# Direct product of Z6 and Z3

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## Definition

This group is defined in the following equivalent ways:

1. It is the direct product of the cyclic group of order six and the cyclic group of order three.
2. It is the direct product of the elementary abelian group of order nine and cyclic group of order two.
3. It is the direct product of the cyclic group of order two and two copies of the cyclic group of order three.

## Arithmetic functions

Function Value Explanation
order 18
exponent 6
derived length 1
nilpotency class 1

## GAP implementation

### Group ID

This finite group has order 18 and has ID 5 among the groups of order 18 in GAP's SmallGroup library. For context, there are groups of order 18. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(18,5)

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

gap> G := SmallGroup(18,5);

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

IdGroup(G) = [18,5]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.