Element structure of groups of order 16

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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]

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]

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]

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]

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]

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