which can be divided into 5 distinct conjugate classes. A regular icosahedron (see Fig. \[fig:icosahedron\]) consists of 20 equilateral triangles, 30 edges and 12 vertices. The rotations of the icosahedron can be classified into five types, including $0$ or $2\pi$ rotation (the identity), $\pi$ rotations about the midpoint of each edge, $2\pi/5$ and $4\pi/5$ rotations about an axis through each vertex, and rotations by $2\pi/3$ about axes through the center of each face. These five types of rotations form five conjugate classes denoted by $$\begin{aligned}
C_{1},~C_{15},~C_{12},~C'_{12},~C_{20}.\end{aligned}$$ Hence the icosahedral symmetry group $I$ also has 60 elements. One can show that the icosahedron symmetry group $I$ and the even permutation group $A_5$ are isomorphic. The orthogonality relations for $A_{5}$ imply $\sum^{5}_{i=1}n^2_{i} = 60$, where $n_i$ denotes the dimensionality of the irreducible representation $R_i$, and we know that the number of classes of a discrete group is equal to its number of irreducible representation. Therefore, we have five irreducible representations (irreps): one trivial singlet $\textbf{1}$, two triplets $\textbf{3}$ and $\textbf{3}'$, one quartet $\textbf{4}$ and one quintet $\textbf{5}$. The character table and multiplication rules for $A_{5}$ are shown in Table [\[character\]]{} and \[multiplication\] respectively.

The sixty elements of $A_5$ can be generated by 