Faber polynomials, generated by a conformal mapping Φ(z) of the exterior of a set E onto the exterior of a circle, have well-known classical applications in numerical analysis as basis sets for polynomial and rational approximations in the complex plane. The structure of the Faber polynomials of a given set E is essential for such applications. In this paper we study the Faber polynomials associated with m-fold symmetric domains. A new determinant representation which relates the zeros of Faber polynomials to the eigenvalues of a certain matrix is derived and numerical computations on the zeros of Faber polynomials associated with symmetric lunes and m-gons are illustrated.