# Cyclic group:Z12

From Groupprops

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## Contents

## Definition

This group, denoted or , is defined in the following equivalent ways:

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

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 12#Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 12 | groups with same order | |

exponent of a group | 12 | groups with same order and exponent of a group | groups with same exponent of a group | |

nilpotency class | 1 | groups with same order and nilpotency class | groups with same nilpotency class | cyclic implies abelian |

derived length | 1 | groups with same order and derived length | groups with same derived length | cyclic implies abelian |

Frattini length | 2 | groups with same order and Frattini length | groups with same Frattini length | |

Fitting length | 1 | groups with same order and Fitting length | groups with same Fitting length |

## GAP implementation

### Group ID

This finite group has order 12 and has ID 2 among the groups of order 12 in GAP's SmallGroup library. For context, there are 5 groups of order 12. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(12,2)`

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

`gap> G := SmallGroup(12,2);`

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

`IdGroup(G) = [12,2]`

or just do:

`IdGroup(G)`

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

### Other descriptions

Description | Functions used |
---|---|

CyclicGroup(12) |
CyclicGroup |

DirectProduct(CyclicGroup(4),CyclicGroup(3)) |
CyclicGroup, DirectProduct |