Coverings of groups, regular Dessins, and surfaces

A coset geometry representation of regular dessins is established, and employed to describe quotients and coverings of regular dessins and surfaces. A characterization is then given of face-quasiprimitive regular dessins as coverings of unicellular regular dessins. It shows that there are exactly three O'Nan-Scott-Praeger types of face-quasiprimitive regular dessins which are smooth coverings of unicellular regular dessins, leading to new constructions of interesting families of regular dessins. Finally, a problem of determining smooth Schur covering of simple groups is initiated by studying coverings between $\SL(2,p)$ and $\PSL(2,p)$, giving rise to interesting regular dessins like Fibonacci coverings.