nLab octahedral group

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Contents

Context

Group Theory

Exceptional structures

Idea
Properties

Basic properties
Character table
Group cohomology

References

Idea

The octahedral group, a finite group, is the group of 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
$A_{n \geq 1}$		cyclic group $\mathbb{Z}_{n+1}$	cyclic group $\mathbb{Z}_{n+1}$	special unitary group $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic group of order 3 $\mathbb{Z}_3$	cyclic group of order 3 $\mathbb{Z}_3$	SU(3)
A3 = D3		cyclic group of order 4 $\mathbb{Z}_4$	cyclic group of order 4 $2 D_2 \simeq \mathbb{Z}_4$	SU(4) $\simeq$ Spin(6)
D4	dihedron on bigon	Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$	quaternion group $2 D_4 \simeq$ Q8	SO(8), Spin(8)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron on square	dihedral group of order 8 $D_8$	binary dihedral group of order 16 $2 D_{8}$	SO(12), Spin(12)
$D_{n \geq 4}$	dihedron, hosohedron	dihedral group $D_{2(n-2)}$	binary dihedral group $2 D_{2(n-2)}$	special orthogonal group, spin group $SO(2n)$ , $Spin(2n)$
$E_6$	tetrahedron	tetrahedral group $T$	binary tetrahedral group $2T$	E6
$E_7$	cube, octahedron	octahedral group $O$	binary octahedral group $2O$	E7
$E_8$	dodecahedron, icosahedron	icosahedral group $I$	binary icosahedral group $2I$	E8

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

the full octahedral group is the subgroup of O(3)

$O_h \hookrightarrow O(3)$

which is the stabilizer of the standard embedding of the octahedron into Cartesian space $\mathbb{R}^3$ ;
the rotational octahedral group $O \hookrightarrow SO(3)$ is the restriction to orientation-preserving symmetries, hence to SO(3); this is isomorphic to the symmetric group $S_4$ ;
the binary octahedral group is the double cover, hence the lift of $O$ to Spin(3) $\simeq$ SU(2);
next there is a string 2-group lift $String_{2O} \hookrightarrow String_{SU(2)}$ of the octahedral group (Epa 10, Epa & Ganter 16)

\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:

$\vert O_h\vert = 48$

$\vert O \vert = 24$

$\vert 2O \vert = 48$

The subgroup of the octahedral group on the orientation-preserving symmetries is isomorphic to the symmetric group $S_4$ . 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):

2 D_4 = Q_8 \subset 2 O \,.

In fact the only finite subgroups of SU(2) which contain $2 D_4 =Q_8$ 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 $2 O$

$\,$

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

conjugacy classes:	1	-1	$i$	a	c	e	f	g
their cardinality:	1	1	6	8	8	6	6	12

character table over the complex numbers $\mathbb{C}$

irrep	1	-1	$i$	a	c	e	f	g
$\rho_1$	1	1	1	1	1	1	1	1
$\rho_2$	1	1	1	1	1	-1	-1	-1
$\rho_3$	2	2	2	-1	-1	0	0	0
$\rho_4$	3	3	-1	0	0	1	1	-1
$\rho_5$	3	3	-1	0	0	-1	-1	1
$\rho_6$	2	-2	0	1	-1	$\sqrt{2}$	$-\sqrt{2}$	0
$\rho_7$	2	-2	0	1	-1	$-\sqrt{2}$	$\sqrt{2}$	0
$\rho_8$	4	-4	0	-1	1	0	0	0

character table over the real numbers $\mathbb{R}$

irrep	1	-1	$i$	a	c	e	f	g
$\rho_1$	1	1	1	1	1	1	1	1
$\rho_2$	1	1	1	1	1	-1	-1	-1
$\rho_3$	2	2	2	-1	-1	0	0	0
$\rho_4$	3	3	-1	0	0	1	1	-1
$\rho_5$	3	3	-1	0	0	-1	-1	1
$\rho_6 \oplus \rho_6$	4	-4	0	2	-2	$2 \sqrt{2}$	$-2 \sqrt{2}$	0
$\rho_7 \oplus \rho_7$	4	-4	0	2	-2	$-2 \sqrt{2}$	$2 \sqrt{2}$	0
$\rho_8 \oplus \rho_8$	8	-8	0	-2	2	0	0	0

References

GroupNames, CSU(2,3)
James Montaldi, Real representations – Binary cubic – 2O
Groupprops, Linear representation theory of binary octahedral group
Bockland, Character tables and McKay quivers (pdf)

category: character tables

Group cohomology

The group cohomology of the orientation-preserving octahedral group is discussed in Groupprops, Tomoda & Zvengrowski 08, Sec. 4.2, Kirdar 13, Epa & Ganter 16, p. 12.

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

Groupprops, Linear representation theory of binary octahedral group