The Klein map


genus c3, orientable
Schläfli formula c{7,3}
V / F / E c 56 / 24 / 84
notesreplete singular is a polyhedral map permutes its vertices evenly
vertex, face multiplicity c1, 1
Petrie polygons
21, each with 8 edges
antipodal sets28 of ( 2v ), 8 of ( 3f ), 21 of ( 4e )
rotational symmetry groupPSL(2,7), with 168 elements
full symmetry groupPSL(2,7)×C2, with 336 elements
its presentation c< r, s, t | t2, s‑3, (sr)2, (st)2, (rt)2, r‑7, (sr‑2)4 >
C&D number cR3.1′
The statistics marked c are from the published work of Professor Marston Conder.

Relations to other Regular Maps

Its dual is the dual Klein map.

Its Petrie dual is N9.1′.

It can be 2-split to give R17.2′.

It can be rectified to give the quasi-Klein map.

It is the result of pyritifying (type 14/3/7/3) S3:{14,3}a.

List of regular maps in orientable genus 3.

Underlying Graph

Its skeleton is cubic Klein graph.


This regular map, embedded in 3-space, can be bought from shapeways, in the form seen to the right (click on the image for a larger version). It is not so clear from a photograph how this physical object portrays the Klein map; if you have one in your hands where you can use binocular vision on it, it works better. This model was designed and uploaded to shapeways by Henry Segerman.

This and its dual are the smallest regular maps to reach the Hurwitz limit B97, which states that the number of edges of a regular map cannot exceed -21 × the Euler characteristic of the surface.

This regular map features in Jarke J. van Wijk's movie Symmetric Tiling of Closed Surfaces: Visualization of Regular Maps, 1:40 seconds from the start. It is shown as a "wireframe diagram", on K4. The wireframe is arranged as the skeleton of the tetrahedron.

Cayley Graphs based in this Regular Map

Type II


Type III


Other Regular Maps

General Index

The images on this page are copyright © 2010 N. Wedd