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Polyhedral group

Geometric polyhedral group

In geometry, the polyhedral groups are 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

Coxeternotation	Order	Abstractstructure	Rotationpoints#valence	Diagrams	Orthogonal	Stereographic	T(332)	Th(3*2)	O(432)	I(532)
[3,3]+	12	A4	43[[File:3-fold rotation axis.svg	12px]] [[File:Purple Fire.svg	12px]]32[[File:Rhomb.svg	12px]]	[[File:Sphere_symmetry_group_t.svg	120px]]		[[File:Tetrakis_hexahedron_stereographic_D4_gyrations.png	120px]]		[[File:Tetrakis_hexahedron_stereographic_D3_gyrations.png	120px]]		[[File:Tetrakis_hexahedron_stereographic_D2_gyrations.png	120px]]
[4,3+]	24	A4 × C2	43[[File:3-fold rotation axis.svg	12px]]3*2		[[File:Sphere_symmetry_group_th.svg	120px]]	[[File:Disdyakis_dodecahedron_stereographic_D4_pyritohedral.png	120px]]	[[File:Disdyakis_dodecahedron_stereographic_D3_pyritohedral.png	120px]]	[[File:Disdyakis_dodecahedron_stereographic_D2_pyritohedral.png	120px]]
[4,3]+	24	S4	34[[File:Monomino.png	12px]]43[[File:3-fold rotation axis.svg	12px]]62[[File:Rhomb.svg	12px]]	[[File:Sphere_symmetry_group_o1.svg	120px]]	[[File:Disdyakis_dodecahedron_stereographic_D4_gyrations.png	120px]]	[[File:Disdyakis_dodecahedron_stereographic_D3_gyrations.png	120px]]	[[File:Disdyakis_dodecahedron_stereographic_D2_gyrations.png	120px]]
[5,3]+	60	A5		65[[File:Patka piechota.png	12px]]103[[File:3-fold rotation axis.svg	12px]]152[[File:Rhomb.svg	12px]]	[[File:Sphere_symmetry_group_i.svg	120px]]	[[File:Disdyakis_triacontahedron_stereographic_d5_gyrations.png	120px]]	[[File:Disdyakis_triacontahedron_stereographic_d3_gyrations.png	120px]]	[[File:Disdyakis_triacontahedron_stereographic_d2_gyrations.png	120px]]

Full polyhedral groups

Coxeternotation	Order	Abstractstructure	Coxeternumber(h)	Mirrors(m)	Mirror diagrams	Orthogonal	Stereographic	A3Td(*332)	B3Oh(*432)	H3Ih(*532)
[3,3]	24	S4	4	6	[[File:Spherical tetrakis hexahedron.svg	120px]]	[[File:Tetrakis hexahedron stereographic D4.svg	120px]]	[[File:Tetrakis hexahedron stereographic D3.png	120px]]	[[File:Tetrakis hexahedron stereographic D2.png	120px]]
[4,3]	48	S4 × C2	8	36	[[File:Spherical disdyakis dodecahedron.svg	120px]]	[[File:Disdyakis dodecahedron stereographic D4.png	120px]]	[[File:Disdyakis dodecahedron stereographic D3.png	120px]]	[[File:Disdyakis dodecahedron stereographic D2.png	120px]]
[5,3]	120	A5 × C2	10	15	[[File:Spherical disdyakis triacontahedron.svg	120px]]	[[File:Disdyakis triacontahedron stereographic d5.svg	120px]]	[[File:Disdyakis triacontahedron stereographic d3.svg	120px]]	[[File:Disdyakis triacontahedron stereographic d2.svg	120px]]

References

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. (The Polyhedral Groups. §3.5, pp. 46–47)
N.W. Johnson: Geometries and Transformations, (2015) Chapter 11: Finite symmetry groups --

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