# Direct product of Z8 and Z2

From Groupprops

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## Contents

## Definition

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

- It is the external direct product of the cyclic group of order eight (denoted or ) and the cyclic group of order two (denoted or ).
- It is the unique abelian group (up to isomorphism) of order sixteen and exponent eight.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

## Group properties

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

cyclic group | No | |

elementary abelian group | No | |

homocyclic group | No | |

metacyclic group | Yes | |

abelian group | Yes |

## GAP implementation

### Group ID

This finite group has order 16 and has ID 5 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(16,5)`

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

`gap> G := SmallGroup(16,5);`

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

`IdGroup(G) = [16,5]`

or just do:

`IdGroup(G)`

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

### Other descriptions

It can also be defined using GAP's DirectProduct function:

DirectProduct(CyclicGroup(8),CyclicGroup(2));