# Direct product of Z16 and V4

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## Contents

## Definition

This group is defined as the external direct product of the cyclic group of order 16 and Klein four-group (i.e., the elementary abelian group of order four). In other words, it is given by the presentation:

.

It is the abelian group of prime power order corresponding to the prime and the partition .

## 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-fourth order and elementary abelian group of prime-square order |

3 | direct product of Z81 and E9 |

5 | direct product of Z625 and E25 |

## 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 | No | |

metacyclic group | No | |

elementary abelian group | No | |

abelian group | Yes | |

nilpotent group | Yes | |

solvable group | Yes |

## GAP implementation

### Group ID

This finite group has order 64 and has ID 183 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(64,183)`

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

`gap> G := SmallGroup(64,183);`

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

`IdGroup(G) = [64,183]`

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 DirectProduct, CyclicGroup, and ElementaryAbelianGroup as:

`DirectProduct(CyclicGroup(16),CyclicGroup(2),CyclicGroup(2))`

or

`DirectProduct(CyclicGroup(16),ElementaryAbelianGroup(4))`