Icosahedral Group
The icosahedral group 
is the point
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 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). Its multiplication
table is illustrated above.
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.
| solid | basis vector |
| dodecahedron |  |
| icosahedron |  |
| icosidodecahedron |  |
| small rhombicosidodecahedron |  |
| snub dodecahedron |  |
| truncated dodecahedron |  |
| truncated icosahedron |  |
Contact the MathWorld Team
baby monster group
