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.
