Field of definition and Galois orbits for the Macbeath-Hurwitz curves

Hurwitz curves are Riemann surfaces with 84(g - 1) automorphisms, g the gen us. Defined over some number field they permit an obvious Gal((Q) over bar/ Q) action. We investigate this action for the first known infinite series o f Hurwitz curves, due to Macbeath. using the canonical model of the curves As a result we obtain the minimal field of definition for these curves. The method can be extended to some other infinite series of modular curves for non-congruence subgroups.