Template:M-type 2-group arithmetic function table

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Want to compare and contrast arithmetic function values with other groups of the same order? Check out [[groups of order {{{order}}}#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) {{{order}}} groups with same order
"{{{order}}}" is not a number.
prime-base logarithm of order {{{order p-log}}} groups with same prime-base logarithm of order
"{{{order p-log}}}" is not a number.
max-length of a group {{{order p-log}}} max-length of a group equals prime-base logarithm of order for group of prime power order
chief length {{{order p-log}}} chief length equals prime-base logarithm of order for group of prime power order
composition length {{{order p-log}}} composition length equals prime-base logarithm of order for group of prime power order
exponent of a group {{{degree}}} groups with same order and exponent of a group
  • "{{{degree}}}" is not a number.
  • "{{{order}}}" is not a number.
| groups with same exponent of a group
"{{{degree}}}" is not a number.
cyclic subgroup of order {{{degree}}}.
prime-base logarithm of exponent {{{degree p-log}}} groups with same order and prime-base logarithm of exponent
  • "{{{degree p-log}}}" is not a number.
  • "{{{order}}}" is not a number.
| groups with same prime-base logarithm of order and prime-base logarithm of exponent
  • "{{{degree p-log}}}" is not a number.
  • "{{{order p-log}}}" is not a number.
| groups with same prime-base logarithm of exponent
"{{{degree p-log}}}" is not a number.
nilpotency class 2 groups with same order and nilpotency class
"{{{order}}}" is not a number.
| groups with same prime-base logarithm of order and nilpotency class
"{{{order p-log}}}" is not a number.
| groups with same nilpotency class
derived length 2 groups with same order and derived length
"{{{order}}}" is not a number.
| groups with same prime-base logarithm of order and derived length
"{{{order p-log}}}" is not a number.
| groups with same derived length
the derived subgroup is contained in the cyclic subgroup and is hence abelian
Frattini length {{{degree p-log}}} groups with same order and Frattini length
  • "{{{degree p-log}}}" is not a number.
  • "{{{order}}}" is not a number.
| groups with same prime-base logarithm of order and Frattini length
  • "{{{degree p-log}}}" is not a number.
  • "{{{order p-log}}}" is not a number.
| groups with same Frattini length
"{{{degree p-log}}}" is not a number.
minimum size of generating set 2 groups with same order and minimum size of generating set
"{{{order}}}" is not a number.
| groups with same prime-base logarithm of order and minimum size of generating set
"{{{order p-log}}}" is not a number.
| groups with same minimum size of generating set
subgroup rank of a group 2 groups with same order and subgroup rank of a group
"{{{order}}}" is not a number.
| groups with same prime-base logarithm of order and subgroup rank of a group
"{{{order p-log}}}" is not a number.
| groups with same subgroup rank of a group
All proper subgroups are cyclic, dihedral, or Klein four-groups.
rank of a p-group 2 groups with same order and rank of a p-group
"{{{order}}}" is not a number.
| groups with same prime-base logarithm of order and rank of a p-group
"{{{order p-log}}}" is not a number.
| groups with same rank of a p-group
there exist Klein four-subgroups.
normal rank of a p-group 2 groups with same order and normal rank of a p-group
"{{{order}}}" is not a number.
| groups with same prime-base logarithm of order and normal rank of a p-group
"{{{order p-log}}}" is not a number.
| groups with same normal rank of a p-group
all abelian normal subgroups are cyclic.
characteristic rank of a p-group 2 groups with same order and characteristic rank of a p-group
"{{{order}}}" is not a number.
| groups with same prime-base logarithm of order and characteristic rank of a p-group
"{{{order p-log}}}" is not a number.
| groups with same characteristic rank of a p-group
There is a unique (hence characteristic) Klein four-subgroup.