Direct product of Z5 and S3

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Definition

This group is the external direct product of cyclic group:Z5 and symmetric group:S3.

Properties

Property	Satisfied?	Explanation
Abelian group	No	This can be seen as e.g. symmetric group:S3 is not abelian, which is part of the direct product
Nilpotent group	No

GAP implementation

Group ID

This finite group has order 30 and has ID 1 among the groups of order 30 in GAP's SmallGroup library. For context, there are groups of order 30. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(30,1)

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

gap> G := SmallGroup(30,1);

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

IdGroup(G) = [30,1]

or just do:

IdGroup(G)

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