A first answer to that question of what this symmetry of the Klein quartic would look like e.g. is given in an article by William Thurston at
http://library.msri.org/books/Book35/files/thurston.pdf (1998):
Consider the regular dodecahedron, which is representable within 3D. It has incidence matrix
- Code: Select all
20 | 3 | 3
---+----+---
2 | 30 | 2
---+----+---
5 | 5 | 12
The symmetry of the Klein quartic, taken geometrically in the similar way, then would be described by the following incidence matrix of some abstract regular polyhedron:
- Code: Select all
56 | 3 | 3
---+----+---
2 | 84 | 2
---+----+---
7 | 7 | 24
That figure obviously cannot be represented faithful within 3D.
(Moreover, it not even can be represented faithful within flat hyperbolic space, as there the order 3 tiling of regular heptagons surely has an infinite number of vertices, edges, and faces.)
--- rk