I am trying to construct the character table for the symmetry group of Cube, $O_h$, which as 48 elements. I figured out that there are 10 classes. Now, as a consequence of the great orthogonality theorem, I have $\sum l_i^2 = 48$ ...(1), and the number of irreducible representations is equal to the number of classes. So , I tried solving the equation (1), for $a_1, a_2,...a_{10}$ keeping a condition that $a_i \leq a_{i+1}$, to avoid repetitions. i am getting only 6 possible solutions. Integer solutions satisfying the conditions: [2, 2, 2, 2, 2, 2, 2, 2, 4] [1, 2, 2, 2, 2, 2, 3, 3, 3] [1, 1, 1, 2, 2, 2, 2, 2, 5] [1, 1, 1, 1, 2, 2, 2, 4, 4] [1, 1, 1, 1, 1, 3, 3, 3, 4] [1, 1, 1, 1, 1, 1, 1, 4, 5]
But I need 10 IR's. So how do I find the remaining ones?
Also, In the examples I have come across, they usually deal with $C_{nv}$ or $D_n$, which are relatively smaller groups, so they usually have one solution to equation (1). Here, there are 10, so do I construct 10 different character tables?