Polyhedral group
Introduction 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 ''A''4. ** 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 ''S''4. **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 ''A''5. **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 2These 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 planeThe 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 axisThe 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 groupsName(Orb. )CoxeternotationOrderAbstractstructureRotationpoints#valenceDiagramsOrthogonalStereographicT(332) 3,3+  12A44312px 12px3212px120px120px120px120pxTh(3*2) 4,3+ 24A4×2 4312px3*2120px120px120px120pxO(432) 4,3+ 24S4 3412px4312px6212px120px120px120px120pxI(532) 5,3+ 60A5|6512px10312px15212px120px120px120px120px
Full polyhedral groups + Full polyhedral groupsWeylSchoe.(Orb. )CoxeternotationOrderAbstractstructureCoxeternumber(h)Mirrors(m)Mirror diagramsOrthogonalStereographicA3Td(*332) 3,3  24 S4 4 6120px120px120px120pxB3Oh(*432) 4,3 48 S4×28 36120px120px120px120pxH3Ih(*532) 5,3 120 A5×210 15120px120px120px120px
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 *