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I read in 'Angular momentum in Quantum Mechanics' by A.R Edmonds that the symmetry group of the 6j symbol is isomorphic to the symmetry group of a regular tetahedron.

Is there an easy way of seeing this? I've tried working out what the symmetry relations of the 6j symbol do to the associated tetahedron, but I can't see the bigger picture. I have also tried constructing an isomorphism to S4 (which is isomorphic to the full symmetry group of a regular tetahedron), but no succes.

Can anyone help me out?