 |

nilpotency class exactly two, i.e., class two group that is non-abelian |
6 | 6 |

nilpotency class exactly three |
3 | 4 |

Total | 14 | 15 |

### The five abelian groups

The nature and classification of the five *abelian* groups of order is the same for both the and odd cases; the abelian groups are classified by the set of unordered integer partitions of the number 4.

Partition of 4 | Corresponding abelian group (in general) | Corresponding abelian group case | GAP ID (2nd part) case | GAP ID (2nd part) case odd |
---|---|---|---|---|

4 | cyclic group of prime-fourth order | cyclic group:Z16 | 1 | 1 |

3 + 1 | direct product of cyclic group of prime-cube order and cyclic group of prime order | direct product of Z8 and Z2 | 5 | 5 |

2 + 2 | direct product of cyclic group of prime-square order and cyclic group of prime-square order | direct product of Z4 and Z4 | 2 | 2 |

2 + 1 + 1 | direct product of cyclic group of prime-square order and elementary abelian group of prime-square order | direct product of Z4 and V4 | 10 | 11 |

1 + 1 + 1 + 1 | elementary abelian group of prime-fourth order | elementary abelian group:E16 | 14 | 15 |