# Direct product of S5 and Z2

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]

## Definition

This group is defined in the following equivalent ways:

- It is the direct product of the symmetric group of degree five and cyclic group of order two.
- It is the automorphism group of general linear group:GL(2,4).

## GAP implentation

### Group ID

This finite group has order 240 and has ID 189 among the groups of order 240 in GAP's SmallGroup library. For context, there are 208 groups of order 240. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(240,189)`

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

`gap> G := SmallGroup(240,189);`

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

`IdGroup(G) = [240,189]`

or just do:

`IdGroup(G)`

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

### Other descriptions

Description | |
---|---|

DirectProduct(SymmetricGroup(5),CyclicGroup(2)) |
DirectProduct, SymmetricGroup, and CyclicGroup |

AutomorphismGroup(GL(2,4)) |
AutomorphismGroup and GeneralLinearGroup |