Dihedral group:D8

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Definition

Definition by presentation

The dihedral group $D_{8}$ , sometimes called $D_{4}$ , 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:

$\langle x,a\mid a^{4}=x^{2}=e,xax^{-1}=a^{-1}\rangle$

Here, the element $a$ is termed the rotation or the generator of the cyclic piece and $x$ is termed the reflection. An alternative presentation is:

$\langle a,b\mid a^{4}=b^{4}=abab=e\rangle$

In terms of the previous presentation, we can set $a=a,b=ax$ .

Geometric definition

The dihedral group $D_{8}$ (also called $D_{4}$ ) 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.

Definition as a permutation group

The group is the subgroup of the symmetric group on $\{1,2,3,4\}$ given by:

$\{(),(1,2,3,4),(1,3)(2,4),(1,4,3,2),(1,3),(2,4),(1,4)(2,3),(1,3)(2,4)\}$

This can be related to the geometric definition by thinking of $1,2,3,4$ as the vertices of the square and considering an element of $D_{8}$ in terms of its induced action on the vertices. It relates to the presentation via setting $a=(1,2,3,4)$ and $x=(1,3)$ .

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 $a^{2}$ , as in the above presentation.

Element	$\!e$	$\!a$	$\!a^{2}$	$\!a^{3}$	$\!x$	$\!ax$	$\!a^{2}x$	$\!a^{3}x$
$\!e$	$\!e$	$\!a$	$\!a^{2}$	$\!a^{3}$	$\!x$	$\!ax$	$\!a^{2}x$	$\!a^{3}x$
$\!a$	$\!a$	$\!a^{2}$	$\!a^{3}$	$\!e$	$\!ax$	$\!a^{2}x$	$\!a^{3}x$	$\!x$
$\!a^{2}$	$\!a^{2}$	$\!a^{3}$	$\!e$	$\!a$	$\!a^{2}x$	$\!a^{3}x$	$\!x$	$\!ax$
$\!a^{3}$	$\!a^{3}$	$\!e$	$\!a$	$\!a^{2}$	$\!a^{3}x$	$\!x$	$\!ax$	$\!a^{2}x$
$\!x$	$\!x$	$\!a^{3}x$	$\!a^{2}x$	$\!ax$	$\!e$	$\!a^{3}$	$\!a^{2}$	$\!a$
$\!ax$	$\!ax$	$\!x$	$\!a^{3}x$	$\!a^{2}x$	$\!a$	$\!e$	$\!a^{3}$	$\!a^{2}$
$\!a^{2}x$	$\!a^{2}x$	$\!ax$	$\!x$	$\!a^{3}x$	$\!a^{2}$	$\!a$	$\!e$	$\!a^{3}$
$\!a^{3}x$	$\!a^{3}x$	$\!a^{2}x$	$\!ax$	$\!x$	$\!a^{3}$	$\!a^{2}$	$\!a$	$\!e$

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

Elements

Upto conjugacy

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

The identity element
The rotation by $\pi$ , which is given as $a^{2}$ in the presentation
The two-element conjugacy class comprising rotations by $\pm \pi /2$ , namely $a$ and $a^{3}$
The two-element conjugacy class comprising the two reflections: $x,xa^{2}$
The two-element conjugacy class comprising the two reflections: $xa,xa^{3}$

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
prime-base logarithm of order	3	The order is $2^{3}$
exponent	4	Cyclic subgroup of order four.
prime-base logarithm of exponent	2
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	$1\mapsto 1,2\mapsto 5,4\mapsto 2$	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	$\langle x\rangle \triangleleft \langle a^{2},x\rangle$ , which is normal, but $\langle x\rangle$ 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:

The trivial subgroup (1)
The center, which is the unique minimal normal subgroup, and is a two-element subgroup generated by $a^{2}$ . This is isomorphic to the cyclic group of order two. (1) Further information: Center of dihedral group:D8
The two-element subgroups generated by $x$ , $ax$ , $a^{2}x$ and $a^{3}x$ . 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 $a^{2}x$ form one conjugacy class; the subgroups generaetd by $ax$ and $a^{3}x$ form another conjugacy class. (4) Further information: 2-subnormal subgroups of dihedral group:D8
The four-element subgroup generated by $a^{2}$ and $x$ . This comprises elements $e,a^{2},x,a^{2}x$ . It is normal and is isomorphic to the Klein four-group. A similar four-element subgroup is obtained as that generated by $a^{2}$ and $ax$ . (2) Further information: Klein four-subgroups of dihedral group:D8
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
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 $a^{n}x$ , 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 $a^{2}x$ , and the other comprising the subgroups generated by $ax$ and by $a^{3}x$ . (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
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:

It is a subgroup in a dihedral group of order $2n$ where $n$ 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 $p^{2}$ 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 groups of order 8 in GAP's SmallGroup library. For context, there are 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 $G$ :

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

Conversely, to check whether a given group $G$ 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.

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);

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