# Direct product of E8 and Z4

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## Contents

## Definition

This group is defined as the direct product of the elementary abelian group of order eight 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 32#Arithmetic functions

## Group properties

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

abelian group | Yes | |

cyclic group | No | |

elementary abelian group | No | |

metacyclic group | No | |

homocyclic group | No | |

group of prime power order | Yes | |

nilpotent group | Yes | |

solvable group | Yes | |

directly indecomposable group | No |

## GAP implementation

### Group ID

This finite group has order 32 and has ID 45 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(32,45)`

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

`gap> G := SmallGroup(32,45);`

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

`IdGroup(G) = [32,45]`

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, ElementaryAbelianGroup, and CyclicGroup functions:

`DirectProduct(ElementaryAbelianGroup(8),CyclicGroup(4))`