Representation Theory: Using the Schur orthogonality relations, we obtain an orthonormal basis of L^2(G) using matrix coefficients of irreducible representations. This shows the sum of squares of dimensions of irreducibles equals |G|. We also obtain an orthonormal basis of Class(G) using irreducible characters, and from this we see that the number of irreducible classes equals the number of conjugacy classes in G. We also obtain character formulas for multiplicities.
Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
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