# Central product of D8 and Z16

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## Definition

This group can be defined in the following equivalent ways:

1. It is the central product of dihedral group:D8 and cyclic group:Z16 over a commonly identified cyclic central subgroup cyclic group:Z2.
2. It is the central product of quaternion group and cyclic group:Z16 over a commonly identified cyclic central subgroup cyclic group:Z2.

## GAP implementation

### Group ID

This finite group has order 64 and has ID 185 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(64,185)

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

gap> G := SmallGroup(64,185);

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

IdGroup(G) = [64,185]

or just do:

IdGroup(G)

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

### Description by presentation

```gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^4,F.2^2,F.1*F.2*F.1*F.2,F.1^2*F.3^8,F.1*F.3*F.1^(-1)*F.3^(-1),F.2*F.3*F.2^(-1)*F.3^(-1)];
<fp group on the generators [ f1, f2, f3 ]>```