How to generate the icosahedral groups $I$ and $I_h$? The icosahedral groups $I$ with 60 elements and $I_h = I \times Z_2$ are also  three dimensional point groups. However, ever unlike other point groups, it seems there is rarely reference to give their representation (e.g., matrix representation for each elements). Can any one help me how to generate them? 

The explicit matrix representations of all 5 irreducible representations of $I$ are given in Hu, Yong, Zhao, and Shu, "The irreducible representation matrices of the icosahedral point groups I and Ih", Superlattices and Microstructures, Volume 3, Issue 4, 1987, pages 391-398, DOI:10.1016/0749-6036(87)90212-6.  You can construct the irreps of $I_h$ via the direct product structure with a point inversion.
