# Nontrivial semidirect product of Z9 and Z9

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

This group is defined by the following presentation (here, denotes the identity element):

It is the case of the nontrivial semidirect product of cyclic groups of prime-square order.

## Arithmetic functions

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

order | 81 | |

prime-base logarithm of order | 4 | |

exponent | 9 | |

prime-base logarithm of exponent | 2 | |

Frattini length | 2 | |

derived length | 2 | |

nilpotency class | 2 | |

minimum size of generating set | 2 | |

subgroup rank | 2 | |

rank as p-group | 2 | |

normal rank | 2 | |

characteristic rank | 2 |

## Group properties

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

abelian group | No | |

metabelian group | Yes | |

metacylcic group | Yes | |

group of nilpotency class two | Yes | |

Frattini-in-center group | Yes | |

group of prime power order | Yes |

## GAP implementation

### Group ID

This finite group has order 81 and has ID 4 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(81,4)`

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

`gap> G := SmallGroup(81,4);`

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

`IdGroup(G) = [81,4]`

or just do:

`IdGroup(G)`

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

### Other descriptions

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