Concretely, the binary icosahedral group is a subgroup of Spin(3), and covers the icosahedral group, which is a subgroup of SO(3). Abstractly, the icosahedral group is isomorphic to the symmetries of the 4-simplex, which is a subgroup of SO(4), and the binary icosahedral group is isomorphic to the double cover of this in Spin(4). Note that the symmetric group formula_9 "does" have a 4-dimensional representation (its usual lowest-dimensional irreducible representation as the full symmetries of the formula_10-simplex), and that the full symmetries of the 4-simplex are thus formula_11 not the full icosahedral group (these are two different groups of order 120).