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 $\infty$. 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 $\mathbb{Q}$.