Direct product of SmallGroup(32,4) and Z2

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Definition

This group is defined as the external direct product of the following two groups: semidirect product of Z8 and Z4 of M-type (GAP ID:(32,4)) and cyclic group:Z2. Explicitly, it is given by the presentation: $G := \langle a,b,c \mid a^8 = b^4 = c^2 = e, ac = ca, bc = cb, bab^{-1} = a^9 \rangle$

GAP implementation

Group ID

This finite group has order 64 and has ID 84 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(64,84)

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

gap> G := SmallGroup(64,84);

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

IdGroup(G) = [64,84]

or just do:

IdGroup(G)

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