The classifying element for quotients of Fermat curves

Suppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and ∞. Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of Q.