Replying to
@OscarCunningham@mathstodon.xyz
@OscarCunningham@mathstodon.xyz i wrote a program that enumerates all the colorings, and it doesn't find any that work: http://ianhenderson.org/2026-08-01-dodecahedron.c.txt (the face numberings were from putting labels on a real-life icosahedron i had handy, so i could have made mistakes during that process, but the group it generates does have order 60 at least)