# Central product of M16 and Z8 over common Z2

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## Definition

This group is defined as the central product of the groups M16 and cyclic group:Z8 over a common cyclic central subgroup cyclic group:Z2. Explicitly, it has the presentation: $G := \langle a,b,c \mid a^8 = b^2 = c^8 = e, a^4 = c^4, ac = ca, bc = cb, bab^{-1} = a^5 \rangle$

## GAP implementation

### Group ID

This finite group has order 64 and has ID 86 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(64,86)

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

gap> G := SmallGroup(64,86);

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

IdGroup(G) = [64,86]

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.1^8,F.2^2,F.3^4*F.1^4,F.2*F.1*F.2^(-1)*F.1^(-5),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 ]>