# Direct product of Z27 and Z9

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## Definition

This group is defined as the external direct product of the cyclic group of order 27 and the cyclic group of order 9. It is thus the abelian group for the prime $p = 3$ corresponding to the partition $5 = 3 + 2$.

## Arithmetic functions

Function Value Explanation
order of a group 243
prime-base logarithm of order 5
exponent of a group 27
prime-base logarithm of exponent 3
minimum size of generating set 2
subgroup rank 2
rank as p-group 2
normal rank 2
characteristic rank 2
derived length 1
nilpotency class 1
Frattini length 3

## GAP implementation

### Group ID

This finite group has order 243 and has ID 10 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(243,10)

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

gap> G := SmallGroup(243,10);

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

IdGroup(G) = [243,10]

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 be described using GAP's DirectProduct and CyclicGroup functions:

DirectProduct(CyclicGroup(27),CyclicGroup(9))