M81
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Contents
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 :
gap> G := SmallGroup(81,6);
Conversely, to check whether a given group 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);