# M27

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## Definition

This group, sometimes denoted or , is defined as the semidirect product of the cyclic group of order nine and a cyclic group of order three, acting on it by nontrivial automorphisms.

It is given by the presentation (with denoting the identity element):

It is part of the family of groups: semidirect product of cyclic group of prime-square order and cyclic group of prime order.

## Arithmetic functions

## Group properties

Property | Satisfied | Explanation |
---|---|---|

abelian group | No | |

group of prime power order | Yes | |

nilpotent group | Yes | |

extraspecial group | Yes | |

Frattini-in-center group | Yes |

## GAP implementation

### Group ID

This finite group has order 27 and has ID 4 among the groups of order 27 in GAP's SmallGroup library. For context, there are groups of order 27. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(27,4)`

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

`gap> G := SmallGroup(27,4);`

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

`IdGroup(G) = [27,4]`

or just do:

`IdGroup(G)`

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