# Direct product of S3 and Z4

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

This group is defined as the external direct product of the symmetric group of degree three 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 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 | 12 | 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 | |

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

Frattini length | 2 | groups with same order and Frattini length | groups with same Frattini 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 5 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(24,5)`

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

`gap> G := SmallGroup(24,5);`

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

`IdGroup(G) = [24,5]`

or just do:

`IdGroup(G)`

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

### Short descriptions

Description | Functions used | Mathematical comments |
---|---|---|

DirectProduct(SymmetricGroup(3),CyclicGroup(4)) |
SymmetricGroup, CyclicGroup |