# 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

## 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 centralizerBounding facts: size of conjugacy class is bounded by order of derived subgroupCounting 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 | |

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 | |

3 | 2 | 3 | 2 | 7 | 3 | dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16 | 7, 8, 9 | 12--14 |

## 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.

## 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 or is odd, while all the other numbers are even. The total number of roots is even for all .

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 cumulative statistics, which gives the number of 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 |

### Group properties based on order statistics

### 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 |

### 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 , and generalized Baer IIP Lie rings have the same number of elements of order , 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 , and generalized Baer IIP Lie rings have the same number of elements of order , 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 |

### 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 |

## 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 |