Polyhedral group

Short description: Geometric polyhedral group

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

## Contents

* 1 Groups
* 2 Chiral polyhedral groups
* 3 Full polyhedral groups
* 4 See also
* 5 References
* 6 External links

## 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

(332)  |
[3,3]+ | 12 | A4 | 43 12px
3212px |  |  |  |

(3*2)  |

[4,3+] | 24 | A4×2 | 43
3*2 |  |  |  |

(432)  |
[4,3]+ | 24 | S4 | 34
4312px
6212px |  |  |  |

(532)  |
[5,3]+ | 60 | A5 | 65
10312px
15212px |  |  |  |

## Full polyhedral groups

Full polyhedral groups  Weyl
Schoe.
(Orb.)  | Coxeter
notation  | Order  | Abstract
structure  | Coxeter
number
(h)  | Mirrors
(m)  | Mirror diagrams

|
|
|
|
|
|

Orthogonal  | Stereographic

(*332)  |

[3,3] | 24 | S4 | 4 | 6 |  | 120px | 120px |

(*432)  |

[4,3] | 48 | S4×2 | 8 | 3
6 |  | 120px | 120px |

(*532)  |

[5,3] | 120 | A5×2 | 10 | 15 |  | 120px | 120px |

## See also

* Wythoff symbol
* List of spherical symmetry groups

## References

* Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. (The Polyhedral Groups. §3.5, pp. 46-47)

## External links

* Weisstein, Eric W.. "PolyhedralGroup". http://mathworld.wolfram.com/PolyhedralGroup.html.

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