# Central product of D8 and Z8

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## Contents

## Definition

This group can be described as the central product of dihedral group:D8 and cyclic group:Z8 over a common cyclic central subgroup of order two.

## Position in classifications

Get more information about other groups of the same order at Groups of order 32#The list

Type of classification | Position/number in classification |
---|---|

GAP ID | , i.e., among groups of order 32 |

Hall-Senior number | 17 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

Function | Value | Explanation |
---|---|---|

cyclic group | No | |

abelian group | No | |

group of nilpotency class two | Yes | |

metabelian group | Yes |

## GAP implementation

### Group ID

This finite group has order 32 and has ID 38 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(32,38)`

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

`gap> G := SmallGroup(32,38);`

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

`IdGroup(G) = [32,38]`

or just do:

`IdGroup(G)`

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

### Description by presentation

Here is the description by a presentation:

gap> F := FreeGroup(3); <free group on the generators [ f1, f2, f3 ]> gap> G := F/[F.1^8,F.2^2,F.3^2,F.1*F.2*F.1^(-1)*F.2^(-1)]; <fp group on the generators [ f1, f2, f3 ]> gap> G := F/[F.1^8,F.2^2,F.3^2,F.1*F.2*F.1^(-1)*F.2^(-1),F.3*F.1*F.3^(-1)*F.1^(-5),F.3*F.2*F.3^(-1)*F.1^4*F.2]; <fp group on the generators [ f1, f2, f3 ]>