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), or sometimes the octic group, is defined by the following presentation, with $e$ denoting the identity element:

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

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

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 $D_{8}$ (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

Further information: D8 in S4

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

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

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.

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

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:

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 unitriangular matrix group of degree three $UT(3,2)$ over field:F2, $2$-Sylow subgroup of PSL(3,2).
9. The extraspecial group of order $2^{3}$ 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	$8\Gamma _{2}a_{1}$

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 $a$ and $x$	Geometric description	Permutation on vertices	Order of the element
$e$ (identity element)	does nothing, i.e., leaves the square invariant	$()$	1
$a$	rotation by angle of $\pi /2$ (i.e., $90\,^{\circ }$) counterclockwise	$(1,2,3,4)$	4
$a^{2}$	rotation by angle of $\pi$ (i.e., $180\,^{\circ }$), also called a half turn	$(1,3)(2,4)$	2
$a^{3}$	rotation by angle of $3\pi /2$ (i.e., $270\,^{\circ }$) counter-clockwise, or equivalently, by $\pi /2$ (i.e., $90\,^{\circ }$) clockwise	$(1,4,3,2)$	4
$x$	reflection about the diagonal joining vertices "2" and "4"	$(1,3)$	2
$ax=xa^{3}$	reflection about the line joining midpoints of opposite sides "14" and "23"	$(1,4)(2,3)$	2
$a^{2}x$	reflection about the diagonal joining vertices "1" and "3"	$(2,4)$	2
$a^{3}x=xa$	reflection about the line joining midpoints of opposite sides "12" and "34"	$(1,2)(3,4)$	2

Below is the conjugacy and automorphism class structure:

Conjugacy class in terms of $a,x$	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
$\{e\}$	identity element, does nothing	$\{()\}$	1	1	whole group
$\{a^{2}\}$	half turn, rotation by $\pi$	$\{(1,3)(2,4)\}$	1	2	whole group
$\{x,a^{2}x\}$	reflections about diagonals	$\{(1,3),(2,4)\}$	2	2	$\{e,a^{2},x,a^{2}x\}$ -- one of the Klein four-subgroups of dihedral group:D8
$\{ax,a^{3}x\}$	reflections about lines joining midpoints of opposite sides	$\{(1,4)(2,3)\ ,\ (1,2)(3,4)\}$	2	2	$\{e,a^{2},ax,a^{3}x\}$ -- one of the Klein four-subgroups of dihedral group:D8
$\{a,a^{3}\}$	rotations by odd multiples of $\pi /2$	$\{(1,2,3,4)\ ,\ (1,4,3,2)\}$	2	4	$\{e,a,a^{2},a^{3}\}$ -- 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 $a,x$	Geometric description of equivalence class	Equivalence class as permutations	Size of equivalence class	Number of conjugacy classes in it	Size of each conjugacy class
$\{e\}$	identity element, does nothing	$\{()\}$	1	1	1
$\{a^{2}\}$	half turn	$\{(1,3)(2,4)\}$	1	1	1
$\{x,ax,a^{2}x,a^{3}x\}$	reflections	$\{(1,3)\ ,\ (2,4)\ ,\ (1,4)(2,3)\ ,\ (1,2)(3,4)\}$	4	2	2
$\{a,a^{3}\}$	rotations by odd multiples of $\pi /2$	$\{(1,2,3,4)\ ,\ (1,4,3,2)\}$	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 $\mathbb {Z} _{4}$ and $\mathbb {Z} _{2}$: the order is the product of the orders of $\mathbb {Z} _{4}$ and $\mathbb {Z} _{2}$, which is $4\times 2=8$ As a wreath product of $\mathbb {Z} _{2}$ and $\mathbb {Z} _{2}$: the order is $2^{2}\cdot 2=8$
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 $2n$ has exponent equal to $\operatorname {lcm} \{n,2\}$.
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 $\{e,a^{2}\}$ -- 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 $\{e,a^{2}\}$, 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 $\{e,a^{2}\}$, 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 $D_{2n}$, $n$ even: $\!(n+6)/2=(4+6)/2=5$. See element structure of dihedral groups and element structure of dihedral group:D8 As unitriangular matrix group $UT(3,q),q=2$: $q^{2}+q-1=2^{2}+2-1=5$ 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 $\!D_{2n},n=4$ number of subgroups is $\!d(n)+\sigma (n)=d(4)+\sigma (4)=3+7=10$, where $d$ is the divisor count function and $\sigma$ 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	$1\mapsto 1,2\mapsto 5,4\mapsto 2$	Of the five elements of order two, one is central. The other four are automorphic to each other. See element structure of dihedral group:D8 and element structure of dihedral groups
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

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	$a$ and $x$ don't commute	Smallest non-abelian group of prime power order
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

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.
A-group	No	The only Sylow subgroup is itself, which is not abelian.	Joint smallest non-A-group, alongside quaternion group.
Group isomorphic to its automorphism group	Yes
2-Engel group	Yes

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	$\{e\}$	trivial group	1	8	1	1	1	dihedral group:D8	1	0
center	$\{e,a^{2}\}$	cyclic group:Z2	2	4	1	1	1	Klein four-group	1	1
other subgroups of order two	$\{e,x\},\{e,a^{2}x\}$ $\{e,ax\},\{e,a^{3}x\}$	cyclic group:Z2	2	4	2	2	4	--	2	1
Klein four-subgroups	$\{e,x,a^{2},a^{2}x\}$, $\{e,ax,a^{2},a^{3}x\}$	Klein four-group	4	2	2	1	2	cyclic group:Z2	1	1
cyclic maximal subgroup	$\{e,a,a^{2},a^{3}\}$	cyclic group:Z4	4	2	1	1	1	cyclic group:Z2	1	1
whole group	$\{e,a,a^{2},a^{3},x,ax,a^{2}x,a^{3}x\}$	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: $\{e,a^{2}\}$	cyclic group:Z2	2	inner automorphism group	Klein four-group	4
derived subgroup	subgroup generated by all commutators	center of dihedral group:D8: $\{e,a^{2}\}$	cyclic group:Z2	2	abelianization	Klein four-group	4
Frattini subgroup	intersection of all maximal subgroups	center of dihedral group:D8: $\{e,a^{2}\}$	cyclic group:Z2	2	Frattini quotient	Klein four-group	4
Jacobson radical	intersection of all maximal normal subgroups	center of dihedral group:D8: $\{e,a^{2}\}$	cyclic group:Z2	2	?	Klein four-group	4
socle	join of all minimal normal subgroups	center of dihedral group:D8: $\{e,a^{2}\}$	cyclic group:Z2	2	socle quotient	Klein four-group	4
Baer norm	intersection of normalizers of all subgroups	center of dihedral group:D8: $\{e,a^{2}\}$	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: $\{e,a^{2}\}$	cyclic group:Z2	2	?	Klein four-group	4
epicenter	intersection of images of centers for all central extensions	trivial subgroup: $\{e\}$	trivial group	1	largest quotient group that is a capable group	dihedral group:D8	8

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 $\mathbb {C}$ or ${\overline {\mathbb {Q} }}$)	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)	$\mathbb {Z}$; same as ring generated by character values
Minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero)	$\mathbb {Q}$ (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 $p\neq 0,2$	The prime field $\mathbb {F} _{p}$
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	$\{e\}$ (size 1)	$\{a^{2}\}$ (size 1)	$\{a,a^{-1}\}$ (size 2)	$\{x,a^{2}x\}$ (size 2)	$\{ax,a^{3}x\}$ (size 2)
Trivial representation	1	1	1	1	1
$\langle a\rangle$-kernel	1	1	1	-1	-1
$\langle a^{2},x\rangle$-kernel	1	1	-1	1	-1
$\langle a^{2},ax\rangle$-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.

Galois theory

Galois extensions

Further information: Galois extensions for dihedral group:D8

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 $p^{2}$ is neither zero nor $1$ modulo $p$. Contrast this with the case of odd $p$, 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 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.

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 $G$:

gap> G := Gap3CatalogueGroup(8,4);

Conversely, to check whether a given group $G$ 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 $8$, degree $4$
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 $2$ and order $2^{3}$
SylowSubgroup(SymmetricGroup(4),2)	SylowSubgroup and SymmetricGroup	The $2$-Sylow subgroup of the symmetric group of degree four
SylowSubgroup(GL(3,2),2)	SylowSubgroup, GL	The $2$-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 $G$ constructed here is the dihedral group of order $8$. The first generator $F.1$ maps to the rotation element of order four and the second generator $F.2$ 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 $C$ 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, $G$ 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.

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 $G$ 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]

gap> DerivedLength(G);
2
gap> FrattiniLength(G);
2
gap> Rank(G);
2
gap> SubgroupRank(G);
2
gap> RankAsPGroup(G);
2
gap> NormalRank(G);
2
gap> CharacteristicRank(G);
1
gap> Length(ConjugacyClasses(G));
5
gap> Length(RationalClasses(G));
5
gap> Length(Subgroups(G));
10
gap> Length(ConjugacyClassesSubgroups(G));
8
gap> Length(NormalSubgroups(G));
6
gap> Length(CharacteristicSubgroups(G));
4