Semidirect product of Z8 and Z8 of M-type

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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The group is defined by means of the presentation:

\! G := \langle a,b \mid a^8 = b^8 = e, bab^{-1} = a^5 \rangle

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 64#Arithmetic functions

GAP implementation

Group ID

This finite group has order 64 and has ID 3 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(64,3);

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

IdGroup(G) = [64,3]

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.1^8,F.2^8,F.2*F.1*F.2^(-1)*F.1^3];
<fp group on the generators [ f1, f2 ]>
gap> IdGroup(G);
[ 64, 3 ]