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tetrahedral group

Contents

Contents

Idea

The tetrahedral group is the finite symmetry group of a tetrahedron.

As a symmetry group of one of the Platonic solids, the tetrahedral group participates in the ADE pattern:

ADE classification and McKay correspondence

Dynkin diagram/
Dynkin quiver
Platonic solidfinite 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)
D4Klein 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)
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
SO(2n)SO(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 tetrahedral group corresponding to the stages of the Whitehead tower of O(3):

String 2T String SU(2) 2T Spin(3)SU(2) TA 4 SO(3) T dS 5 O(3) \array{ String_{2T} &\hookrightarrow& String_{SU(2)} \\ \downarrow && \downarrow \\ 2 T &\hookrightarrow & Spin(3) \simeq SU(2) \\ \downarrow && \downarrow \\ T \simeq A_4 &\hookrightarrow& SO(3) \\ \downarrow && \downarrow \\ T_d \simeq S_5 &\hookrightarrow & O(3) }

Properties

Basic properties

The full tetrahedral group is isomorphic to the symmetric group S 4S_4 of permutations of four elements (see Full tetrahedral group is isomorphic to S4).

The subgroup of orientation-preserving symmetries is isomorphic to the alternating group A 4A_4.

The group order is:

|T d|=24\vert T_d\vert = 24

|T|=12\vert T\vert = 12

|2T|=24\vert 2T\vert = 24

Proposition

(quaternion group inside binary tetrahedral group)

The binary tetrahedral group contains the quaternion group of order 8, hence the binary dihedral group of order 8, as a subgroup, in fact as a normal subgroup:

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

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.

As part of the ADE pattern

ADE classification and McKay correspondence

Dynkin diagram/
Dynkin quiver
Platonic solidfinite 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)
D4Klein 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)
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
SO(2n)SO(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

Character table

linear representation theory of binary tetrahedral group 2T2 T

\,

group order: |2T|=24\vert 2T\vert = 24

conjugacy classes:1-1iiabcd
their cardinality:1164444

\,

let ζ 3\zeta_3 be a third root of unity, (ζ 3) 3=1(\zeta_3)^3 = 1

e.g. ζ 3=12(1+3i)\zeta_3 = \tfrac{1}{2}(-1 + \sqrt{3} i), notice that ζ 3+ζ 3 2=1\zeta_3 + \zeta_3^2 = 1

character table over the complex numbers \mathbb{C}

irrep1-1iiabcd
ρ 1\rho_11111111
ρ 2\rho_2111ζ 3\zeta_3ζ 3 2\zeta_3^2ζ 3 2\zeta_3^2ζ 3\zeta_3
ρ 2 *\rho_2^\ast111ζ 3 2\zeta^2_3ζ 3\zeta_3ζ 3\zeta_3ζ 3 2\zeta_3^2
ρ 3\rho_333-10000
ρ 4\rho_42-20ζ 3\zeta_3ζ 3 2\zeta_3^2ζ 3 2-\zeta_3^2ζ 3-\zeta_3
ρ 4 *\rho_4^\ast2-20ζ 3 2\zeta_3^2ζ 3\zeta_3ζ 3-\zeta_3ζ 3 2-\zeta_3^2
ρ 5\rho_5 2-2011-1-1

character table over the real numbers \mathbb{R}

irrep1-1iiabcd
ρ 1\rho_11111111
ρ 2ρ 2 *\rho_2 \oplus \rho_2^\ast222-1-1-1-1
ρ 3\rho_333-10000
ρ 4ρ 4 *\rho_4 \oplus \rho_4^\ast4-40-1-111
ρ 5ρ 5\rho_5 \oplus \rho_54-4022-2-2

References

Group cohomology

The group cohomology of the tetrahedral group is discussed in Groupprops, Kirdar 13.

References

Discussion in the context of classification of finite rotation groups goes back to:

  • Felix Klein, chapter I.4 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

Exposition is in

Discussion of higher central extension to Platonic 2-groups is in

See also

Last revised on October 8, 2018 at 09:59:01. See the history of this page for a list of all contributions to it.