SmallGroup(81,8)

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Definition

This is a group of order 81 given by the following presentation (with $e$ denoting the identity element):

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

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	81	groups with same order
exponent of a group	9	groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of order	4	groups with same prime-base logarithm of order
prime-base logarithm of exponent	2	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class	3	groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length	2	groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length
Frattini length	2	groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
rank of a p-group	2	groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group
normal rank of a p-group	2	groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group

Elements

Further information: element structure of SmallGroup(81,8)

1-isomorphism

The group is 1-isomorphic to the abelian group direct product of Z9 and E9. In other words, there is a bijection between the groups that restricts to an isomorphism on cyclic subgroups of both sides.

GAP implementation

Group ID

This finite group has order 81 and has ID 8 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(81,8)

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

gap> G := SmallGroup(81,8);

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

IdGroup(G) = [81,8]

or just do:

IdGroup(G)

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

Description by presentation

The group can be described using a presentation by means of the following GAP command/code:

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.3*F.1*F.3^(-1)*F.1^(-1)*F.2^(-1),F.3*F.2*F.3^(-1)*F.1^(-3)*F.2^(-1)];
<fp group on the generators [ f1, f2, f3 ]>
gap> IdGroup(G);
[ 81, 8 ]