# General affine group:GA(1,8)

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## Definition

This group is defined as the general affine group of degree one over the field of eight elements. In other words, it is the semidirect product of the additive group of this field and the multiplicative group of the field.

## Arithmetic functions

Function Value Explanation
order 56
exponent 14
Frattini length 1
Fitting length 2
derived length 2
minimum size of generating set 2
subgroup rank 3

## Group properties

Property Satisfied Explanation
abelian group No
nilpotent group No
metacyclic group No
supersolvable group No
solvable group No

## GAP implementation

### Group ID

This finite group has order 56 and has ID 11 among the groups of order 56 in GAP's SmallGroup library. For context, there are 13 groups of order 56. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(56,11)

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

gap> G := SmallGroup(56,11);

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

IdGroup(G) = [56,11]

or just do:

IdGroup(G)

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