# Dihedral group:D8

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## Definition

### Definition by presentation

The **dihedral group** , sometimes called , also called the dihedral group of order eight or the dihedral group of degree four (since its natural action is on four elements), is defined by the following presentation:

Here, the element is termed the *rotation* or the *generator of the cyclic piece* and is termed the *reflection".*

### Geometric definition

The **dihedral group** (also called ) is defined as the group of all symmetries of the square (the regular 4-gon). This has a cyclic subgroup comprising rotations (which is the cyclic subgroup generated by ) and has four *reflections* each being an involution: reflections about lines joining midpoints of opposite sides, and reflections about diagonals.

### Definition as a permutation group

`Further information: D8 in S4`

The group is (up to isomorphism) the subgroup of the symmetric group on given by:

This can be related to the geometric definition by thinking of as the vertices of the square and considering an element of in terms of its induced action on the vertices. It relates to the presentation via setting and .

### Multiplication table

Here, denotes the identity element, is an element of order 4, and is an element of order two that isn't equal to , as in the above presentation.

Element | ||||||||
---|---|---|---|---|---|---|---|---|

### Other definitions

The dihedral group can be described in the following ways:

- The dihedral group of order eight.
- The generalized dihedral group corresponding to the cyclic group of order four.
- The holomorph of the cyclic group of order four.
- The external wreath product of the cyclic group of order two with the cyclic group of order two, acting via the regular action.
- The -Sylow subgroup of the symmetric group on four letters.
- The -Sylow subgroup of the symmetric group on five letters.
- The -Sylow subgroup of the alternating group on six letters.
- The -Sylow subgroup of PSL(3,2).

## Position in classifications

Type of classification | Name in that classification |
---|---|

GAP ID | (8,3), i.e., the third among the groups of order 8 |

Hall-Senior number | 4 among groups of order 8 |

Hall-Senior symbol |

## Elements

`Further information: element structure of dihedral group:D8`

Below, we list all the elements, also giving the interpretation of each element under the geometric description of the dihedral group as the symmetries of a 4-gon, and for the corresponding permutation representation (see D8 in S4). Note that for different conventions, one can obtain somewhat different correspondences, so this may not match up with other correspondences elsewhere. *Note that the descriptions below assume the left action convention for functions and the corresponding convention for composition, and hence some of the entries may become different if you adopt the right action convention.*:

Element in terms of and | Geometric description | Permutation on vertices | Order of the element |
---|---|---|---|

(identity element) | does nothing, i.e., leaves the square invariant | 1 | |

rotation by angle of (i.e., ) counterclockwise | 4 | ||

rotation by angle of (i.e., ), also called a half turn |
2 | ||

rotation by angle of (i.e., ) counter-clockwise, or equivalently, by (i.e., ) clockwise | 4 | ||

reflection about the diagonal joining vertices "2" and "4" | 2 | ||

reflection about the line joining midpoints of opposite sides "14" and "23" | 2 | ||

reflection about the diagonal joining vertices "1" and "3" | 2 | ||

reflection about the line joining midpoints of opposite sides "12" and "34" | 2 |

Below is the conjugacy and automorphism class structure:

Conjugacy class in terms of | Geometric description of conjugacy class | Conjugacy class as permutations | Size of conjugacy class | Order of elements in conjugacy class | Centralizer of first element of class |
---|---|---|---|---|---|

identity element, does nothing | 1 | 1 | whole group | ||

half turn, rotation by | 1 | 2 | whole group | ||

reflections about diagonals | 2 | 2 | -- one of the Klein four-subgroups of dihedral group:D8 | ||

reflections about lines joining midpoints of opposite sides | 2 | 2 | -- one of the Klein four-subgroups of dihedral group:D8 | ||

rotations by odd multiples of | 2 | 4 | -- the cyclic maximal subgroup of dihedral group:D8 | ||

Total (5) | -- | -- | 8 | -- | -- |

The equivalence classes up to automorphisms are:

Equivalence class under automorphisms in terms of | Geometric description of equivalence class | Equivalence class as permutations | Size of equivalence class | Number of conjugacy classes in it | Size of each conjugacy class |
---|---|---|---|---|---|

identity element, does nothing | 1 | 1 | 1 | ||

half turn | 1 | 1 | 1 | ||

reflections | 4 | 2 | 2 | ||

rotations by odd multiples of | 2 | 1 | 2 | ||

Total (4) | -- | -- | 8 | 5 | -- |

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 8#Arithmetic functions

### Arithmetic functions of a counting nature

### Lists of numerical invariants

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

conjugacy class sizes | Two central elements, all others in conjugacy classes of size two. See element structure of dihedral group:D8 and element structure of dihedral groups. | |

order statistics | Of the five elements of order two, one is central. The other four are automorphic to each other. See element structure of dihedral group:D8 and element structure of dihedral groups | |

degrees of irreducible representations | See linear representation theory of dihedral group:D8 | |

orders of subgroups | See subgroup structure of dihedral group:D8 |

Numerical invariants arising from subgroup series-defining functions:

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

prime-base logarithms of orders of successive quotient groups of upper central series | ||

prime-base logarithms of orders of successive quotient groups of lower central series | ||

prime-base logarithms of orders of successive quotient groups of derived series | ||

prime-base logarithms of orders of successive quotient groups of Frattini series |

### Action-based/automorphism group realization invariants

Function | Value | Explanation |
---|---|---|

minimum degree of faithful representation | 2 | |

minimum degree of nontrivial irreducible representation | 2 | |

smallest size of set with faithful action | 4 | |

smallest size of set with faithful transitive action | 4 | |

symmetric genus | ? |

## Group properties

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 8#Group properties

## Subgroups

`Further information: subgroup structure of dihedral group:D8`

The dihedral group has ten subgroups:

- The trivial subgroup (1)
- The center, which is the unique minimal normal subgroup, and is a two-element subgroup generated by . This is isomorphic to the cyclic group of order two. (1)
`Further information: Center of dihedral group:D8` - The two-element subgroups generated by , , and . All of these are isomorphic to the cyclic group of order two.These are in two conjugacy classes: the subgroups generated by and by form one conjugacy class; the subgroups generated by and form another conjugacy class. (4)
`Further information: 2-subnormal subgroups of dihedral group:D8` - The four-element subgroups and . Both are normal, related by an outer automorphism, and isomorphic to the Klein four-group. (2)
`Further information: Klein four-subgroups of dihedral group:D8` - The four-element subgroup generated by . This is normal and is isomorphic to the cyclic group of order four. (1)
`Further information: Cyclic maximal subgroup of dihedral group:D8` - The whole group. (1)

### Normal subgroups

All subgroups except those in header (3) above, are normal. The subgroups in header (3), which are two-element subgroups generated by some , are 2-subnormal, as each of these is contained in a Klein four-group. Of the subgroups in header (3), there are two conjugacy classes: one comprising the subgroups generated by and by , and the other comprising the subgroups generated by and by . (These conjugacy classes are related by an outer automorphism).

### Characteristic subgroups

The subgroups in headers (1), (2), (5) and (6) are characteristic. The subgroups in header (4) are normal but not characteristic, and in fact, the two subgroups are automorphs of each other.

## Quotient groups

- The group itself: this is obtained as the quotient by the trivial subgroup. (1)
- The Klein four-group, which is obtained as the quotient by the center. (1)
- The cyclic group:Z2, which is obtained as the quotient by either of the two Klein four-subgroups. (2)
- The cyclic group:Z2, which is obtained as the quotient by the cyclic maximal subgroup. (1)
- The trivial group, which is obtained as the quotient by the group itself. (1)

## Subgroup-defining functions

Subgroup-defining function | Subgroup type in list | Page on subgroup embedding | Isomorphism class | Comment |
---|---|---|---|---|

Center | (2) | Center of dihedral group:D8 | Cyclic group:Z2 | Prime power order implies not centerless |

Derived subgroup | (2) | Center of dihedral group:D8 | Cyclic group:Z2 | |

Frattini subgroup | (2) | Center of dihedral group:D8 | Cyclic group:Z2 | The three maximal subgroups of order four intersect here. |

Socle | (2) | Center of dihedral group:D8 | Cyclic group:Z2 | This subgroup is the unique minimal normal subgroup, i.e.,the monolith, and the group is monolithic. Also, minimal normal implies central in nilpotent. |

Join of abelian subgroups of maximum order | (6) | -- | whole group | The group is generated by abelian subgroups of maximum order. |

ZJ-subgroup | (2) | Center of dihedral group:D8 | Cyclic group:Z2 | Since the group equals the join of abelian subgroups of maximum order, the ZJ-subgroup equals the center. |

Join of abelian subgroups of maximum rank | (6) | -- | whole group | The group is generated by abelian subgroups of maximum rank. |

Join of elementary abelian subgroups of maximum order | (6) | -- | whole group | The group is generated by abelian subgroups of maximum rank. |

## Quotient-defining functions

Quotient-defining function | Isomorphism class | Comment |
---|---|---|

Inner automorphism group | Klein four-group | It is the quotient by the center, which is of order two. |

Abelianization | Klein four-group | It is the quotient by the commutator subgroup, which is cyclic of order two. |

Frattini quotient | Klein four-group | It is the quotient by the Frattini subgroup, which is cyclic of order two. |

## Other associated constructs

## In larger groups

### Occurrence as a subgroup

`Further information: Supergroups of dihedral group:D8`

The dihedral group of order eight occurs as a subgroup in bigger groups. Here are some examples:

- It is a subgroup in a dihedral group of order where is a multiple of 4.
- It occurs as a Sylow subgroup in a number of groups: for instance, in the symmetric group on four letters.

### Occurrence as a quotient

The dihedral group of order eight also occurs as a quotient; for instance, it is a quotient of the dicyclic group of order 16, by its center (which has order two).

## Distinguishing features

### Smallest of its kind

- This is the unique non-T-group of smallest order, i.e., the unique smallest example of a group in which normality is not transitive.
- This is a non-abelian nilpotent group of smallest order, though not the only one. The other such group is the quaternion group.

### Different from others of the same order

- It is the only group of its order that is isomorphic to its automorphism group.
- It is the only group of its order that is not a T-group.
- It is the only group of its order having two Klein four-subgroups. In particular, it gives an example of a situation where the number of elementary abelian subgroups of order is neither zero nor modulo . Contrast this with the case of odd , where we have the elementary abelian-to-normal replacement theorem for prime-square order.

## GAP implementation

### Group ID

This finite group has order 8 and has ID 3 among the groups of order 8 in GAP's SmallGroup library. For context, there are 5 groups of order 8. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(8,3)`

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

`gap> G := SmallGroup(8,3);`

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

`IdGroup(G) = [8,3]`

or just do:

`IdGroup(G)`

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

### Short descriptions

Description | GAP functions used | Mathematical translation of description |
---|---|---|

DihedralGroup(8) |
DihedralGroup | dihedral group of order , degree |

WreathProduct(CyclicGroup(2),CyclicGroup(2)) |
WreathProduct, CyclicGroup | external wreath product of two copies of cyclic group of order two |

ExtraspecialGroup(2^3,'+') |
ExtraspecialGroup | extraspecial group of '+' type for the prime and order |

SylowSubgroup(SymmetricGroup(4),2) |
SylowSubgroup and SymmetricGroup | The -Sylow subgroup of the symmetric group of degree four |

SylowSubgroup(GL(3,2),2) |
SylowSubgroup, GL | The -Sylow subgroup of GL(3,2) |

### Description using a presentation

Here is the code:

gap> F := FreeGroup(2);; gap> G := F/[F.1^4, F.2^2, F.2 * F.1 * F.2 * F.1]; <fp group on the generators [ f1, f2 ]>

The group constructed here is the dihedral group of order . The first generator maps to the *rotation* element of order four and the second generator maps to a *reflection* element of order two.

### Long descriptions

It can be described as the holomorph of the cyclic group of order four. For this, first define to be the cyclic group of order four (using CyclicGroup), and then use SemidirectProduct and AutomorphismGroup:

C := CyclicGroup(4); G := SemidirectProduct(AutomorphismGroup(C),C);

Here, is the dihedral group of order eight. We can also construct it as a semidirect product of the Klein four-group and an automorphism of order two.

K := DirectProduct(CyclicGroup(2),CyclicGroup(2)); A := AutomorphismGroup(K); S := SylowSubgroup(A,2); G := SemidirectProduct(S,K);

Then, is isomorphic to the dihedral group of order eight.