SmallGroup(81,8)

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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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^3b \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 15 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 ]