Let X and Y be compact Riemann surfaces and let φ : X ⇒ Y be a ramified covering of a finite degree n. Let [Pscr ]_Y ⊂ Y be a finite set of points that includes all branch points of φ and let [Pscr ]_X = φ⁻¹([Pscr ]_Y). Let X₀ = X \ [Pscr ]_X and Y₀ = Y \ [Pscr ]_Y. Pick a base point y ∊ Y₀ and let x ∊ φ⁻¹(y). Since the restriction of φ to X₀ is a covering, it induces an embedding φ^* of π₁(X₀, x) into π₁(Y₀, y) as a subgroup of index n. We describe an algorithm that, given canonical generators of π₁(Y₀, y), computes canonical generators of π₁(X₀, x). The monodromy group G of the covering φ is naturally isomorphic to the factor group of π₁(Y₀, y) over its largest normal subgroup contained in φ^*(π₁(X₀, x)). In light of this our algorithm can be used to compute standard generators for subgroups of G. The algorithm is implemented in GAP, and it was used to determine the containment among the Hurwitz loci of Riemann surfaces of low genus.