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, with denoting the identity element:

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:

  1. The dihedral group of order eight.
  2. The generalized dihedral group corresponding to the cyclic group of order four.
  3. The holomorph of the cyclic group of order four.
  4. The external wreath product of the cyclic group of order two with the cyclic group of order two, acting via the regular action.
  5. The -Sylow subgroup of the symmetric group on four letters.
  6. The -Sylow subgroup of the symmetric group on five letters.
  7. The -Sylow subgroup of the alternating group on six letters.
  8. The unitriangular matrix group of degree three over field:F2, -Sylow subgroup of PSL(3,2).
  9. 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
Function Value Similar groups Explanation for function value
underlying prime of p-group 2
order (number of elements, equivalently, cardinality or size of underlying set) 8 groups with same order As a semidirect product of and : the order is the product of the orders of and , which is

As a wreath product of and : the order is
prime-base logarithm of order 3 groups with same prime-base logarithm of order
max-length of a group 3 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 3 chief length equals prime-base logarithm of order for group of prime power order
composition length 3 composition length equals prime-base logarithm of order for group of prime power order
exponent 4 groups with same order and exponent | groups with same exponent As a dihedral group: the dihedral group of order has exponent equal to .
prime-base logarithm of exponent 2
nilpotency class 2 groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class The derived subgroup is -- and it is the same as the center. See center of dihedral group:D8. Also see element structure of dihedral group:D8#Commutator map
derived length 2 groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length The derived subgroup is , which is abelian. See center of dihedral group:D8.
Frattini length 2 groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length The Frattini subgroup is , which is of prime order, hence its Frattini subgroup is trivial.
Fitting length 1 All groups of prime power order are nilpotent, hence have Fitting length 1.
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set Generator of cyclic subgroup of order four and element of order two outside.
subgroup rank of a group 2 groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group All proper subgroups are cyclic or Klein four-groups.
rank of a p-group 2 groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group There exist Klein four-subgroups.
normal rank of a p-group 2 groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group There exist normal Klein four-subgroups.
characteristic rank of a p-group 1 groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group All abelian characteristic subgroups are cyclic.

Arithmetic functions of an element-counting nature

Further information: element structure of dihedral group:D8

Function Value Similar groups Explanation for function value GAP verification (set G := DihedralGroup(8);) -- See more at #GAP implementation
number of conjugacy classes 5 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As dihedral group , even:
. See element structure of dihedral groups and element structure of dihedral group:D8
As unitriangular matrix group :

See element structure of unitriangular matrix group of degree three over a finite field
Length(ConjugacyClasses(G)); using ConjugacyClasses
number of equivalence classes under real conjugacy 5 groups with same order and number of equivalence classes under real conjugacy | groups with same number of equivalence classes under real conjugacy Same as number of conjugacy classes, because the group is an ambivalent group. See dihedral groups are ambivalent
number of conjugacy classes of real elements 5 groups with same order and number of conjugacy classes of real elements | groups with same number of conjugacy classes of real elements Same as number of conjugacy classes, because the group is an ambivalent group. See dihedral groups are ambivalent
number of equivalence classes under rational conjugacy 5 groups with same order and number of equivalence classes under rational conjugacy | groups with same number of equivalence classes under rational conjugacy Same as number of conjugacy classes, because the group is a rational group. Length(RationalClasses(G)); using RationalClasses
number of conjugacy classes of rational elements 5 groups with same order and number of conjugacy classes of rational elements | groups with same number of conjugacy classes of rational elements Same as number of conjugacy classes, because the group is a rational group.

Arithmetic functions of a subgroup-counting nature

Further information: subgroup structure of dihedral group:D8

Function Value Similar groups Explanation GAP verification (set G := DihedralGroup(8);) -- See more at #GAP verification
number of subgroups 10 As a dihedral group number of subgroups is , where is the divisor count function and is the divisor sum function. See subgroup structure of dihedral group:D8, subgroup structure of dihedral groups Length(Subgroups(G)); using Subgroups
number of conjugacy classes of subgroups 8 See subgroup structure of dihedral groups, subgroup structure of dihedral group:D8 Length(ConjugacyClassesSubgroups(G)); using ConjugacyClassesSubgroups
number of normal subgroups 6 groups with same order and number of normal subgroups | groups with same number of normal subgroups See subgroup structure of dihedral groups, subgroup structure of dihedral group:D8#Lattice of normal subgroups Length(NormalSubgroups(G)); using NormalSubgroups
number of automorphism classes of subgroups 6
number of characteristic subgroups 4 Length(CharacteristicSubgroups(G)); using CharacteristicSubgroups

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

Property Satisfied? Explanation Comment
one-headed group No Three distinct maximal normal subgroups of order four
SC-group No
ACIC-group Yes Every automorph-conjugate subgroup is characteristic
algebra group Yes It is isomorphic to the unitriangular matrix group of degree three over field:F2, which is clearly an algebraic group.
ambivalent group Yes dihedral groups are ambivalent Also see generalized dihedral groups are ambivalent
rational group Yes Any two elements that generate the same cyclic group are conjugate Thus, all characters are integer-valued.
rational-representation group Yes All representations over characteristic zero are realized over the rationals. Contrast with quaternion group, that is rational but not rational-representation.
extraspecial group Yes The center, derived subgroup, and Frattini subgroup all coincide and are cyclic of prime order
special group Yes (via extraspecial): the center, derived subgroup, and Frattini subgroup all coincide
Frattini-in-center group Yes (via extraspecial): the Frattini subgroup is contained in the center
group of nilpotency class two Yes (via special): the derived subgroup is contained in the center
UL-equivalent group Yes (via special): the upper central series and lower central series coincide
maximal class group Yes
Frobenius group No Frobenius groups are centerless, and this group isn't
Camina group Yes extraspecial implies Camina
Every element is automorphic to its inverse Yes Follows from being an ambivalent group
any two elements generating the same cyclic subgroup are automorphic Yes
every element is order-automorphic No
directly indecomposable group Yes
centrally indecomposable group Yes
splitting-simple group No
stem group Yes the center equals the derived subgroup, and hence, in particular, is contained in the derived subgroup.
Schur-trivial group No the Schur multiplier is cyclic group:Z2; see group cohomology of dihedral group:D8.

Subgroups

Further information: subgroup structure of dihedral group:D8

Lattice of subgroups of the dihedral group


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

Subgroup-defining function What it means Value as subgroup Value as group Order Associated quotient-defining function Value as group Order (= index of subgroup)
center elements that commute with every group element center of dihedral group:D8: cyclic group:Z2 2 inner automorphism group Klein four-group 4
derived subgroup subgroup generated by all commutators center of dihedral group:D8: cyclic group:Z2 2 abelianization Klein four-group 4
Frattini subgroup intersection of all maximal subgroups center of dihedral group:D8: cyclic group:Z2 2 Frattini quotient Klein four-group 4
Jacobson radical intersection of all maximal normal subgroups center of dihedral group:D8: cyclic group:Z2 2  ? Klein four-group 4
socle join of all minimal normal subgroups center of dihedral group:D8: cyclic group:Z2 2 socle quotient Klein four-group 4
Baer norm intersection of normalizers of all subgroups center of dihedral group:D8: cyclic group:Z2 2  ? Klein four-group 4
join of all abelian normal subgroups subgroup generated by all the abelian normal subgroups whole group dihedral group:D8 8  ? trivial group 1
join of abelian subgroups of maximum order join of all abelian subgroups of maximum order among abelian subgroups whole group dihedral group:D8 8  ? trivial group 1
join of abelian subgroups of maximum rank join of all abelian subgroups of maximum rank among abelian subgroups whole group dihedral group:D8 8  ? trivial group 1
join of elementary abelian subgroups of maximum order join of all elementary abelian subgroups of maximum order among elementary abelian subgroups whole group dihedral group:D8 8  ? trivial group 1
ZJ-subgroup center of the join of abelian subgroups of maximum order center of dihedral group:D8: cyclic group:Z2 2  ? Klein four-group 4
epicenter intersection of images of centers for all central extensions trivial subgroup: trivial group 1 largest quotient group that is a capable group dihedral group:D8 8

Some more notes:


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

GAP implementation

ACCESS GAP IMPLEMENTATION ONLINE USING SAGE: The following public document on SAGE contains the full GAP implementation of the group: Public document 5013 (NOTE: As of September 2012, SAGE had disabled public worksheets. However, this restriction will probably be lifted eventually, in which case the link should work)

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);
2
More: [SHOW MORE]