# Holomorph of Z9

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## Definition

This group is the holomorph of the cyclic group of order nine. In other words, it is the semidirect product of the cyclic group of order nine with its automorphism group, which is cyclic of order six.

## Arithmetic functions

Function Value Explanation
order 54
exponent 18
Frattini length 2
Fitting length 2
derived length 2
minimum size of generating set 2
subgroup rank 2

## Group properties

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

## GAP implementation

### Group ID

This finite group has order 54 and has ID 6 among the groups of order 54 in GAP's SmallGroup library. For context, there are groups of order 54. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(54,6)

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

gap> G := SmallGroup(54,6);

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

IdGroup(G) = [54,6]

or just do:

IdGroup(G)

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