# Direct product of Z27 and Z3

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## Contents

## Definition

This group is defined as the direct product of the cyclic group of order 27 and the cyclic group of order three.

## 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 | |

abelian group | Yes | |

metacyclic group | Yes | |

homocyclic group | No | |

group of prime power order | Yes | |

nilpotent group | Yes |

## GAP implementation

### Group ID

This finite group has order 81 and has ID 5 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(81,5)`

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

`gap> G := SmallGroup(81,5);`

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

`IdGroup(G) = [81,5]`

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 defined using GAP's DirectProduct and CyclicGroup functions:

`DirectProduct(CyclicGroup(27),CyclicGroup(3))`