# Cyclic group:Z243

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## Contents

## Definition

This group is defined as the cyclic group of order , and is denoted , or .

## 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 :

`gap> G := SmallGroup(243,1);`

Conversely, to check whether a given group 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)`