SmallGroup(81,3)

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined by the following presentation:

$G:=\langle a,b,c\mid a^{9}=b^{3}=c^{3}=e,ab=ba,bc=cb,cac^{-1}=ab\rangle$

Arithmetic functions

Want to compare with other groups of the same order? Check out groups of order 81#Arithmetic functions

Function	Value	Explanation
order	81
exponent	9
Frattini length	2
Fitting length	1
derived length	2
nilpotency class	2
minimum size of generating set	2
rank as p-group	3
normal rank as p-group	3
characteristic rank as p-group	3
number of conjugacy classes	33
number of subgroups	41
number of conjugacy classes of subgroups	23

Group properties

Want to compare with other groups of the same order? Check out groups of order 81#Group properties.

Property	Satisfied	Explanation
group of prime power order	Yes
abelian group	No
nilpotent group	Yes	prime power order implies nilpotent
group of nilpotency class two	Yes
metabelian group	Yes
T-group	No
ambivalent group	No	odd-order and ambivalent implies trivial

Families

This group is part of the family SmallGroup(p^4,3).

GAP implementation

Group ID

This finite group has order 81 and has ID 3 among the groups of order 81 in GAP's SmallGroup library. For context, there are groups of order 81. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(81,3)

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

gap> G := SmallGroup(81,3);

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

IdGroup(G) = [81,3]

or just do:

IdGroup(G)

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

Other descriptions

gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^9,F.2^3,F.3^3,F.1*F.2*F.1^(-1)*F.2^(-1),F.2*F.3*F.2^(-1)*F.3^(-1),F.3*F.1*F.3^(-1)*F.1^(-1)*F.2^(-1)];
<fp group on the generators [ f1, f2, f3 ]>