# Direct product of E9 and Z4

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## Contents

## Definition

This group is defined in the following equivalent ways:

- It is the direct product of the elementary abelian group of order nine and the cyclic group of order four.
- It is the direct product of the cyclic group of order twelve and the cyclic group of order three.

## Arithmetic functions

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

order | 36 | |

exponent | 12 |

## GAP implementation

### Group ID

This finite group has order 36 and has ID 8 among the groups of order 36 in GAP's SmallGroup library. For context, there are groups of order 36. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(36,8)`

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

`gap> G := SmallGroup(36,8);`

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

`IdGroup(G) = [36,8]`

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

`DirectProduct(ElementaryAbelianGroup(9),CyclicGroup(4))`