Acronym	...
Name	Shephard's p-generalised hexadecachoron, complex polychoron x2-3-o2-3-o2-4-op, βp4
	© p=3 p=4 p=5
Vertex figure	x2-3-o2-4-op
Coordinates	(εpn, 0, 0, 0)   & all permutations, each for any 1≤n≤p, where εp=exp(2πi/p)
Dual	xp-4-o2-3-o2-3-o2
Face vector	4p, 6p2, 4p3, p4
Especially	x2-3-o2-3-o2-4-o3 (p=3)   x2-3-o2-3-o2-4-o4 (p=4)   x2-3-o2-3-o2-4-o5 (p=5)
Confer	general polytopal classes: complex polytopes
External links

Those complex polychora are somewhat special in so far as its edges, faces, and calls all are degenerate, i.e. remain real space polytopes only. For p=2 it surely becomes fully depenerate and then is nothing but the real space hex. For larger values of p however, its vertex figure is the truely complex polyhedron x₂-3-o₂-4-o_p. Thence it still remains embeddable into a real space polyzetton, in fact into the tegum product of 4 (fully orthogonal) p-gons. In fact the to be chosen cells are just the lacing tets, which have one vertex on each of those p-gons.

Incidence matrix according to Dynkin symbol

x2-3-o2-3-o2-4-op

.    .    .    .  | 4p ♦ 3p  | 3p2 | p3
-----------------+----+-----+-----+---
x2   .    .    .  |  2 | 6p2 ♦ 2p  | p2
-----------------+----+-----+-----+---
x2-3-o2   .    .  |  3 |  3  | 4p3 | p 
-----------------+----+-----+-----+---
x2-3-o2-3-o2   .  ♦  4 |  6  |  4  | p4