Abstract. Dessins d’enfants are combinatorial structures on compact Riemann surfaces defined over algebraic number fields, and regular dessins are the most symmetric of them. If is a finite group, there are only finitely many regular dessins with automorphism group . It is shown how to enumerate them, how to represent them all as quotients of a single regular dessin , and how certain hypermap operations act on them. For example, if is a cyclic group of order then is a map on the Fermat curve of degree and genus . On the other hand, if then has genus 274218830047232000000000000000001. For other non-abelian finite simple groups, the genus is much larger.
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