# Direct product of D8 and V4

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## Contents

## Definition

The group is defined in the following ways:

- It is the direct product of the dihedral group of order eight and the Klein four-group.
- It is the generalized dihedral group corresponding to the direct product of Z4 and V4.

## Position in classifications

Type of classification | Name in that classification |
---|---|

GAP ID | (32,46), i.e., the 46th among the groups of order 32 |

Hall-Senior number | 8 among groups of order 32 |

Hall-Senior symbol |

## 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 | No | |

metabelian group | Yes | |

group of nilpotency class two | Yes | |

directly indecomposable group | No |

## GAP implementation

### Group ID

This finite group has order 32 and has ID 46 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(32,46)`

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

`gap> G := SmallGroup(32,46);`

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

`IdGroup(G) = [32,46]`

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 constructed using GAP's DirectProduct, DihedralGroup, and GAP:ElementaryAbelianGroup functions:

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