Burnside group:B(4,3)

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Definition

This group is defined as the Burnside group with parameters 4,3. In other words, it is the quotient of free group:F4 by the subgroup generated by all cubes in the group.

Arithmetic functions

Function	Value	Similar groups	Explanation
underlying prime of p-group	3
order (number of elements, equivalently, cardinality or size of underlying set)	4782969	groups with same order	As $B(n,3)$: $3^{n+{\binom {n}{2}}+{\binom {n}{3}}}=3^{4+{\binom {4}{2}}+{\binom {4}{3}}}=3^{4+6+4}=3^{14}=4782969$
prime-base logarithm of order	14	groups with same prime-base logarithm of order
max-length of a group	14		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	14		chief length equals prime-base logarithm of order for group of prime power order
composition length	14		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	3	groups with same order and exponent of a group | groups with same exponent of a group	Follows from definition as a Burnside group
prime-base logarithm of exponent	1	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	All groups $B(n,3)$, $n\geq 3$, have class precisely three. See also exponent three implies class three
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	Follows from nilpotency class being three
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	All groups $B(n,3)$, $n\geq 2$, have Frattini length 2