# Nontrivial semidirect product of Z3 and Z8

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## Contents

## Definition

This group is defined as the external semidirect product of cyclic group:Z3 by cyclic group:Z8, where the generator of the latter acts on the former via the inverse map. Explicitly, it is given by:

where denotes the identity element.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 24#Arithmetic functions

Function | Value | Similar groups | Explanation for function value |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 24 | groups with same order | |

exponent of a group | 24 | groups with same order and exponent of a group | groups with same exponent of a group | |

derived length | 2 | groups with same order and derived length | groups with same derived length | |

Frattini length | 3 | groups with same order and Frattini length | groups with same Frattini length | |

Fitting length | 2 | groups with same order and Fitting length | groups with same Fitting length | |

minimum size of generating set | 2 | groups with same order and minimum size of generating set | groups with same minimum size of generating set |

## GAP implementation

### Group ID

This finite group has order 24 and has ID 1 among the groups of order 24 in GAP's SmallGroup library. For context, there are 15 groups of order 24. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(24,1)`

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

`gap> G := SmallGroup(24,1);`

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

`IdGroup(G) = [24,1]`

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 a description using the presentation given in the definition:

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