# SmallGroup(81,3)

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## Definition

This group is defined by the following presentation:

$G := \langle a,b,c \mid a^9 = b^3 = c^3 = e, ab = ba, bc = cb, cac^{-1} = ab \rangle$

## Families

This group is part of the family SmallGroup(p^4,3).

## GAP implementation

### Group ID

This finite group has order 81 and has ID 3 among the groups of order 81 in GAP's SmallGroup library. For context, there are 15 groups of order 81. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(81,3)

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

gap> G := SmallGroup(81,3);

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

IdGroup(G) = [81,3]

or just do:

IdGroup(G)

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

### Other descriptions

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.2*F.3*F.2^(-1)*F.3^(-1),F.3*F.1*F.3^(-1)*F.1^(-1)*F.2^(-1)];
<fp group on the generators [ f1, f2, f3 ]>