A geometric realization without self-intersections does exist for Dyck's regular map
Dyck's map \(\{3,8\}_ 6\) is a regular triangulation of the orientable surface of genus 3 with \(f_ 0=12\) vertices, \(f1=48\) edges and \(f_ 2=32\) triangular faces. The author constructs an embedding into \(E^ 3\) such that each face is a planar euclidean triangle. This polyhedron in \(E^ 3\) has no euclidean symmetries. For such an embedding with the maximal possible symmetry group of order 6 see \textit{U. Brehm}, Mathematika 34, No.2, 229-236 (1987; Zbl 0663.52004).
scientific article; zbMATH DE number 4129532
scientific article