alternatively and more subtly, by chopping up each square face into 2 triangles and each octagonal face into 6 triangles, the small cubicuboctahedron can be interpreted as a non-regular "coloring" of the combinatorially "regular" (not just "uniform") tiling of the genus 3 surface by 56 equilateral triangles, meeting at 24 vertices, each with degree 7. this regular tiling is significant as it is a tiling of the klein quartic, the genus 3 surface with the most symmetric metric (automorphisms of this tiling equal isometries of the surface), and the orientation-preseserving automorphism group of this surface is isomorphic to the projective special linear group psl(2,7), equivalently gl(3,2) (the order 168 group of all orientation-preserving isometries). note that the small cubicuboctahedron is "not" a realization of this abstract polyhedron, as it only has 24 orientation-preserving symmetries (not every abstract automorphism is realized by a euclidean isometry) – the isometries of the small cubicuboctahedron preserve not only the triangular tiling, but also the coloring, and hence are a proper subgroup of the full isometry group.