# Direct product of Z9 and E9

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## Contents

## Definition

This group is defined as the direct product of the cyclic group of order 9 and elementary abelian group of order 9. Equivalently, it is the direct product of the cyclic group of order and two copies of the cyclic group of order 3.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 81#Arithmetic functions

## Group properties

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

cyclic group | No | |

metacyclic group | No | |

homocyclic group | No | |

abelian group | Yes | |

group of prime power order | Yes | |

nilpotent group | Yes |

## GAP implementation

### Group ID

This finite group has order 81 and has ID 11 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(81,11)`

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

`gap> G := SmallGroup(81,11);`

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

`IdGroup(G) = [81,11]`

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 described using GAP's DirectProduct, CyclicGroup and ElementaryAbelianGroup functions:

`DirectProduct(CyclicGroup(9),ElementaryAbelianGroup(9))`

Alternatively:

`DirectProduct(CyclicGroup(9),CyclicGroup(3),CyclicGroup(3))`