# SmallGroup(81,3)

From Groupprops

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:

## Arithmetic functions

Want to compare with other groups of the same order? Check out groups of order 81#Arithmetic functions

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

order | 81 | |

exponent | 9 | |

Frattini length | 2 | |

Fitting length | 1 | |

derived length | 2 | |

nilpotency class | 2 | |

minimum size of generating set | 2 | |

rank as p-group | 3 | |

normal rank as p-group | 3 | |

characteristic rank as p-group | 3 | |

number of conjugacy classes | 33 | |

number of subgroups | 41 | |

number of conjugacy classes of subgroups | 23 |

## Group properties

Want to compare with other groups of the same order? Check out groups of order 81#Group properties.

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

group of prime power order | Yes | |

abelian group | No | |

nilpotent group | Yes | prime power order implies nilpotent |

group of nilpotency class two | Yes | |

metabelian group | Yes | |

T-group | No | |

ambivalent group | No | odd-order and ambivalent implies trivial |

## Families

This group is part of the family SmallGroup(p^4,3).

## GAP implementation

### Group ID

This finite group has order 81 and has ID 3 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(81,3)`

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

`gap> G := SmallGroup(81,3);`

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

`IdGroup(G) = [81,3]`

or just do:

`IdGroup(G)`

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

### Other descriptions

gap> F := FreeGroup(3); <free group on the generators [ f1, f2, f3 ]> gap> G := F/[F.1^9,F.2^3,F.3^3,F.1*F.2*F.1^(-1)*F.2^(-1),F.2*F.3*F.2^(-1)*F.3^(-1),F.3*F.1*F.3^(-1)*F.1^(-1)*F.2^(-1)]; <fp group on the generators [ f1, f2, f3 ]>