A detailed write up of what you are looking for is also available in Chapter 13 of Thurston's notes. Specifically, formula 13.3.4 on page 331, has the desired formula
where is the orbifold, is it's underlying space, there are N points fixed locally by dihedral groups of orders and points fixed locally by rotations of orders .
However, I believe the question addressed (at least in part) by Hurwitz in the proof of Hurwitz's theorem, since the bound the order of an automorphism group H acting on surface of genus g is 168(g − 1) if H includes orientation reversing symmetries and 84(g-1) in the orientable case. The key observation is that the Gauss-Bonnet extends to orbifolds as well.
Computing the Euler characteristic for the quotient of by the (full) (2,3,7) triangle group is a good way to see this in action.
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