# M81

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## Definition

This group is defined as the semidirect product of the cyclic group of order 27 with the unique cyclic subgroup of order 3 in its automorphism group.

## Arithmetic functions

Function Value Explanation
order 81
exponent 27
Frattini length 3
derived length 2
nilpotency class 2
subgroup rank 2
minimum size of generating set 2
rank as p-group 2
normal rank as p-group 2

## Group properties

Property Satisfied Explanation
group of prime power order Yes
abelian group No
nilpotent group Yes
group of nilpotency class two Yes
maximal class group No

## GAP implementation

### Group ID

This finite group has order 81 and has ID 6 among the groups of order 81 in GAP's SmallGroup library. For context, there are 15 groups of order 81. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(81,6)

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

gap> G := SmallGroup(81,6);

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

IdGroup(G) = [81,6]

or just do:

IdGroup(G)

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

### Other descriptions

The group can be constructed using the following series of GAP instructions (note that double semicolons are to suppress output display after each command):

```gap> H := CyclicGroup(27);;
gap> A := AutomorphismGroup(H);;
gap> B := Filtered(NormalSubgroups(A),K->Order(K) = 3);;
gap> G := SemidirectProduct(B,H);```