# General affine group:GA(1,9)

## Definition

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

## Arithmetic functions

Function Value Explanation
order 72
exponent 24
Frattini length 1
Fitting length 2
minimum size of generating set 2
subgroup rank 2

## Group properties

Property Satisfied Explanation
abelian group No
nilpotent group No
supersolvable group No
solvable group Yes
Frobenius group Yes

## GAP implementation

### Group ID

This finite group has order 72 and has ID 39 among the groups of order 72 in GAP's SmallGroup library. For context, there are 50 groups of order 72. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(72,39)

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

gap> G := SmallGroup(72,39);

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

IdGroup(G) = [72,39]

or just do:

IdGroup(G)

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