# Direct product of Z9 and Z9 and Z3

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## Definition

This group is defined as the external direct product of two copies of the cyclic group of order 9 and one copy of the cyclic group of order 3. In other words, it is:

$\Z_9 \times \Z_9 \times \Z_3$

## Arithmetic functions

Function Value Explanation
order 243
prime-base logarithm of order 5
exponent 9
prime-base logarithm of exponent 2
minimum size of generating set 3
subgroup rank 3
rank as p-group 3
normal rank 3
characteristic rank 3
derived length 1
nilpotency class 1
Frattini length 2

## GAP implementation

### Group ID

This finite group has order 243 and has ID 31 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(243,31)

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

gap> G := SmallGroup(243,31);

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

IdGroup(G) = [243,31]

or just do:

IdGroup(G)

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

### Other descriptions

The group can also be described using GAP's CyclicGroup and DirectProduct functions as:

DirectProduct(CyclicGroup(9),CyclicGroup(9),CyclicGroup(3))