Element structure of groups of order 16

This article gives specific information, namely, element structure, about a family of groups, namely: groups of order 16.
View element structure of group families | View element structure of groups of a particular order |View other specific information about groups of order 16

Group	Second part of GAP ID	Hall-Senior number	Nilpotency class	Element structure page
Cyclic group:Z16	1	5	1	element structure of cyclic group:Z16
Direct product of Z4 and Z4	2	4	1	element structure of direct product of Z4 and Z4
SmallGroup(16,3)	3	9	2	element structure of SmallGroup(16,3)
Nontrivial semidirect product of Z4 and Z4	4	10	2	element structure of nontrivial semidirect product of Z4 and Z4
Direct product of Z8 and Z2	5	3	1	element structure of direct product of Z8 and Z2
M16	6	11	2	element structure of M16
Dihedral group:D16	7	12	3	element structure of dihedral group:D16
Semidihedral group:SD16	8	13	3	element structure of semidihedral group:SD16
Generalized quaternion group:Q16	9	14	3	element structure of generalized quaternion group:Q16
Direct product of Z4 and V4	10	2	1	element structure of direct product of Z4 and V4
Direct product of D8 and Z2	11	6	2	element structure of direct product of D8 and Z2
Direct product of Q8 and Z2	12	7	2	element structure of direct product of Q8 and Z2
Central product of D8 and Z4	13	8	2	element structure of central product of D8 and Z4
Elementary abelian group:E16	14	1	1	element structure of elementary abelian group:E16

Conjugacy class sizes

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

Because of the small order, it turns out that the nilpotency class completely determines the number of conjugacy classes of each size. Further information: Nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order

In particular, we have:

Nilpotency class	Number of conjugacy classes of size 1	Number of conjugacy classes of size 2	Number of conjugacy classes of size 4	Total number of conjugacy classes	Number of groups	List of groups	List of GAP IDs	List of Hall-Senior numbers	List of Hall-Senior families
1	16	0	0	16	5	cyclic group:Z16, direct product of Z4 and Z4, direct product of Z8 and Z2, direct product of Z4 and V4, elementary abelian group:E16	1, 2, 5, 10, 14	1--5	$\Gamma_1$
2	4	6	0	10	6	SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16, direct product of D8 and Z2, direct product of Q8 and Z2, central product of D8 and Z4	3, 4, 6, 11, 12, 13	6--11	$\Gamma_2$
3	2	3	2	7	3	dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16	7, 8, 9	12--14	$\Gamma_3$

Action of automorphisms and endomorphisms

Automorphism class sizes and conjugacy class sizes

The automorphism group acts on the group, permuting conjugacy classes, and the inner automorphism group sends every element to within its conjugacy class. We thus get an action of the outer automorphism group on the set of conjugacy classes.

In the table below, the column "Sizes of orbits of size 2 conjugacy classes" gives the sizes of the orbits under the action on size 2 conjugacy classes. Each orbit on elements is twice the size, and the row underneath gives that data. Similarly for orbits of size 4.

Group	Second part of GAP ID	Hall-Senior number	Nilpotency class	Automorphism group	Outer automorphism group	Sizes of orbits of size 1 conjugacy classes Sizes of orbits of elements in them	Sizes or orbits of size 2 conjugacy classes Sizes of orbits of elements in them	Sizes or orbits of size 4 conjugacy classes Sizes of orbits of elements in them	Total number of conjugacy classes Total number of elements	Total number of orbits under automorphism group
cyclic group:Z16	1	5	1	direct product of Z4 and Z2	direct product of Z4 and Z2	1,1,2,4,8 1,1,2,4,8	no orbits	no orbits	16 16	5
direct product of Z4 and Z4	2	4	1			1,3,12 1,3,12	no orbits	no orbits	16 16	3
SmallGroup(16,3)	3	9	2			1,1,2 1,1,2	2,4 4,8	no orbits	10 16	5
nontrivial semidirect product of Z4 and Z4	4	10	2			1,1,1,1 1,1,1,1	2,4 4,8	no orbits	10 16	6
direct product of Z8 and Z2	5	3	1			1,1,2,2,2,8 1,1,2,2,2,8	no orbits	no orbits	16 16	6
M16	6	11	2			1,1,2 1,1,2	1,1,4 2,2,8	no orbits	10 16	6
dihedral group:D16	7	12	3			1,1 1,1	1,2 2,4	2 8	7 16	5
semidihedral group:SD16	8	13	3			1,1 1,1	1,2 2,4	1,1 4,4	7 16	6
generalized quaternion group:Q16	9	14	3			1,1 1,1	1,2 2,4	2 8	7 16	5
direct product of Z4 and V4	10	2	1			1,1,6,8 1,1,6,8	no orbits	no orbits	16 16	4
direct product of D8 and Z2	11	6	2			1,1,2 1,1,2	2,4 4,8	no orbits	10 16	5
direct product of Q8 and Z2	12	7	2			1,1,2 1,1,2	6 12	no orbits	10 16	4
central product of D8 and Z4	13	8	2			1,1,2 1,1,2	3,3 6,6	no orbits	10 16	5
elementary abelian group:E16	14	1	1			1,15 1,15	no orbits	no orbits	16 16	2

Grouping by automorphism class sizes

Automorphism class sizes (i.e., orbit sizes under automorphism group)	Number of orbits under automorphism group	Number of groups	List of groups	List of GAP IDs (second part)
1,1,2,4,8	5	5	cyclic group:Z16, SmallGroup(16,3), dihedral group:D16, generalized quaternion group:Q16, direct product of D8 and Z2	1,3,7,9,11
1,3,12	3	1	direct product of Z4 and Z4	2
1,1,1,1,4,8	6	1	nontrivial semidirect product of Z4 and Z4	4
1,1,2,2,2,8	6	2	direct product of Z8 and Z2, M16	5,6
1,1,2,4,4,4	6	1	semidihedral group:SD16	8
1,1,6,8	4	1	direct product of Z4 and V4	10
1,1,2,12	4	1	direct product of Q8 and Z2	12
1,1,2,6,6	5	1	central product of D8 and Z4	13
1,15	2	1	elementary abelian group:E16	14

1-isomorphism

Recall that a 1-isomorphism of groups is a bijection that induces an isomorphism on cyclic subgroups on either side. Two groups are 1-isomorphic if they admit a 1-isomorphism between them.

Of the 14 groups of order 16, 10 are not 1-isomorphic to any other group. The remaining four come in pairs. These two pairs are described below.

Non-abelian member of pair	GAP ID	Abelian member of pair	GAP ID	The 1-isomorphism arises as a ...	Description of the 1-isomorphism	Best perspective	Alternative perspective
central product of D8 and Z4	13	direct product of Z4 and V4	10	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring	generalized Baer correspondence between central product of D8 and Z4 and direct product of Z4 and V4	second cohomology group for trivial group action of V4 on Z4#Generalized Baer Lie rings	second cohomology group for trivial group action of E8 on Z2#Direct sum decomposition
M16	6	direct product of Z8 and Z2	5	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring	generalized Baer correspondence between M16 and direct product of Z8 and Z2	second cohomology group for trivial group action of V4 on Z4#Generalized Baer Lie rings	second cohomology group for trivial group action of direct product of Z4 and Z2 on Z2#Generalized Baer Lie rings

Order statistics

FACTS TO CHECK AGAINST:

ORDER STATISTICS (cf. order statistics, order statistics-equivalent finite groups): number of nth roots is a multiple of n | Finite abelian groups with the same order statistics are isomorphic | Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring | Frobenius conjecture on nth roots

1-ISOMORPHISM (cf. 1-isomorphic groups): Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring | order statistics-equivalent not implies 1-isomorphic

Order statistics raw data

Note that because number of nth roots is a multiple of n, we see that the number of elements whose order is $1$ or $2$ is odd, while all the other numbers are even. The total number of $n^{th}$ roots is even for all $n = 2^k, k \ge 1$.

Group	Second part of GAP ID	Hall-Senior number	Number of elements of order 1	Number of elements of order 2	Number of elements of order 4	Number of elements of order 8	Number of elements of order 16
Cyclic group:Z16	1	5	1	1	2	4	8
Direct product of Z4 and Z4	2	4	1	3	12	0	0
SmallGroup(16,3)	3	9	1	7	8	0	0
Nontrivial semidirect product of Z4 and Z4	4	10	1	3	12	0	0
Direct product of Z8 and Z2	5	3	1	3	4	8	0
M16	6	11	1	3	4	8	0
Dihedral group:D16	7	12	1	9	2	4	0
Semidihedral group:SD16	8	13	1	5	6	4	0
Generalized quaternion group:Q16	9	14	1	1	10	4	0
Direct product of Z4 and V4	10	2	1	7	8	0	0
Direct product of D8 and Z2	11	6	1	11	4	0	0
Direct product of Q8 and Z2	12	7	1	3	12	0	0
Central product of D8 and Z4	13	8	1	7	8	0	0
Elementary abelian group:E16	14	1	1	15	0	0	0

Here are the GAP commands to generate these order statistics: [SHOW MORE]

gap> F := List(AllSmallGroups(16),G -> List(Set(G),Order));;
gap> K := List(F,L->[Length(Filtered(L,x -> x = 1)),
> Length(Filtered(L,x -> x = 2)),Length(Filtered(L,x -> x = 4)),
> Length(Filtered(L,x -> x = 8)),Length(Filtered(L,x -> x = 16))]);;
gap> M := List([1..14], i ->[i,K[i]]);

GAP's output is:

[ [ 1, [ 1, 1, 2, 4, 8 ] ], [ 2, [ 1, 3, 12, 0, 0 ] ], [ 3, [ 1, 7, 8, 0, 0 ] ], [ 4, [ 1, 3, 12, 0, 0 ] ], [ 5, [ 1, 3, 4, 8, 0 ] ],
  [ 6, [ 1, 3, 4, 8, 0 ] ], [ 7, [ 1, 9, 2, 4, 0 ] ], [ 8, [ 1, 5, 6, 4, 0 ] ], [ 9, [ 1, 1, 10, 4, 0 ] ], [ 10, [ 1, 7, 8, 0, 0 ] ],
  [ 11, [ 1, 11, 4, 0, 0 ] ], [ 12, [ 1, 3, 12, 0, 0 ] ], [ 13, [ 1, 7, 8, 0, 0 ] ], [ 14, [ 1, 15, 0, 0, 0 ] ] ]

A shorter version of this can be done by first writing the function OrderStatistics and then writing:

List(AllSmallGroups(16),G -> [IdGroup(G)[2],OrderStatistics(G)]);

which yields the same output.

Here are the cumulative statistics, which gives the number of $n^{th}$ roots:

Group	Second part of GAP ID	Hall-Senior number	Number of 1st roots	Number of 2nd roots	Number of 4th roots	Number of 8th rots	Number of 16th roots
Cyclic group:Z16	1	5	1	2	4	8	16
Direct product of Z4 and Z4	2	4	1	4	16	16	16
SmallGroup(16,3)	3	9	1	8	16	16	16
Nontrivial semidirect product of Z4 and Z4	4	10	1	4	16	16	16
Direct product of Z8 and Z2	5	3	1	4	8	16	16
M16	6	11	1	4	8	16	16
Dihedral group:D16	7	12	1	10	12	16	16
Semidihedral group:SD16	8	13	1	6	12	16	16
Generalized quaternion group:Q16	9	14	1	2	12	16	16
Direct product of Z4 and V4	10	2	1	8	16	16	16
Direct product of D8 and Z2	11	6	1	12	16	16	16
Direct product of Q8 and Z2	12	7	1	4	16	16	16
Central product of D8 and Z4	13	8	1	8	16	16	16
Elementary abelian group:E16	14	1	1	16	16	16	16

Here are the GAP commands to generate these cumulative statistics: [SHOW MORE]

gap> F := List(AllSmallGroups(16),G -> List(Set(G),Order));;
gap> J := List(F,L->[Length(Filtered(L,x -> x = 1)),
> Length(Filtered(L,x -> x <= 2)),Length(Filtered(L,x -> x <= 4)),
> Length(Filtered(L,x -> x <= 8)),Length(Filtered(L,x -> x <= 16))]);;
gap> N := List([1..14], i ->[i,J[i]]);

Here is GAP's output:

[ [ 1, [ 1, 2, 4, 8, 16 ] ], [ 2, [ 1, 4, 16, 16, 16 ] ], [ 3, [ 1, 8, 16, 16, 16 ] ], [ 4, [ 1, 4, 16, 16, 16 ] ], [ 5, [ 1, 4, 8, 16, 16 ] ],
  [ 6, [ 1, 4, 8, 16, 16 ] ], [ 7, [ 1, 10, 12, 16, 16 ] ], [ 8, [ 1, 6, 12, 16, 16 ] ], [ 9, [ 1, 2, 12, 16, 16 ] ], [ 10, [ 1, 8, 16, 16, 16 ] ],
  [ 11, [ 1, 12, 16, 16, 16 ] ], [ 12, [ 1, 4, 16, 16, 16 ] ], [ 13, [ 1, 8, 16, 16, 16 ] ], [ 14, [ 1, 16, 16, 16, 16 ] ] ]

We can also use the OrderStatisticsCumulative function, followed by:

List(AllSmallGroups(16),G -> [IdGroup(G)[2],OrderStatisticsCumulative(G)]);

Group properties based on order statistics

Group	GAP ID	Hall-Senior number	abelian group of prime power order	group of prime power order admitting a bijective quasihomomorphism to an abelian group	group of prime power order 1-isomorphic to an abelian group	group of prime power order order statistics-equivalent to an abelian group	finite p-group in which the number of nth roots is a power of p for all n	order statistics-unique finite group
Cyclic group:Z16	1	5	Yes	Yes	Yes	Yes	Yes	Yes
Direct product of Z4 and Z4	2	4	Yes	Yes	Yes	Yes	Yes	No
SmallGroup(16,3)	3	9	No	No	No	Yes	Yes	No
Nontrivial semidirect product of Z4 and Z4	4	10	No	No	No	Yes	Yes	No
Direct product of Z8 and Z2	5	3	Yes	Yes	Yes	Yes	Yes	No
M16	6	11	No	Yes	Yes	Yes	Yes	No
Dihedral group:D16	7	12	No	No	No	No	No	Yes
Semidihedral group:SD16	8	13	No	No	No	No	No	Yes
Generalized quaternion group:Q16	9	14	No	No	No	No	No	Yes
Direct product of Z4 and V4	10	2	Yes	Yes	Yes	Yes	Yes	No
Direct product of D8 and Z2	11	6	No	No	No	No	No	Yes
Direct product of Q8 and Z2	12	7	No	No	No	Yes	Yes	No
Central product of D8 and Z4	13	8	No	Yes	Yes	Yes	Yes	No
Elementary abelian group:E16	14	1	Yes	Yes	Yes	Yes	Yes	Yes

Equivalence classes based on order statistics

Here, we discuss the equivalence classes of groups of order 16 up to being order statistics-equivalent finite groups and up to the stronger notion of being 1-isomorphic groups (which means there is a bijection that restricts to isomorphisms on cyclic subgroups). See also order statistics-equivalent not implies 1-isomorphic.

Order statistics	Order statistics (cumulative)	Number of groups	Number of equivalence classes up to 1-isomorphism	Members of first equivalence class	Members of second equivalence class	Members of third equivalence class	Abelian group with these order statistics?	Cumulative order statistics all powers of 2?
1,1,2,4,8	1,2,4,8,16	1	1	Cyclic group:Z16 (ID:1)			Yes	Yes
1,1,10,4,0	1,2,12,16,16	1	1	Generalized quaternion group:Q16 (ID:9)			No	No
1,3,4,8,0	1,4,8,16,16	2	1	Direct product of Z8 and Z2 (ID:5) and M16 (ID:6)			Yes	Yes
1,3,12,0,0	1,4,16,16,16	3	3	Direct product of Z4 and Z4 (ID:2) (characterized by having three squares of order 2)	Nontrivial semidirect product of Z4 and Z4 (ID:4) (characterized by having two squares of order 2)	Direct product of Q8 and Z2 (ID:12)	Yes	Yes
1,5,6,4,0	1,6,12,16,16	1	1	Semidihedral group:SD16 (ID:8)			No	No
1,7,8,0,0	1,8,16,16,16	3	2	Direct product of Z4 and V4 (ID:10) and Central product of D8 and Z4 (ID:13) (characterized by having exactly one square of order 2)	SmallGroup(16,3) (ID:3)(characterized by having two squares of order 2)		Yes	Yes
1,9,2,4,0	1,10,12,16,16	1	1	Dihedral group:D16 (ID:7)			No	No
1,11,4,0,0	1,12,16,16,16	1	1	Direct product of D8 and Z2 (ID:11)			No	No
1,15,0,0,0	1,16,16,16,16	1	1	Elementary abelian group:E16 (ID:14)			Yes	Yes

Here are the GAP commands to produce a list sorted by order statistics: [SHOW MORE]

gap> F := List(AllSmallGroups(16),G -> List(Set(G),Order));;
gap> K := List(F,L->[Length(Filtered(L,x -> x = 1)),
> Length(Filtered(L,x -> x = 2)),Length(Filtered(L,x -> x = 4)),
> Length(Filtered(L,x -> x = 8)),Length(Filtered(L,x -> x = 16))]);;
gap> M := List([1..14], i ->[K[i],i]);;
gap> S := SortedList(M);

Here is GAP's output (note that this does not distinguish based on 1-isomorphism):

[ [ [ 1, 1, 2, 4, 8 ], 1 ], [ [ 1, 1, 10, 4, 0 ], 9 ], [ [ 1, 3, 4, 8, 0 ], 5 ], [ [ 1, 3, 4, 8, 0 ], 6 ], [ [ 1, 3, 12, 0, 0 ], 2 ],
  [ [ 1, 3, 12, 0, 0 ], 4 ], [ [ 1, 3, 12, 0, 0 ], 12 ], [ [ 1, 5, 6, 4, 0 ], 8 ], [ [ 1, 7, 8, 0, 0 ], 3 ], [ [ 1, 7, 8, 0, 0 ], 10 ],
  [ [ 1, 7, 8, 0, 0 ], 13 ], [ [ 1, 9, 2, 4, 0 ], 7 ], [ [ 1, 11, 4, 0, 0 ], 11 ], [ [ 1, 15, 0, 0, 0 ], 14 ] ]

Explanation of the equivalences

Some of the equivalences between order statistics arise on account of 1-isomorphisms, discussed in the 1-isomorphism section. Here, we focus on the equivalences in order statistics that do not arise from 1-isomorphisms. There are three such equivalences that need to be "explained":

Abelian member	Second part of GAP ID	Non-abelian member	Second part of GAP ID	The equivalence arises because of ...	More information on the equivalence/correspondence
direct product of Z4 and Z4	2	nontrivial semidirect product of Z4 and Z4	4	generalized Baer IIP Lie ring for central subgroup cyclic group:Z2 and quotient group direct product of Z4 and Z2. Note that because the exponent is $4$, and generalized Baer IIP Lie rings have the same number of elements of order $2$, the order statistics all match up.	See also second cohomology group for trivial group action of direct product of Z4 and Z2 on Z2
direct product of Z4 and V4	10	SmallGroup(16,3)	3	generalized Baer IIP Lie ring for central subgroup cyclic group:Z2 and quotient group direct product of Z4 and Z2. Note that because the exponent is $4$, and generalized Baer IIP Lie rings have the same number of elements of order $2$, the order statistics all match up.	See also second cohomology group for trivial group action of direct product of Z4 and Z2 on Z2
direct product of Z4 and Z4	2	direct product of Q8 and Z2	Mystery	Mystery

Conjugacy class-cum-order statistics

Nilpotency class one: abelian groups

There are 16 conjugacy classes of size 1. See the order statistics section for the order statistics of these groups.

Nilpotency class exactly two

There are 4 conjugacy classes of size 1, comprising the central elements, and 6 conjugacy classes of size two. Listed below are the conjugacy classes sorted by order. Note that the numbers listed are the numbers of conjugacy classes with the given size/order specifications. To obtain the number of elements, multiply by the size of the conjugacy class.

Group	Second part of GAP ID	Hall-Senior number	size 1, order 1	size 1, order 2	size 1, order 4	size 2, order 2	size 2, order 4	size 2, order 8
SmallGroup(16,3)	3	9	1	3	0	2	4	0
Nontrivial semidirect product of Z4 and Z4	4	10	1	3	0	0	6	0
M16	6	11	1	1	2	1	1	4
Direct product of D8 and Z2	11	6	1	3	0	4	2	0
Direct product of Q8 and Z2	12	7	1	3	0	0	6	0
Central product of D8 and Z4	13	8	1	1	2	3	3	0

Here is the GAP code to generate this: [SHOW MORE]

gap> B := Filtered(AllSmallGroups(16),G -> NilpotencyClassOfGroup(G) = 2);;
gap> F := List(B,G->SortedList(List(ConjugacyClasses(G),c -> [Size(c),Order(Representative(c))])));;
gap> K := List(F,L->[Length(Filtered(L,x->x=[1,1])),
> Length(Filtered(L,x->x=[1,2])),Length(Filtered(L,x->x=[1,4])),
> Length(Filtered(L,x->x=[2,2])),Length(Filtered(L,x->x=[2,4])),
> Length(Filtered(L,x->x=[2,8]))]);;
gap> List([1..6],i -> [IdGroup(B[i])[2],K[i]]);

GAP's output is:

[ [ 3, [ 1, 3, 0, 2, 4, 0 ] ], [ 4, [ 1, 3, 0, 0, 6, 0 ] ], [ 6, [ 1, 1, 2, 1, 1, 4 ] ], [ 11, [ 1, 3, 0, 4, 2, 0 ] ], [ 12, [ 1, 3, 0, 0, 6, 0 ] ],
  [ 13, [ 1, 1, 2, 3, 3, 0 ] ] ]

Nilpotency class exactly three

There are 2 conjugacy classes of size 1, comprising the central elements, 3 conjugacy classes of size 2, and 2 conjugacy classes of size 4. Listed below are the conjugacy classes sorted by order. Note that the numbers listed are the numbers of conjugacy classes with the given size/order specifications. To obtain the number of elements, multiply by the size of the conjugacy class.

Group	Second part of GAP ID	Hall-Senior number	size 1, order 1	size 1, order 2	size 2, order 2	size 2, order 4	size 2, order 8	size 4, order 2	size 4, order 4
Dihedral group:D16	7	12	1	1	0	1	2	2	0
Semidihedral group:SD16	8	13	1	1	0	1	2	1	1
Generalized quaternion group:Q16	9	14	1	1	0	1	2	0	2

Here is the GAP code to generate this: [SHOW MORE]

gap> B := Filtered(AllSmallGroups(16),G -> NilpotencyClassOfGroup(G) = 3);;
gap> F := List(B,G->SortedList(List(ConjugacyClasses(G),c -> [Size(c),Order(Representative(c))])));;
gap> K := List(F,L->[Length(Filtered(L,x->x=[1,1])),
> Length(Filtered(L,x->x=[1,2])),Length(Filtered(L,x->x=[2,2])),
> Length(Filtered(L,x->x=[2,4])),Length(Filtered(L,x->x=[2,8])),
> Length(Filtered(L,x->x=[4,2])),Length(Filtered(L,x->x=[4,4]))]);;
gap> List([1..3],i -> [IdGroup(B[i])[2],K[i]]);

Here is GAP's output:

[ [ 7, [ 1, 1, 0, 1, 2, 2, 0 ] ], [ 8, [ 1, 1, 0, 1, 2, 1, 1 ] ], [ 9, [ 1, 1, 0, 1, 2, 0, 2 ] ] ]

Power statistics

Cumulative power statistics

Note that the number of first powers is always 16 and the number of 16th powers is always 1.

Group	Second part of GAP ID	Hall-Senior number	Number of 1st powers	Number of squares	Number of fourth powers	Number of eighth powers	Number of 16th powers
Cyclic group:Z16	1	5	16	8	4	2	1
Direct product of Z4 and Z4	2	4	16	4	1	1	1
SmallGroup(16,3)	3	9	16	3	1	1	1
Nontrivial semidirect product of Z4 and Z4	4	10	16	3	1	1	1
Direct product of Z8 and Z2	5	3	16	4	2	1	1
M16	6	11	16	4	2	1	1
Dihedral group:D16	7	12	16	4	2	1	1
Semidihedral group:SD16	8	13	16	4	2	1	1
Generalized quaternion group:Q16	9	14	16	4	2	1	1
Direct product of Z4 and V4	10	2	16	2	1	1	1
Direct product of D8 and Z2	11	6	16	2	1	1	1
Direct product of Q8 and Z2	12	7	16	2	1	1	1
Central product of D8 and Z4	13	8	16	2	1	1	1
Elementary abelian group:E16	14	9	16	1	1	1	1

Non-cumulative power statistics

Group	Second part of GAP ID	Hall-Senior number	Number of non-squares	Number of squares that are not 4th powers	Number of fourth powers that are not eight powers	Number of eighth powers that are not sixteenth powers	Number of sixteenth powers
Cyclic group:Z16	1	5	8	4	2	1	1
Direct product of Z4 and Z4	2	4	12	3	0	0	1
SmallGroup(16,3)	3	9	13	2	0	0	1
Nontrivial semidirect product of Z4 and Z4	4	10	13	2	0	0	1
Direct product of Z8 and Z2	5	3	12	2	1	0	1
M16	6	11	12	2	1	0	1
Dihedral group:D16	7	12	12	2	1	0	1
Semidihedral group:SD16	8	13	12	2	1	0	1
Generalized quaternion group:Q16	9	14	12	2	1	0	1
Direct product of Z4 and V4	10	2	14	1	0	0	1
Direct product of D8 and Z2	11	6	14	1	0	0	1
Direct product of Q8 and Z2	12	7	14	1	0	0	1
Central product of D8 and Z4	13	8	14	1	0	0	1
Elementary abelian group:E16	14	9	15	0	0	0	1

Equivalence classes based on power statistics

Here, we discuss the equivalence classes of groups of order 16 up to being power statistics-equivalent finite groups and up to the stronger notion of being 1-isomorphic groups (which means there is a bijection that restricts to isomorphisms on cyclic subgroups). See also power statistics-equivalent not implies 1-isomorphic.

Power statistics (cumulative)	Power statistics (non-cumulative)	Number of groups	Number of equivalence classes up to 1-isomorphism	Members of first equivalence class	Members of second equivalence class	Members of third equivalence class	Members of fourth equivalence class	Abelian group with these power statistics?
16,8,4,2,1	8,4,2,1,1	1	1	Cyclic group:Z16 (GAP ID:1)				Yes
16,4,1,1,1	12,3,0,0,1	1	1	Direct product of Z4 and Z4 (GAP ID:2)				Yes
16,3,1,1,1	13,2,0,0,1	2	2	SmallGroup(16,3) (GAP ID:3)	Nontrivial semidirect product of Z4 and Z4 (GAP ID:4)			No
16,4,2,1,1	12,2,1,0,1	5	4	Direct product of Z8 and Z2 (GAP ID:5) and M16 (GAP ID:6)	Dihedral group:D16 (GAP ID:7)	Semidihedral group:SD16 (GAP ID:8)	Generalized quaternion group (GAP ID:9)	Yes
16,2,1,1,1	14,1,0,0,1	4	3	Direct product of Z4 and V4 (GAP ID:10) and Central product of D8 and Z4 (GAP ID:13)	Direct product of D8 and Z2 (GAP ID:11)	Direct product of Q8 and Z2 (GAP ID:12)		Yes
16,1,1,1,1	15,0,0,0,1	1	1	Elementary abelian group:E16 (GAP ID:14)				Yes