# Direct product of Z8 and Z8

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## Contents

## Definition

This group can be defined as the external direct product of two copies of the cyclic group of order eight. In other words, it has the presentation:

.

## 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 | direct product of cyclic group of prime-cube order and cyclic group of prime-cube order |

3 | direct product of Z27 and Z27 |

5 | direct product of Z125 and Z125 |

## 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

Property | Satisfied? | Explanation |
---|---|---|

cyclic group | No | |

homocyclic group | Yes | |

metacyclic group | Yes | |

elementary abelian group | No | |

abelian group | Yes | |

group of prime power order | Yes | |

nilpotent group | Yes | |

solvable group | Yes |

## GAP implementation

### Group ID

This finite group has order 64 and has ID 2 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(64,2)`

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

`gap> G := SmallGroup(64,2);`

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

`IdGroup(G) = [64,2]`

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 the DirectProduct and CyclicGroup functions as:

DirectProduct(CyclicGroup(8),CyclicGroup(8))