# Direct product of E9 and Z4

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## Definition

This group is defined in the following equivalent ways:

1. It is the direct product of the elementary abelian group of order nine and the cyclic group of order four.
2. 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 $G$:

gap> G := SmallGroup(36,8);

Conversely, to check whether a given group $G$ 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))