Direct product of S3 and Z3
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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.
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:
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:
to have GAP output the group ID, that we can then compare to what we want.