SmallGroup(81,8)

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Definition

This is a group of order 81 given by the following presentation (with $e$ denoting the identity element):

$G:=⟨a,b,c\mid a^9=b^3=c^3=e,ab=ba,ca{c}^{-1}=ab,cb{c}^{-1}=a^{3}b⟩$

GAP implementation

Group ID

This finite group has order 81 and has ID 8 among the groups of order 81 in GAP's SmallGroup library. For context, there are groups of order 81. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(81,8)

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

gap> G := SmallGroup(81,8);

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

IdGroup(G) = [81,8]

or just do:

IdGroup(G)

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

Description by presentation

The group can be described using a presentation by means of the following GAP command/code:

gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^9,F.2^3,F.3^3,F.1*F.2*F.1^(-1)*F.2^(-1),F.3*F.1*F.3^(-1)*F.1^(-1)*F.2^(-1),F.3*F.2*F.3^(-1)*F.1^(-3)*F.2^(-1)];
<fp group on the generators [ f1, f2, f3 ]>
gap> IdGroup(G);
[ 81, 8 ]