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)[1];;
gap> G := SemidirectProduct(B,H);