Does anybody know if there is a convex polytope in $R^4$ with vertices at the [binary octahedral group][1] (identifying $H$ with $R^4$).  

The binary tetrahedral group lies at the vertices of the so-called 24-cell, and the binary octahedral group is just a direct some of two binary tetrahedral groups, but it is not clear how to interpret that geometrically.  

Experimentally, I have found that, for each octahedron in the 24-cell, each vertex in that octahedron is equidistance from exactly one point in binoct not in bintet.  I don't know if this is relevant at all.  


  [1]: https://en.wikipedia.org/wiki/Binary_octahedral_group