SmallGroup(243,19)

Definition
This group of order $$243 = 3^5$$ is defined by means of the following presentation:

$$G := \langle a_1,a_2,a_3 \mid a_1^{27} = a_2^3 = a_3^3 = e, [a_1,a_2] = a_3, [a_1,a_3] = e, [a_2,a_3] = a_1^9 \rangle$$

Here, $$e$$ denotes the identity element and $$[ \, \ ]$$ denotes the commutator of two elements. Note that the left and right conventions for the commutator give different presentations but define isomorphic groups.

Description by presentation
gap> F := FreeGroup(3);; gap> G :=F/[F.1^(27),F.2^3,F.3^3,Comm(F.1,F.2)*F.3^(-1),Comm(F.1,F.3),Comm(F.2,F.3)*F.1^(-9)];  gap> IdGroup(G); [ 243, 19 ]