# Cyclic group:Z64

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## Contents

## Definition

This group is defined as the cyclic group of order .

## As an abelian group of prime power order

This group is the abelian group of prime power order corresponding to the partition:

In other words, it is the group .

Value of prime number | Corresponding group |
---|---|

generic prime | cyclic group of prime-sixth order |

3 | cyclic group:Z729 |

5 | cyclic group:15625 |

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 64#Arithmetic functions

## Group properties

Function | Satisfied? | Explanation |
---|---|---|

cyclic group | Yes | |

abelian group | Yes | |

elementary abelian group | No | |

metacyclic group | Yes | |

homocyclic group | Yes | |

nilpotent group | Yes | |

group of prime power order | Yes | |

solvable group | Yes |

## Subgroups

Since the group is cyclic, there is a unique subgroup of every order dividing its order, which is a cyclic group and the quotient group is cyclic as well. `Further information: cyclicity is subgroup-closed, cyclicity is quotient-closed`

- The trivial subgroup. (1)
- The unique subgroup of order two. Isomorphic to cyclic group:Z2. The quotient group is cyclic group:Z32. (1)
- The unique subgroup of order four. Isomorphic to cyclic group:Z4. The quotient group is cyclic group:Z16. (1)
- The unique subgroup of order eight. Isomorphic to cyclic group:Z8. The quotient group is cyclic group:Z8. (1)
- The unique subgroup of order sixteen. Isomorphic to cyclic group:Z16. The quotient group is cyclic group:Z4. (1)
- The unique subgroup of order thirty-two. Isomorphic to cyclic group:Z32. The quotient group is cyclic group:Z2. (1)
- The whole group. (1)

## GAP implementation

### Group ID

This finite group has order 64 and has ID 1 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(64,1)`

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

`gap> G := SmallGroup(64,1);`

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

`IdGroup(G) = [64,1]`

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 described using GAP's CyclicGroup function:

`CyclicGroup(64)`