# M81

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## Contents

## Definition

This group is defined as the semidirect product of the cyclic group of order 27 with the unique cyclic subgroup of order 3 in its automorphism group.

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order | 81 | |

exponent | 27 | |

Frattini length | 3 | |

derived length | 2 | |

nilpotency class | 2 | |

subgroup rank | 2 | |

minimum size of generating set | 2 | |

rank as p-group | 2 | |

normal rank as p-group | 2 |

## Group properties

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

group of prime power order | Yes | |

abelian group | No | |

nilpotent group | Yes | |

group of nilpotency class two | Yes | |

maximal class group | No |

## GAP implementation

### Group ID

This finite group has order 81 and has ID 6 among the groups of order 81 in GAP's SmallGroup library. For context, there are 15 groups of order 81. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(81,6)`

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

`gap> G := SmallGroup(81,6);`

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

`IdGroup(G) = [81,6]`

or just do:

`IdGroup(G)`

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

### Other descriptions

The group can be constructed using the following series of GAP instructions (note that double semicolons are to suppress output display after each command):

gap> H := CyclicGroup(27);; gap> A := AutomorphismGroup(H);; gap> B := Filtered(NormalSubgroups(A),K->Order(K) = 3)[1];; gap> G := SemidirectProduct(B,H);