Given the multiplication table of a (finite) group there are a couple of "obvious" representations you can pick. Firstly theres always the trivial representati

Given the multiplication table of a (finite) group there are a couple of "obvious" representations you can pick. Firstly theres always the trivial representation that just sends everything to the identity. More interestingly you can use the group elements as labels for an orthonormal basis for a vector space (e.g. your orthonormal set is {v_g for all g in G}). Then have the representation act by the usual group operation on the labels R_h[v_g] = v_{hg}. Then once you have an operation defined on a basis extending it linearly to the rest of the vector space is easy. This is (left) regular representation. https://en.wikipedia.org/wiki/Regular_representation https://en.wikipedia.org/wiki/Regular_representation Edit: Intuition for why the trace is important comes from looking at permutation matrices - the trace is the sum of the diagonal elements of the matrix and the number of 1s on the diagonal is exactly the number of points that are invariant under the permutation - e.g. the permutation [[1,0,0,0], [0,1,0,0], [0,0,0,1], [0,0,1,0]] has trace 2, and the permutation leaves the first two elements unchanged and flips the last two.