M16

Definition

The group, sometimes denoted $M_{16}$, is defined as follows:

$M_{16}=⟨a,x\mid a^8=x^2=e,xax=a^5⟩$.

Here, $e$ denotes the identity element.

Arithmetic functions

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	16	groups with same order
prime-base logarithm of order	4	groups with same prime-base logarithm of order
max-length of a group	4		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	4		chief length equals prime-base logarithm of order for group of prime power order
composition length	4		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	8	groups with same order and exponent of a group | groups with same prime-base logarithm of 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
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
nilpotency class	2	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
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
subgroup rank of a group	2	groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group
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
characteristic rank of a p-group	2	groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group

Group properties

Property	Satisfied	Explanation	Comment
Abelian group	No	$a,x$ do not commute
Nilpotent group	Yes	prime power order implies nilpotent
Metacyclic group	Yes
Supersolvable group	Yes
Solvable group	Yes

GAP implementation

Group ID

This group has ID $6$ among the groups of order sixteen. It can thus be defined using GAP's SmallGroup function as follows:

SmallGroup(16,6)