I am trying to solve a problem from Miranda's book, Algebraic Curves and Riemann Surfaces. On page 84, problem K gives the Klein curve $X$ as a smooth projective plane curve defined by the equation $xy^3+yz^3+zx^3=0$. The problem asks us to show that this Riemann surface, of genus g=3, realizes the Hurwitz bound by finding 168 automorphisms of $X$.
I have found an automorphism subgroup of order 3 (cyclically permuting the variables), and one of order 7 (multiplying the coordinates by appropriate 7th roots of unity), but I just can't figure out how to get a subgroup of order 8, or 4, or 2. Could somebody please give me a hint.
I've spent the last little while reading up on this subject, and most of the discussions involve using a heptagonal tiling etc.. Given where this problem is placed in the book, I can't appeal to that kind of reasoning, so I'm asking for a way to find these automorphisms directly from the defining equation of $X$.
Any help or hint would be appreciated. Even an involution :)