# SmallGroup(81,8)

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 is a group of order 81 given by the following presentation (with denoting the identity element):

## Arithmetic functions

## Elements

`Further information: element structure of SmallGroup(81,8)`

### 1-isomorphism

The group is 1-isomorphic to the abelian group direct product of Z9 and E9. In other words, there is a bijection between the groups that restricts to an isomorphism on cyclic subgroups of both sides.

## GAP implementation

### Group ID

This finite group has order 81 and has ID 8 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(81,8)`

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

`gap> G := SmallGroup(81,8);`

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

`IdGroup(G) = [81,8]`

or just do:

`IdGroup(G)`

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

### Description by presentation

The group can be described using a presentation by means of the following GAP command/code:

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.3*F.1*F.3^(-1)*F.1^(-1)*F.2^(-1),F.3*F.2*F.3^(-1)*F.1^(-3)*F.2^(-1)]; <fp group on the generators [ f1, f2, f3 ]> gap> IdGroup(G); [ 81, 8 ]