Central product of Z9 and wreath product of Z3 and Z3

Definition
This group is defined in the following equivalent ways:


 * It is the central product of defining ingredient::cyclic group:Z9 and defining ingredient::wreath product of Z3 and Z3, with the common shared central subgroup being the unique central cyclic group:Z3 in both.
 * It is the central product of defining ingredient::cyclic group:Z9 and defining ingredient::SmallGroup(81,8), with the common shared central subgroup being the unique central cyclic group:Z3 in both.
 * It is the central product of defining ingredient::cyclic group:Z9 and defining ingredient::SmallGroup(81,9), with the common shared central subgroup being the unique central cyclic group:Z3 in both.
 * It is the central product of defining ingredient::cyclic group:Z9 and defining ingredient::SmallGroup(81,10), with the common shared central subgroup being the unique central cyclic group:Z3 in both.

The group can be define by the following presentation, where $$e$$ denotes the identity element and $$[ \, \ ]$$denotes the commutator -- note that although the left and right conventions give different presentations, these define isomorphic groups.

$$G := \langle a_1,a_2,a_3,a_4|a_1^3=a_2^3=a_3^3=a_4^9 = e,[a_1,a_2] = a_3, [a_1,a_3] = e, [a_2,a_3] = a_4^3,[a_1,a_4] = [a_2,a_4] = [a_3,a_4] = e\rangle$$

Description by presentation
gap> F := FreeGroup(4);  gap> G := F/[F.1^3,F.2^3,F.3^3,F.4^9,Comm(F.1,F.2)*F.3^(-1),Comm(F.1,F.3),Comm(F.2,F.3)*F.4^(-3),Comm(F.1,F.4),Comm(F.2,F.4),Comm(F.3,F.4)];  gap> IdGroup(G); [ 243, 55 ]