# Direct product of E4 and Z9

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## Definition

This group is defined in the following equivalent ways:

1. It is the direct product of the Klein four-group and the cyclic group of order nine.
2. It is the direct product of the cyclic group of order eighteen and the cyclic group of order two.

## Arithmetic functions

Function Value Explanation
order 36
exponent 18

## GAP implementation

### Group ID

This finite group has order 36 and has ID 5 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(36,5)

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

gap> G := SmallGroup(36,5);

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

IdGroup(G) = [36,5]

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

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