SmallGroup(243,20)

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Definition

This group of order 243 can be defined by means of the following presentation:

G := \langle a_1,a_2,a_3 \mid a_1^{27} = a_2^3 = a_3^3 = e, [a_1,a_2] = a_3, [a_1,a_3] = e, [a_2,a_3] = a_1^{-9} \rangle

Here, e denotes the identity element and [ \ , \ ] denotes the commutator of two elements. Note that the left and right conventions for the commutator give different presentations but define isomorphic groups.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 243#Arithmetic functions
Function Value Similar groups Explanation
underlying prime of p-group 3
order (number of elements, equivalently, cardinality or size of underlying set) 243 groups with same order
prime-base logarithm of order 5 groups with same prime-base logarithm of order
exponent of a group 27 groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent 3 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 3 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

GAP implementation

Group ID

This finite group has order 243 and has ID 20 among the groups of order 243 in GAP's SmallGroup library. For context, there are groups of order 243. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(243,20)

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

gap> G := SmallGroup(243,20);

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

IdGroup(G) = [243,20]

or just do:

IdGroup(G)

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


Description by presentation

gap> F := FreeGroup(3);;
gap> G :=F/[F.1^(27),F.2^3,F.3^3,Comm(F.1,F.2)*F.3^(-1),Comm(F.1,F.3),Comm(F.2,F.3)*F.1^(9)];
<fp group on the generators [ f1, f2, f3 ]>
gap> IdGroup(G);
[ 243, 20 ]