Nilpotent Symmetric dessins of class two

Abstract

A dessin is a cellular embedding of a connected bipartite graph into an orientable closed surface with a fixed colouring of vertices and prescribed global orientation. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts regularly on the set of edges, and a regular dessin is symmetric if it admits an external symmetry transposing the vertex colours. The symmetric dessins whose automorphism groups are nilpotent of class two are classified.