# Central product of UT(3,3) and Z9

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## Contents

## Definition

This group can be defined in the following equivalent ways:

- It is the central product of prime-cube order group:U(3,3) (which is a non-abelian group of order 27 and exponent 3) and cyclic group:Z9, where they share a common central subgroup of order three.
- It is the central product of semidirect product of Z9 and Z3 (which is a non-abelian group of order 27 and exponent 9) and cyclic group:Z9, where they share a common central subgroup of order three.

## Arithmetic functions

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

order | 81 | |

prime-base logarithm of order | 4 | |

exponent | 9 | |

prime-base logarithm of exponent | 2 | |

nilpotency class | 2 | |

derived length | 2 | |

Frattini length | 2 | |

Fitting length | 1 | |

minimum size of generating set | 3 | |

subgroup rank | 3 | |

rank as p-group | 2 | |

normal rank as p-group | 2 | |

characteristic rank as p-group | 1 |

## Group properties

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

abelian group | No | |

metabelian group | Yes | |

group of nilpotency class two | Yes | |

directly indecomposable group | Yes | |

centrally indecomposable group | No |

## GAP implementation

### Group ID

This finite group has order 81 and has ID 14 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(81,14)`

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

`gap> G := SmallGroup(81,14);`

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

`IdGroup(G) = [81,14]`

or just do:

`IdGroup(G)`

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