You might not want to know, but we actually intended to involve some Gaussian map of the monkey saddle this time. Since we got lost in our construction, this strange graph was the result. It also has a right to live, so we didn't eliminate it.
vertices: 29
faces: 29
edges: 58
genus: 1
characteristic polynomial:
(-1 + x5) (1 + x5) (1 - 3x + x2)2 (1 + 3x + x2)2 (-10 - x + x2) (-10 + x + x2)
dual's characteristic polynomial:
x (-4 + x) (-8 + x2) (-1 - 5x - 6x2 + x4)2 (5 - 5x - 4x2 + 2x3 + x4)2
chromatic number: 3
lattice group (lossy):
Z(2) 58 × A(58)