Maximally symmetric polyhedral realizations of Dyck's regular map

TLDR

The authors construct genus‑three realizations of Dyck's regular map as polyhedra in ℝ³. The resulting polyhedra exhibit maximal symmetry: one has a 3‑fold axis and three 2‑fold axes, the others have three 2‑fold axes, and no realization can have a symmetry group larger than order six.

Abstract

We construct realizations of Dyck's regular map of genus three as polyhedra in ℝ3. One of these has one axis of symmetry of order three and three axes of symmetry of order two. The other polyhedra have three axes of symmetry. We show that a polyhedron realizing Dyck's regular map cannot have a symmetry group of order larger than six. Thus the symmetry groups of our realizations are maximal.