Semidirect product of Z16 and Z4 via cube map

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This group is defined as an external semidirect product where the base normal subgroup is cyclic group:Z16 and the acting quotient group is cyclic group:Z4 and the latter acts on the former via the cube map:

G := \langle a,b \mid a^{16} = b^4 = e, bab^{-1} = a^3 \rangle

GAP implementation

Group ID

This finite group has order 64 and has ID 46 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:


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

gap> G := SmallGroup(64,46);

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

IdGroup(G) = [64,46]

or just do:


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

Description by presentation

gap> F := FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> G := F/[F.1^(16),F.2^4,F.2*F.1*F.2^(-1)*F.1^(-3)];
<fp group on the generators [ f1, f2 ]>
gap> IdGroup(G);
[ 64, 46 ]