Section 1.2.7.4.4. Determination of the vector representation

For a point group G, its isomorphism class and its character table are known. For each conjugacy class, a representative element is given as word where the 's correspond to generators. Replacing the letters by the generating matrices, one obtains as product a matrix for which the trace is the character of the vector representation in the conjugacy class. The characters of all conjugacy classes being known, the representation can be decomposed into irreducible components by means of where labels the irreducible representations (the row number in the character table), the number of times the representation occurs, the order of the group G, the number of elements in the ith conjugacy class (given as the second row in the character table), the cyclotomic in the ith row and th column of the character table, and the calculated character in the ith conjugacy class.