# General affine group:GA(1,7)

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## Definition

The group is defined in the following equivalent ways:

1. It is the holomorph of the cyclic group of order seven.
2. 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 $G$:

gap> G := SmallGroup(42,1);

Conversely, to check whether a given group $G$ 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.