Template:Dihedral 2-group arithmetic function table

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	GAP verification (set G := DihedralGroup({{{order}}});) -- See more at #GAP implementation
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.		Order(G); using Order.
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<ul><li>"{{{degree}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same exponent of a group"{{{degree}}}" is not a number.	cyclic subgroup of order {{{degree}}}.	Exponent(G); using Exponent.
prime-base logarithm of exponent	{{{degree p-log}}}	groups with same order and prime-base logarithm of exponent<ul><li>"{{{degree p-log}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same prime-base logarithm of order and prime-base logarithm of exponent<ul><li>"{{{degree p-log}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same prime-base logarithm of exponent"{{{degree p-log}}}" is not a number.
nilpotency class	{{{degree p-log}}}	groups with same order and nilpotency class<ul><li>"{{{degree p-log}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same prime-base logarithm of order and nilpotency class<ul><li>"{{{degree p-log}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same nilpotency class"{{{degree p-log}}}" is not a number.		NilpotencyClassOfGroup(G); using NilpotencyClassOfGroup.
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	DerivedLength(G); using DerivedLength.
Frattini length	{{{degree p-log}}}	groups with same order and Frattini length<ul><li>"{{{degree p-log}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same prime-base logarithm of order and Frattini length<ul><li>"{{{degree p-log}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same Frattini length"{{{degree p-log}}}" is not a number.	The first Frattini subgroup is cyclic generated by $a^{2}$, each subsequent Frattini subgroup moves to the set of squares of its predecessor.	FrattiniLength(G); using FrattiniLength.
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		Rank(G); using Rank.
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.	SubgroupRank(G); using SubgroupRank.
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.	RankAsPGroup(G); using RankAsPGroup.
normal rank of a p-group	1	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.	NormalRank(G); using NormalRank.
characteristic rank of a p-group	1	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	All abelian characteristic subgroups are cyclic.	CharacteristicRank(G); using CharacteristicRank.