# Cyclic group:Z24

From Groupprops

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## Contents

## Definition

This group is defined in the following equivalent ways:

- It is the cyclic group of order .
- It is the direct product of the cyclic group of order eight and the cyclic group of order three.

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order | 24 | |

exponent | 24 | |

nilpotency class | 1 | |

derived length | 1 |

## GAP implementation

### Group ID

This finite group has order 24 and has ID 2 among the groups of order 24 in GAP's SmallGroup library. For context, there are 15 groups of order 24. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(24,2)`

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

`gap> G := SmallGroup(24,2);`

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

`IdGroup(G) = [24,2]`

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:

`CyclicGroup(24)`