# Central product of Z9 and wreath product of Z3 and Z3

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## Definition

This group is defined in the following equivalent ways:

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.

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

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 243#Arithmetic functions

## GAP implementation

### Group ID

This finite group has order 243 and has ID 55 among the groups of order 243 in GAP's SmallGroup library. For context, there are groups of order 243. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(243,55)

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

gap> G := SmallGroup(243,55);

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

IdGroup(G) = [243,55]

or just do:

IdGroup(G)

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

### Description by presentation

gap> F := FreeGroup(4);
<free group on the generators [ f1, f2, f3, f4 ]>
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)];
<fp group on the generators [ f1, f2, f3, f4 ]>
gap> IdGroup(G);
[ 243, 55 ]