

genus ^{c}  3, orientable 
Schläfli formula ^{c}  {7,3} 
V / F / E ^{c}  56 / 24 / 84 
notes  
vertex, face multiplicity ^{c}  1, 1 
21, each with 8 edges  
antipodal sets  28 of ( 2v ), 8 of ( 3f ), 21 of ( 4e ) 
rotational symmetry group  PSL(2,7), with 168 elements 
full symmetry group  PGL(2,7), with 336 elements 
its presentation ^{c}  < r, s, t  t^{2}, s^{‑3}, (sr)^{2}, (st)^{2}, (rt)^{2}, r^{‑7}, (sr^{‑2})^{4} > 
C&D number ^{c}  R3.1′ 
The statistics marked ^{c} are from the published work of Professor Marston Conder. 
Its Petrie dual is
It can be 2split to give
It can be rectified to give
It is the result of pyritifying (type 14/3/7/3)
List of regular maps in orientable genus 3.
Its skeleton is cubic Klein graph.
This regular map, embedded in 3space, 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 K_{4}. The wireframe is arranged as the skeleton of
PSL(2,7) 
PGL(2,7) 
Orientable  
Nonorientable 
The images on this page are copyright © 2010 N. Wedd