# General affine group:GA(1,5)

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 general affine group of degree one over the field of five elements. In other words, it is the semidirect product of the additive and multiplicative groups of this field. It is denoted .
- It is the holomorph of the cyclic group of order five.
- It is the Suzuki group or the Suzuki group where . Note: This is the only non-simple Suzuki group.

The group can be given by the presentation, with denoting the identity element:

## Arithmetic functions

## Group properties

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

abelian group | No | |

nilpotent group | No | |

metacyclic group | Yes | |

supersolvable group | Yes | |

solvable group | Yes | |

Frobenius group | Yes | |

Camina group | Yes |

## GAP implementation

### Group ID

This finite group has order 20 and has ID 3 among the groups of order 20 in GAP's SmallGroup library. For context, there are 5 groups of order 20. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(20,3)`

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

`gap> G := SmallGroup(20,3);`

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

`IdGroup(G) = [20,3]`

or just do:

`IdGroup(G)`

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