The icosahedral group is the group of symmetries of the icosahedron and dodecahedron having order 120, equivalent to the group direct product of the alternating group and cyclic group . The icosahedral group consists of the conjugacy classes 1, , , , , , , , , and (Cotton 1990, pp. 49 and 436). Its multiplication table is illustrated above. The icosahedral group is a subgroup of the special orthogonal group . The icosahedal group is implemented in the Wolfram Language as FiniteGroupData["Icosahedral", "PermutationGroupRepresentation"].
Icosahedral symmetry is possible as a rotational group but is not compatible with translational symmetry. As a result, there are no crystals with this symmetry and so, unlike the octahedral group and tetrahedral group , is not one of the 32 point groups.
The great rhombicosidodecahedron can be generated using the matrix representation of using the basis vector , where is the golden ratio.
The icosahedral group has a pure rotation subgroup denoted that is isomorphic to the alternating group . is of order 60 and has conjugacy classes 1, , , , and (Cotton 1990, pp. 50 and 436). Like , is not a point group. Its multiplication table is illustrated above. The group is currently not implemented as a separate group in the Wolfram Language.
Platonic and Archimedean solids that can be generated by group are illustrated above, with the corresponding basis vector summarized in the following table, where is the golden ratio and and are the largest positive roots of two sixth-order polynomials.