On the wikipedia page of [tetrahedron][1], there is a list of eight symmetry groups for a (possibly irregular) $3$-simplex (with unmarked faces).  There is also a list on the page of [5-cell][2] but doesn't seem complete.

This seems elementary, but I can not find a source to cite.  Also, is there already a general classification for symmetry groups of unmarked (possibly irreglar) d-simplices (or an algorithm for enumerating it)?

I believe this is equivalent to asking about the symmetry of edge-colored complete graphs.  I find a lot of literature on those with transitive groups, but not general cases.


  [1]: https://en.wikipedia.org/wiki/Tetrahedron#Isometries_of_irregular_tetrahedra
  [2]: https://en.wikipedia.org/wiki/5-cell#Irregular_5-cell