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octahedral group

Contents

Context

Group Theory

Exceptional structures

Contents

Idea

The octahedral group, a finite group, is the group ofn symmetries of an octahedron.

As a symmetry group of one of the Platonic solids, the octahedral group participates in one of the three exceptional entries cases of the ADE pattern:

ADE classification and McKay correspondence

Dynkin diagram/
Dynkin quiver
dihedron,
Platonic solid
finite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A n1A_{n \geq 1}cyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
SU(n+1)SU(n+1)
A1cyclic group of order 2
2\mathbb{Z}_2
cyclic group of order 2
2\mathbb{Z}_2
SU(2)
A2cyclic group of order 3
3\mathbb{Z}_3
cyclic group of order 3
3\mathbb{Z}_3
SU(3)
A3
=
D3
cyclic group of order 4
4\mathbb{Z}_4
cyclic group of order 4
2D 2 42 D_2 \simeq \mathbb{Z}_4
SU(4)
\simeq
Spin(6)
D4dihedron on
bigon
Klein four-group
D 4 2× 2D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2
quaternion group
2D 42 D_4 \simeq Q8
SO(8), Spin(8)
D5dihedron on
triangle
dihedral group of order 6
D 6D_6
binary dihedral group of order 12
2D 62 D_6
SO(10), Spin(10)
D6dihedron on
square
dihedral group of order 8
D 8D_8
binary dihedral group of order 16
2D 82 D_{8}
SO(12), Spin(12)
D n4D_{n \geq 4}dihedron,
hosohedron
dihedral group
D 2(n2)D_{2(n-2)}
binary dihedral group
2D 2(n2)2 D_{2(n-2)}
special orthogonal group, spin group
SO(2n)SO(2n), Spin(2n)Spin(2n)
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

More in detail, there are variants of the octahedral group corresponding to the stages of the Whitehead tower of O(3):

String 2O String SU(2) 2O Spin(3)SU(2) OS 4 SO(3) O hS 4×/2 O(3) \array{ String_{2O} &\hookrightarrow& String_{SU(2)} \\ \downarrow && \downarrow \\ 2 O &\hookrightarrow & Spin(3) \simeq SU(2) \\ \downarrow && \downarrow \\ O \simeq S_4 &\hookrightarrow& SO(3) \\ \downarrow && \downarrow \\ O_h \simeq S_4 \times \mathbb{Z}/2 &\hookrightarrow & O(3) }

Properties

Basic properties

The group order is:

|O h|=48\vert O_h\vert = 48

|O|=24\vert O \vert = 24

|2O|=48\vert 2O \vert = 48

The subgroup of the octahedral group on the orientation-preserving symmetries is isomorphic to the symmetric group S 4S_4. This also happens to be the full tetrahedral group.

Proposition

(quaternion group inside binary tetrahedral group)

The binary octahedral group contains the quaternion group of order 8, hence the binary dihedral group of order 8, as a subgroup (normal):

2D 4=Q 82O. 2 D_4 = Q_8 \subset 2 O \,.

In fact the only finite subgroups of SU(2) which contain 2D 4=Q 82 D_4 =Q_8 as a proper subgroup are the exceptional ones, hence the binary tetrahedral group, the binary octahedral group and the binary icosahedral group.

See this Prop at quaternion group.

Character table

linear representation theory of binary octahedral group 2O2 O

\,

group order: |2O|=48{\vert 2O\vert} = 48

conjugacy classes:1-1iiacefg
their cardinality:116886612

character table over the complex numbers \mathbb{C}

irrep1-1iiacefg
ρ 1\rho_111111111
ρ 2\rho_211111-1-1-1
ρ 3\rho_3222-1-1000
ρ 4\rho_433-10011-1
ρ 5\rho_533-100-1-11
ρ 6\rho_62-201-12\sqrt{2}2-\sqrt{2}0
ρ 7\rho_72-201-12-\sqrt{2}2\sqrt{2}0
ρ 8\rho_84-40-11000

character table over the real numbers \mathbb{R}

irrep1-1iiacefg
ρ 1\rho_111111111
ρ 2\rho_211111-1-1-1
ρ 3\rho_3222-1-1000
ρ 4\rho_433-10011-1
ρ 5\rho_533-100-1-11
ρ 6ρ 6\rho_6 \oplus \rho_64-402-2222 \sqrt{2}22-2 \sqrt{2}0
ρ 7ρ 7\rho_7 \oplus \rho_74-402-222-2 \sqrt{2}222 \sqrt{2}0
ρ 8ρ 8\rho_8 \oplus \rho_88-80-22000

References

Group cohomology

The group cohomology of the orientation-preserving octahedral group is discussed in Groupprops, Kirdar 13.

References

  • Felix Klein, chapter I.7 of Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, 1884, translated as Lectures on the Icosahedron and the Resolution of Equations of Degree Five by George Morrice 1888, online version

Aspects of the linear representation theory of the binary octahedral group (irreducible representations, character table) is spelled out at

See also

Last revised on May 15, 2019 at 05:17:40. See the history of this page for a list of all contributions to it.