One can express the symmetry types of, say, Escher's "Circle Limit" prints using Conway's orbifold notation, best known in the context of symmetries of Euclidean plane patterns.

For example, Circle Limit III has symmetry type $433$ (with Euler characteristic $-1/12$).

Where can I find an explicit algorithm that produces generators for some appropriate subgroup of the isometries of the Poincaré model of the hyperbolic plane given a suitable Conway notation? Only certain notation give rise to rigid orbifolds, so I'd also like to know how to read off the number of moduli from the Conway notation. Absent rigidity, I'd really like a parametric family of generating sets realizing all the distinct forms of the underlying orbifold.

The popular program Kali facilitates drawing symmetric Euclidean patterns? Does anyone distribute some appropriate hyperbolic counterpart?