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 $B(n,3)$, $n \ge 3$, 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 $B(n,3)$, $n \ge 2$, have Frattini length 2