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From Groupprops

This article is about a particular group, viz a group unique upto isomorphism[SHOW MORE]

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Contents 1 Definition 1.1 Definition by presentation 1.2 Geometric definition 1.3 Multiplication table 1.4 Other definitions 2 Elements 2.1 Upto conjugacy 2.2 Upto automorphism 3 Arithmetic functions 3.1 Lists of numerical invariants 4 Group properties 5 Subgroups 5.1 Normal subgroups 5.2 Characteristic subgroups 6 Quotient groups 7 Subgroup-defining functions 8 Quotient-defining functions 9 Other associated constructs 10 In larger groups 10.1 Occurrence as a subgroup 10.2 Occurrence as a quotient 11 Distinguishing features 11.1 Smallest of its kind 11.2 Different from others of the same order 12 GAP implementation 12.1 Group ID 12.2 Other descriptions 13 Internal links

Definition

Definition by presentation

The dihedral group D₈, sometimes called D₄, 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:

Geometric definition

The dihedral group D₈ (also called D₄) 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 a) and has four reflections each being an involution: reflections about lines joining midpoints of opposite sides, and reflections about diagonals.

Multiplication table

Here, e denotes the identity element, a is an element of order 4, and x is an element of order two that isn't equal to a², as in the above presentation.

Element	e	a	a2	a3	x	ax	a2x	a3x
e	e	a	a2	a3	x	ax	a2x	a3x
a	a	a2	a3	e	ax	a2x	a3x	x
a2	a2	a3	e	a	a2x	a3x	x	ax
a3	a3	e	a	a2	a3x	x	ax	a2x
x	x	a3x	a2x	ax	e	a3	a2	a
ax	ax	x	a3x	a2x	a	e	a3	a2
a2x	a2x	ax	x	a3x	a2	a	e	a3
a3x	a3x	a2x	ax	x	a3	a2	a	e

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 2-Sylow subgroup of the symmetric group on four letters.
6. The 2-Sylow subgroup of the symmetric group on five letters.
7. The 2-Sylow subgroup of the alternating group on six letters.
8. The 2-Sylow subgroup of PSL(3,2).

Elements

Upto conjugacy

There are five conjugacy classes of elements of the dihedral group:

1. The identity element
2. The rotation by π, which is given as a² in the presentation
3. The two-element conjugacy class comprising rotations by , namely a and a³
4. The two-element conjugacy class comprising the two reflections: x,xa²
5. The two-element conjugacy class comprising the two reflections: xa,xa³

Upto automorphism

Under the equivalence relation of automorphisms, the last two conjugacy classes merge into one. There are thus four equivalence classes under the actions of automorphisms, of sizes 1, 1, 2 and 4.

Arithmetic functions

Want to compare with other groups of the same order? Check out groups of order 8#Arithmetic functions

Function	Value	Explanation
order	8
exponent	4	Cyclic subgroup of order four.
nilpotency class	2
derived length	2
Frattini length	2
Fitting length	1
minimum size of generating set	2	Generator of cyclic subgroup of order four and element of order two outside.
subgroup rank	2	All proper subgroups are cyclic or Klein four-groups.
max-length	3
rank as p-group	2	There exist Klein four-subgroups.
normal rank	2	There exist normal Klein four-subgroups.
characteristic rank of a p-group	1	All abelian characteristic subgroups are cyclic.
number of subgroups	10	See subgroup structure of dihedral group:D8
number of conjugacy classes	5
number of conjugacy classes of subgroups	7

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.
order statistics		Of the five elements of order two, one is central. The other four are automorphic to each other.
degrees of irreducible representations	1,1,1,1,2	See linear representation theory of dihedral group:D8
orders of subgroups	1,2,2,2,2,2,4,4,4,8	See subgroup structure of dihedral group:D8

Group properties

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

Property	Satisfied	Explanation	Comment
abelian group	No	a and x don't commute	Smallest non-abelian group of prime power order
nilpotent group	Yes	Prime power order implies nilpotent	Smallest nilpotent non-abelian group, along with quaternion group.
metacyclic group	Yes	Cyclic normal subgroup of order four, cyclic quotient of order two
supersolvable group	Yes	Metacyclic implies supersolvable
solvable group	Yes	Metacyclic implies solvable
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
one-headed group	No	Three distinct maximal normal subgroups of order four
SC-group	No
ACIC-group	Yes	Every automorph-conjugate subgroup is characteristic
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	Center, commutator subgroup, Frattini subgroup coincide, cyclic of order two.
special group	Yes	Center, commutator subgroup, Frattini subgroup coincide
group of nilpotency class two	Yes
Frattini-in-center group	Yes
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

Subgroups

Further information: subgroup structure of dihedral group:D8

The dihedral group has ten subgroups:

1. The trivial subgroup (1)
2. The center, which is the unique minimal normal subgroup, and is a two-element subgroup generated by a². This is isomorphic to the cyclic group of order two. (1) Further information: Center of dihedral group:D8
3. The two-element subgroups generated by x, ax, a²x and a³x. All of these are isomorphic to the cyclic group of order two.These are in two conjugacy classes: the subgroups generated by x and by a²x form one conjugacy class; the subgroups generaetd by ax and a³x form another conjugacy class. (4) Further information: 2-subnormal subgroups of dihedral group:D8
4. The four-element subgroup generated by a² and x. This comprises elements e,a²,x,a²x. It is normal and is isomorphic to the Klein four-group. A similar four-element subgroup is obtained as that generated by a² and ax. (2) Further information: Klein four-subgroups of dihedral group:D8
5. The four-element subgroup generated by a. This is normal and is isomorphic to the cyclic group of order four. (1) Further information: Cyclic maximal subgroup of dihedral group:D8
6. 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 aⁿx, 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 x and by a²x, and the other comprising the subgroups generated by ax and by a³x. (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

1. The group itself: this is obtained as the quotient by the trivial subgroup. (1)
2. The Klein four-group, which is obtained as the quotient by the center. (1)
3. The cyclic group:Z2, which is obtained as the quotient by either of the two Klein four-subgroups. (2)
4. The cyclic group:Z2, which is obtained as the quotient by the cyclic maximal subgroup. (1)
5. 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
Commutator 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

Associated construct	Isomorphism class	Comment
Automorphism group	dihedral group:D8
Outer automorphism group	cyclic group:Z2
Holomorph	holomorph of D8
Inner holomorph	inner holomorph of D8
Extended automorphism group	direct product of D8 and Z2
Quasiautomorphism group	direct product of D8 and Z2
1-automorphism group	direct product of S4 and Z2

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:

1. It is a subgroup in a dihedral group of order 2n where n is a multiple of 4.
2. 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 p² is neither zero nor 1 modulo p. Contrast this with the case of odd p, 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 group of order 8 in GAP's SmallGroup library. 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 G:

gap> G := SmallGroup(8,3);

Other descriptions

The dihedral group can be described using GAP's DihedralGroup command:

DihedralGroup(8)

It can also be described as a wreath product:

WreathProduct(CyclicGroup(2),CyclicGroup(2))

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

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

Here, G 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, G is isomorphic to the dihedral group of order eight.

Finally, it can be constructed as the 2-Sylow subgroup in a number of groups. For instance:

SylowSubgroup(SymmetricGroup(4),2);

or:

SylowSubgroup(GL(3,2),2);

Internal links

• Linear representation theory of dihedral group:D8
• Galois extensions for dihedral group:D8
• Subgroup structure of dihedral group:D8
• Fusion systems for dihedral group:D8
• Supergroups of dihedral group:D8