# 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*.

Confused about presentations in general or this one in particular? If you're new to this stuff, check out constructing dihedral group:D8 from its presentation. Sophisticated group theorists need simply recall that presentation of semidirect product is disjoint union of presentations plus action by conjugation relations

### 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.

The row element is multiplied on the left and the column element is multiplied on the right.

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).
- The extraspecial group of order and type '+'.

## 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 | (8,4), i.e., 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

### Basic 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 an element-counting nature

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

### Arithmetic functions of a subgroup-counting nature

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

### Lists of numerical invariants

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

conjugacy class sizes | 1,1,2,2,2 | Two central elements, all others in conjugacy classes of size two. See element structure of dihedral group:D8 and element structure of dihedral groups. |

sizes of orbits under automorphism group | 1,1,2,4 | Two central elements, one conjugacy class of elements of order four, one orbit of size four, comprising two conjugacy classes of size, with all elements non-central of order two. |

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 |

### 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

### Important 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 | Smallest non-abelian group of prime power order |

T-group | No | , which is normal, but is not normal | Smallest example for normality is not transitive. |

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

### Other properties

[SHOW MORE]## Subgroups

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

In the "List of subgroups" columns below, a row break within the cell indicates that *each row represents one conjugacy class of subgroups*.

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 subgroup is normal) | Subnormal depth (if proper and normal, this equals 1) | Nilpotency class |
---|---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 8 | 1 | 1 | 1 | dihedral group:D8 | 1 | 0 | |

center | cyclic group:Z2 | 2 | 4 | 1 | 1 | 1 | Klein four-group | 1 | 1 | |

other subgroups of order two | |
cyclic group:Z2 | 2 | 4 | 2 | 2 | 4 | -- | 2 | 1 |

Klein four-subgroups | , | Klein four-group | 4 | 2 | 2 | 1 | 2 | cyclic group:Z2 | 1 | 1 |

cyclic maximal subgroup | cyclic group:Z4 | 4 | 2 | 1 | 1 | 1 | cyclic group:Z2 | 1 | 1 | |

whole group | dihedral group:D8 | 8 | 1 | 1 | 1 | 1 | trivial group | 0 | 2 | |

Total (6 rows) | -- | -- | -- | -- | 8 | -- | 10 | -- | -- | -- |

## Subgroup-defining functions and associated quotient-defining functions

`Further information: subgroup structure of dihedral group:D8#Defining functions`

Some more notes:

- The following subgroup-defining functions are equal to the whole group on account of the group being a nilpotent group: Fitting subgroup, hypercenter, solvable radical.
- The following subgroup-defining functions are equal to the trivial subgroup on account of the group being a solvable group: hypocenter, nilpotent residual, perfect core, solvable residual.

## Automorphisms and endomorphisms

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

Construct | Value | Order | Second part of GAP ID (if group) |
---|---|---|---|

endomorphism monoid | ? | 36 | not applicable |

automorphism group | dihedral group:D8 | 8 | 3 |

inner automorphism group | Klein four-group | 4 | 2 |

extended automorphism group | direct product of D8 and Z2 | 16 | 11 |

quasiautomorphism group | direct product of D8 and Z2 | 16 | 11 |

1-automorphism group | direct product of S4 and Z2 | 48 | 48 |

outer automorphism group | cyclic group:Z2 | 2 | 1 |

## Linear representation theory

`Further information: Linear representation theory of dihedral group:D8, linear representation theory of dihedral groups`

### Summary

Item | Value |
---|---|

Degrees of irreducible representations over a splitting field (such as or ) | 1,1,1,1,2 maximum: 2, lcm: 2, number: 5, sum of squares: 8 |

Schur index values of irreducible representations | 1,1,1,1,1 |

Smallest ring of realization for all irreducible representations (characteristic zero) | ; same as ring generated by character values |

Minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero) | (hence, it is a rational representation group) Same as field generated by character values, because all Schur index values are 1. |

Condition for being a splitting field for this group | Any field of characteristic not two is a splitting field. |

Minimal splitting field in characteristic | The prime field |

Smallest size splitting field | field:F3, i.e., the field with three elements. |

Orbits over a splitting field under action of automorphism group | orbit sizes: 1 (degree 1 representation), 1 (degree 1 representation), 2 (degree 1 representations), and 1 (degree 2 representation) number: 4 |

Orbits over a splitting field under the multiplicative action of one-dimensional representations, i.e., up to projective equivalence | orbit sizes: 4 (degree 1 representations), 1 (degree 2 representations) number: 2 |

Other groups with the same character table | quaternion group (see linear representation theory of quaternion group) |

### Character table

This character table works over characteristic zero:

Representation/Conj class | (size 1) | (size 1) | (size 2) | (size 2) | (size 2) |
---|---|---|---|---|---|

Trivial representation | 1 | 1 | 1 | 1 | 1 |

-kernel | 1 | 1 | 1 | -1 | -1 |

-kernel | 1 | 1 | -1 | 1 | -1 |

-kernel | 1 | 1 | -1 | -1 | 1 |

2-dimensional | 2 | -2 | 0 | 0 | 0 |

The same character table works over any characteristic not equal to 2 where the elements 1,-1,0,2,-2 are interpreted over the field.

## Fusion systems

`Further information: fusion systems for dihedral group:D8`

### Summary

Item | Value |
---|---|

Total number of saturated fusion systems on a concrete instance of the group (strict, not up to isomorphism of fusion systems) |
4 |

Total number of saturated fusion systems up to isomorphism | 3 |

List of saturated fusion systems with orbit sizes | inner fusion system (orbit size 1 under isomorphisms), non-inner non-simple fusion system for dihedral group:D8 (orbit size 2 under isomorphisms), simple fusion system for dihedral group:D8 (orbit size 1) |

Number of simple fusion systems | 1 |

Number of maximal saturated fusion systems, i.e., saturated fusion systems not contained in bigger saturated fusion systems | 1 (the simple fusion system for dihedral group:D8) |

### Description of fusion systems

Isomorphism type of fusion system | Number of such fusion systems under strict counting | Can the fusion system be realized using a Sylow subgroup of a finite group? | Does the identity functor control strong fusion? This would mean that all fusion occurs in the normalizer | Is the fusion system simple? | Smallest size embedding realizing this fusion system (if any) |
---|---|---|---|---|---|

inner fusion system | 1 | Yes | Yes | inner fusion system is not simple | as a subgroup of itself |

non-inner non-simple fusion system for dihedral group:D8 | 2 | Yes | No | No | D8 in S4 |

simple fusion system for dihedral group:D8 | 1 | Yes | No | Yes | D8 in PSL(3,2) |

Total (3 rows) | 4 | -- | -- | -- | -- |

## 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 congruence condition on number of elementary abelian subgroups of prime-square order for odd prime.

## 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.

### Hall-Senior number

This group of prime power order has order 8 and has Hall-Senior number 4 among the groups of order 8. This information can be used to construct the group in GAP using the Gap3CatalogueGroup function as follows:

`Gap3CatalogueGroup(8,4)`

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(8,4);`

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

`Gap3CatalogueIdGroup(G) = [8,4]`

or just do:

`Gap3CatalogueIdGroup(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 by 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 ]> gap> IdGroup(G); [ 8, 3 ]

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.

### GAP verification

Below is a GAP implementation verifying the various function values and group properties as stated in this page. Before beginning, set `G := DihedralGroup(8);` or any equivalent way of setting to be dihedral of order eight.

gap> IdGroup(G); [ 8, 3 ] gap> Order(G); 8 gap> Exponent(G); 4 gap> NilpotencyClassOfGroup(G); 2More: [SHOW MORE]