Operations on maps, and outer automorphisms

Abstract By representing maps on surfaces as transitive permutation representations of a certain group Γ, it is shown that there are exactly six invertible operations (such as duality) on maps; they are induced by the outer automorphisms of Γ, and form a group isomorphic to S3. Various consequences are deduced, such as the result that each finite map has a finite reflexible cover which is invariant under all six operations.