# Cyclic group:Z243

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## Definition

This group is defined as the cyclic group of order $243 = 3^5$, and is denoted $\Z_{243}$, $C_{243}$ or $\Z/243\Z$.

## Arithmetic functions

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

## GAP implementation

### Group ID

This finite group has order 243 and has ID 1 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(243,1)

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

gap> G := SmallGroup(243,1);

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

IdGroup(G) = [243,1]

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 defined using GAP's CyclicGroup function as:

CyclicGroup(243)

or

CyclicGroup(3^5)