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A dessin is a 2-cell embedding of a connected bipartite graph into anorientable closed surface. An automorphism of a dessin is a permutation of theedges of the underlying graph which preserves the colouring of the vertices andextends to an orientation-preserving self-homeomorphism of the supportingsurface. A dessin is regular if its automorphism group is transitive on theedges, and a regular dessin is totally symmetric if it is invariant under alldessin operations. Thus totally symmetric dessins possesses the highest levelof external symmetry. In this paper we present a classification of totallysymmetric dessins with a nilpotent automorphism group of class three

Totally symmetric dessins with nilpotent automorphism groups of class three