# General semilinear group:GammaL(1,8)

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## Contents

## Definition

This group is defined in the following equivalent ways:

- It is the general semilinear group of degree one over field:F8.
- It is the external semidirect product of cyclic group:Z7 by cyclic group:Z3 for the unique nontrivial action of the latter on the former.

## Arithmetic functions

### Basic arithmetic functions

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

order (number of elements, equivalently, cardinality or size of underlying set) | 21 | groups with same order | As : As semidirect product of groups of order 7 and 3: (see order of semidirect product is product of orders) |

exponent of a group | 21 | groups with same order and exponent of a group | groups with same exponent of a group | There are elements of order 7 and 3 |

nilpotency class | -- | -- | not a nilpotent group |

derived length | 2 | groups with same order and derived length | groups with same derived length | Derived subgroup is isomorphic to cyclic group:Z7. |

Fitting length | 2 | groups with same order and Fitting length | groups with same Fitting length | Fitting subgroup is same as derived subgroup |

Frattini length | 1 | groups with same order and Frattini length | groups with same Frattini length | Maximal subgroups of orders 7 and 3 intersect trivially |

### Arithmetic functions of a counting nature

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

number of conjugacy classes | 5 | groups with same order and number of conjugacy classes | groups with same number of conjugacy classes | As : See element structure of general semilinear group of degree one over a finite field |

## GAP implementation

### Group ID

This finite group has order 21 and has ID 1 among the groups of order 21 in GAP's SmallGroup library. For context, there are groups of order 21. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(21,1)`

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

`gap> G := SmallGroup(21,1);`

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

`IdGroup(G) = [21,1]`

or just do:

`IdGroup(G)`

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