# Central product of D8 and Z12

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## Definition

This group is defined in the following equivalent ways:

1. It is the central product of dihedral group:D8 and cyclic group:Z12 with a common cyclic central subgroup of order two.
2. It is the external direct product of central product of D8 and Z4 (GAP ID: (16,13)) and cyclic group:Z3.

## GAP implementation

### Group ID

This finite group has order 48 and has ID 47 among the groups of order 48 in GAP's SmallGroup library. For context, there are groups of order 48. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(48,47)

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

gap> G := SmallGroup(48,47);

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

IdGroup(G) = [48,47]

or just do:

IdGroup(G)

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