# Direct product of S4 and Z3

From Groupprops

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 symmetric group:S4 and cyclic group:Z3.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 72#Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 72 | groups with same order | order of direct product is product of orders: the order is , where is the order of symmetric group:S4 and is the order of cyclic group:Z3. |

## GAP implementation

### Group ID

This finite group has order 72 and has ID 42 among the groups of order 72 in GAP's SmallGroup library. For context, there are 50 groups of order 72. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(72,42)`

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

`gap> G := SmallGroup(72,42);`

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

`IdGroup(G) = [72,42]`

or just do:

`IdGroup(G)`

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

### Other descriptions

Description | Functions used |
---|---|

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