# Direct product of S3 and Z3

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

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

- It is the direct product of the symmetric group of degree three and the cyclic group of order three.
- 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 where 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 :

`gap> G := SmallGroup(18,3);`

Conversely, to check whether a given group 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))`