# Direct product of S3 and Z3

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## Definition

This group is defined in the following equivalent ways (equivalent up to isomorphism):

1. It is the direct product of the symmetric group of degree three and the cyclic group of order three.
2. It is the wreath product of the cyclic group of order three and the cyclic group of order two. In other words, it is the semidirect product $(\Z_3 \times \Z_3) \rtimes \Z_2$ where $\Z_2$ acts by coordinate exchange.

## GAP implementation

### Group ID

This finite group has order 18 and has ID 3 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(18,3)

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

gap> G := SmallGroup(18,3);

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

IdGroup(G) = [18,3]

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 described using the DirectProduct, SymmetricGroup, and CyclicGroup functions:

DirectProduct(SymmetricGroup(3),CyclicGroup(3))

It can also be defined using the WreathProduct and CyclicGroup functions:

WreathProduct(CyclicGroup(3),CyclicGroup(2))