  Let G be a finite group. By Riemann's Existence Theorem, braid orbits of
generating systems of G with product 1 correspond to irreducible families of
covers of the Riemann sphere with monodromy group G. Thus many problems on
algebraic curves require the computation of braid orbits. In this paper we
describe an implementation of this computation. We discuss several
applications, including the classification of irreducible families of
indecomposable rational functions with exceptional monodromy group.
