Ever since I first read Coxeter’s definition of a regular compound (which seems to be the most commonly used), I didn’t like it on account of it being completely different than for properly connected polytopes. This made me begin to wonder what the set of regular compounds would be if the same definition (flag-transitivity) was used. After searching the internet and discovering a few of my own, my list includes two infinite families plus 8 others. However, I have no idea if it’s complete. Could there be others?

• 2 n-simplices (Aut(An) symmetry)
• n p/q-gons (Inp symmetry)
• 3 tesseracts (F4 symmetry)
• 3 16-cells (F4 symmetry)
• 2 24-cells (Aut(F4) symmetry)
• 6 tesseracts (Aut(F4) symmmetry)
• 6 16-cells (Aut(F4) symmetry)
• 120 5-cells (H4 symmetry)
• 72 7-simplices (E7 symmetry)
• 1920 8-simplices (E8 symmetry)