Polyhedral Symmetry
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Polyhedral Symmetry
 Polyhedral group, [n,3], (*n32) Involutional symmetryCs, (*)[ ] = Cyclic symmetryCnv, (*nn)[n] = Dihedral symmetryDnh, (*n22)[n,2] = Tetrahedral symmetryTd, (*332)[3,3] = Octahedral symmetryOh, (*432)[4,3] = Icosahedral symmetryIh, (*532)[5,3] =

In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids.

## Groups

There are three polyhedral groups:

• The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. It is isomorphic to A4.
• The conjugacy classes of T are:
• identity
• 4 × rotation by 120°, order 3, cw
• 4 × rotation by 120°, order 3, ccw
• 3 × rotation by 180°, order 2
• The octahedral group of order 24, rotational symmetry group of the cube and the regular octahedron. It is isomorphic to S4.
• The conjugacy classes of O are:
• identity
• 6 × rotation by ±90° around vertices, order 4
• 8 × rotation by ±120° around triangle centers, order 3
• 3 × rotation by 180° around vertices, order 2
• 6 × rotation by 180° around midpoints of edges, order 2
• The icosahedral group of order 60, rotational symmetry group of the regular dodecahedron and the regular icosahedron. It is isomorphic to A5.
• The conjugacy classes of I are:
• identity
• 12 × rotation by ±72°, order 5
• 12 × rotation by ±144°, order 5
• 20 × rotation by ±120°, order 3
• 15 × rotation by 180°, order 2

These symmetries double to 24, 48, 120 respectively for the full reflectional groups. The reflection symmetries have 6, 9, and 15 mirrors respectively. The octahedral symmetry, [4,3] can be seen as the union of 6 tetrahedral symmetry [3,3] mirrors, and 3 mirrors of dihedral symmetry Dih2, [2,2]. Pyritohedral symmetry is another doubling of tetrahedral symmetry.

The conjugacy classes of full tetrahedral symmetry, Td?S4, are:

• identity
• 8 × rotation by 120°
• 3 × rotation by 180°
• 6 × reflection in a plane through two rotation axes
• 6 × rotoreflection by 90°

The conjugacy classes of pyritohedral symmetry, Th, include those of T, with the two classes of 4 combined, and each with inversion:

• identity
• 8 × rotation by 120°
• 3 × rotation by 180°
• inversion
• 8 × rotoreflection by 60°
• 3 × reflection in a plane

The conjugacy classes of the full octahedral group, Oh?S4 × C2, are:

• inversion
• 6 × rotoreflection by 90°
• 8 × rotoreflection by 60°
• 3 × reflection in a plane perpendicular to a 4-fold axis
• 6 × reflection in a plane perpendicular to a 2-fold axis

The conjugacy classes of full icosahedral symmetry, Ih?A5 × C2, include also each with inversion:

• inversion
• 12 × rotoreflection by 108°, order 10
• 12 × rotoreflection by 36°, order 10
• 20 × rotoreflection by 60°, order 6
• 15 × reflection, order 2

## Chiral polyhedral groups

Chiral polyhedral groups
Name
(Orb.)
Coxeter
notation
Order Abstract
structure
Rotation
points
#valence
Diagrams
Orthogonal Stereographic
T
(332)

[3,3]+
12 A4 43
32
Th
(3*2)

[4,3+]
24 A4×2 43
3*2
O
(432)

[4,3]+
24 S4 34
43
62
I
(532)

[5,3]+
60 A5 65
103
152

## Full polyhedral groups

Full polyhedral groups
Weyl
Schoe.
(Orb.)
Coxeter
notation
Order Abstract
structure
Coxeter
number

(h)
Mirrors
(m)
Mirror diagrams
Orthogonal Stereographic
A3
Td
(*332)

[3,3]
24 S4 4 6
B3
Oh
(*432)

[4,3]
48 S4×2 8 3
6
H3
Ih
(*532)

[5,3]
120 A5×2 10 15