# General affine group:GA(1,7)

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

The group is defined in the following equivalent ways:

- It is the holomorph of the cyclic group of order seven.
- it is the general affine group of degree one over the field of seven elements.

## Arithmetic functions

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

order | 42 | |

exponent | 42 | |

Frattini length | 1 | |

Fitting length | 2 | |

derived length | 2 | |

subgroup rank | 2 | |

minimum size of generating set | 2 |

## Group properties

Property | Satisfied | Explanation |
---|---|---|

abelian group | No | |

nilpotent group | No | |

solvable group | Yes | |

metacyclic group | Yes | |

supersolvable group | Yes |

## GAP implementation

### Group ID

This finite group has order 42 and has ID 1 among the groups of order 42 in GAP's SmallGroup library. For context, there are groups of order 42. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(42,1)`

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

`gap> G := SmallGroup(42,1);`

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

`IdGroup(G) = [42,1]`

or just do:

`IdGroup(G)`

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