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, , 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, , 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, , 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 groupsName(Orb.)CoxeternotationOrderAbstractstructureRotationpoints#valenceDiagramsOrthogonalStereographicT(332) [3,3]+ 12A443 32Th(3*2) [4,3+] 24A4 × C2 433*2O(432) [4,3]+ 24S4 344362I(532) [5,3]+ 60A565103152
Full polyhedral groups + Full polyhedral groupsWeylSchoe.(Orb.)CoxeternotationOrderAbstractstructureCoxeternumber(h)Mirrors(m)Mirror diagramsOrthogonalStereographicA3Td(*332) [3,3] 24 S4 4 6B3Oh(*432) [4,3] 48 S4 × C28 3>6H3Ih(*532) [5,3] 120 A5 × C210 15
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)
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