SmallGroup(81,8)

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

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

1-isomorphism
The group is 1-isomorphic to the abelian group direct product of Z9 and E9. In other words, there is a bijection between the groups that restricts to an isomorphism on cyclic subgroups of both sides.

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

gap> F := FreeGroup(3);  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)];  gap> IdGroup(G); [ 81, 8 ]