# Dihedral group:D16

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## Contents

## Definition

### Definition by presentation

The **dihedral group** , sometimes denoted , also called the **dihedral group of order sixteen** or the **dihedral group of degree eight** or the **dihedral group acting on eight elements**, is a dihedral group defined by the presentation:

.

where is the identity element.

The element is termed a *rotation* or a *generator of the cyclic piece* and the element is termed a *reflection*.

Note that the notation is more commonly used to denote the dihedral group of order eight.

### Geometric definition

The **dihedral group** (also called ) is defined as the group of all symmetries of the regular octagon. This has a cyclic subgroup comprising rotations (which is the cyclic subgroup generated by ) and has eight *reflections* each being an involution: four reflections about lines joining midpoints of opposite sides, and four reflections about diagonals.

### Definition as a permutation group

The group can be defined as the subgroup of the symmetric group on generated by a cycle of length and a *reflection*:

## Elements

`Further information: element structure of dihedral group:D16`

### Upto conjugacy

There are seven conjugacy classes of elements:

- The identity element. (1)
- The rotation by , which is given by in the presentation. (1)
- The rotations by , which are given by in the presentation. (2)
- The rotations by , which are given by in the presentation. (2)
- The rotations by , which are given by in the presentation. (2)
- The rotations about axes joining opposite vertices, given by in the presentation. (4)
- The rotations about axes joining midpoints of opposite sides, given by in the presentation. (4)

### Upto automorphism

Under the action of outer automorphisms, the conjugacy classes (4) and (5) combine, and the conjugacy classes (6) and (7) combine. Thus, there are five equivalence classes of sizes .

## Arithmetic functions

### Basic arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

### Arithmetic functions of a counting nature

### Lists of numerical invariants

List | Value | Explanation/comment |
---|---|---|

conjugacy class sizes | See element structure of dihedral group:D16 and element structure of dihedral groups | |

order statistics | See element structure of dihedral group:D16 and element structure of dihedral groups | |

orders of subgroups | , (9 times), | See subgroup structure of dihedral group:D16 and subgroup structure of dihedral groups |

degrees of irreducible representations | See linear representation theory of dihedral group:D16 and linear representation theory of dihedral groups |

## Group properties

Property | Satisfied? | Explanation | Comment |
---|---|---|---|

group of prime power order | Yes | ||

nilpotent group | Yes | prime power order implies nilpotent | |

supersolvable group | Yes | via nilpotent: finite nilpotent implies supersolvable | |

solvable group | Yes | via nilpotent: nilpotent implies solvable | |

Abelian group | No | and don't commute | |

Metacyclic group | Yes | Cyclic normal subgroup of order eight, cyclic quotient of order two | |

T-group | No | , which is normal, but is not normal | |

Monolithic group | Yes | Unique minimal normal subgroup of order two | |

One-headed group | No | Three distinct maximal normal subgroups of order eight | |

SC-group | No | ||

ACIC-group | Yes | Every automorph-conjugate subgroup is characteristic | |

Rational group | No | The elements generate the same cyclic group but are not conjugate | Smallest order for a non-abelian -group that is not rational. |

Rational-representation group | No | The representation as symmetries of a regular octagon is not rational | |

Ambivalent group | Yes | dihedral groups are ambivalent | Also see generalized dihedral groups are ambivalent. |

Directly indecomposable group | Yes | ||

Centrally indecomposable group | Yes | ||

Splitting-simple group | No |

## Elements

`Further information: element structure of dihedral group:D16`

## Subgroups

`Further information: subgroup structure of dihedral group:D16`

In case a single equivalence class of subgroups under automorphisms comprises multiple conjugacy classes of subgroups, outer curly braces are used to bucket the conjugacy classes.

Automorphism class of subgroups | List of subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes (=1 iff automorph-conjugate subgroup) | Size of each conjugacy class (=1 iff normal subgroup) | Total number of subgroups (=1 iff characteristic subgroup) | Isomorphism class of quotient (if exists) | Subnormal depth | Nilpotency class |
---|---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 16 | 1 | 1 | 1 | dihedral group:D16 | 1 | 0 | |

center of dihedral group:D16 | cyclic group:Z2 | 2 | 8 | 1 | 1 | 1 | dihedral group:D8 | 1 | 1 | |

non-normal order two subgroups of dihedral group:D16 | |
cyclic group:Z2 | 2 | 8 | 2 | 4 | 8 | -- | 3 | 1 |

derived subgroup of dihedral group:D16 | cyclic group:Z4 | 4 | 4 | 1 | 1 | 1 | Klein four-group | 1 | 1 | |

Klein four-subgroups of dihedral group:D16 | |
Klein four-group | 4 | 4 | 2 | 2 | 4 | -- | 2 | 1 |

Cyclic maximal subgroup of dihedral group:D16 | cyclic group:Z8 | 8 | 2 | 1 | 1 | 1 | cyclic group:Z2 | 1 | 1 | |

D8 in D16 | , | dihedral group:D8 | 8 | 2 | 2 | 1 | 2 | cyclic group:Z2 | 1 | 2 |

whole group | all elements | dihedral group:D16 | 16 | 1 | 1 | 1 | 1 | trivial group | 0 | 3 |

Total (8 rows) | -- | -- | -- | -- | 11 | -- | 19 | -- | -- | -- |

## GAP implementation

### Group ID

This finite group has order 16 and has ID 7 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(16,7)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(16,7);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [16,7]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Hall-Senior number

This group of prime power order has order 16 and has Hall-Senior number 12 among the groups of order 16. This information can be used to construct the group in GAP using the Gap3CatalogueGroup function as follows:

`Gap3CatalogueGroup(16,12)`

WARNING: There is some disagreement between the GAP 3 catalogue numbers and the Hall-Senior numbers for some abelian groups, but it does not affect this group.

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := Gap3CatalogueGroup(16,12);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's Gap3CatalogueIdGroup function:

`Gap3CatalogueIdGroup(G) = [16,12]`

or just do:

`Gap3CatalogueIdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

Description | Functions used |
---|---|

DihedralGroup(16) |
DihedralGroup |