Direct product of D12 and Z3

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Definition

This group is defined in the following equivalent ways:

1. It is the external direct product of dihedral group:D12 and cyclic group:Z3.
2. It is the external direct product of symmetric group:S3 and cyclic group:Z6.

GAP implementation

Group ID

This finite group has order 36 and has ID 12 among the groups of order 36 in GAP's SmallGroup library. For context, there are 14 groups of order 36. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(36,12)

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

gap> G := SmallGroup(36,12);

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

IdGroup(G) = [36,12]

or just do:

IdGroup(G)

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

Short descriptions

Description Functions used Mathematical comments
DirectProduct(DihedralGroup(12),CyclicGroup(3)) DirectProduct, DihedralGroup, and CyclicGroup
DirectProduct(SymmetricGroup(3),CyclicGroup(6)) DirectProduct, SymmetricGroup, and CyclicGroup