Order-7 triangular tiling

This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a triangulation whose symmetry group equals their automorphism group as Riemann surfaces.

The smallest of these is the Klein quartic, the most symmetric genus 3 surface, together with a tiling by 56 triangles, meeting at 24 vertices, with symmetry group the simple group of order 168, known as PSL(2,7).

The resulting surface can in turn be polyhedrally immersed into Euclidean 3-space, yielding the small cubicuboctahedron.

[1] The dual order-3 heptagonal tiling has the same symmetry group, and thus yields heptagonal tilings of Hurwitz surfaces.

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {3,p}.