# Semidirect product of cyclic group of prime-cube order and cyclic group of prime order

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.

View other such prime-parametrized groups

## Definition

This group is defined as the external semidirect product of the cyclic group of order with the unique subgroup of order in its automorphism group. The group has order and is sometimes denoted .

## Particular cases

Value of prime | Value | Corresponding group |
---|---|---|

2 | 16 | M16 |

3 | 81 | M81 |

## GAP implementation

This finite group has order the fourth power of the prime, i.e., , and has ID 6 among the groups of order in GAP's SmallGroup library. For context, there are 15 groups of order for odd and 14 groups of order for . It can thus be defined using GAP's SmallGroup function as follows, assuming is specified beforehand:

`SmallGroup(p^4,6)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(p^4,6);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [p^4,6]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.