Acronym	...
Name	Shephard's 3-generalised hexadecachoron, complex polychoron x2-3-o2-3-o2-4-o3, β34
	©
Vertex figure	x2-3-o2-4-o3
Coordinates	(ε3n, 0, 0, 0)   & all permutations, each for any 1≤n≤3, where ε3=exp(2πi/3)
Dual	x3-4-o2-3-o2-3-o2
Face vector	12, 54, 108, 81
Confer	more general: x2-3-o2-3-o2-4-op   general polytopal classes: complex polytopes
External links

This complex polychoron is somewhat special in so far as its edges, faces, and calls all are degenerate, i.e. remain real space polytopes only. However, its vertex figure is the truely complex polyhedron x₂-3-o₂-4-o₃. Thence it still remains embeddable into a real space polyzetton, in fact into the tegum product of 4 (fully orthogonal) triangles. In fact the to be chosen cells are just the lacing tets, which have one vertex on each of those triangles.

Incidence matrix according to Dynkin symbol

x2-3-o2-3-o2-4-o3

.    .    .    .  | 12 ♦  9 |  27 | 27
-----------------+----+----+-----+---
x2   .    .    .  |  2 | 54 ♦   6 |  9
-----------------+----+----+-----+---
x2-3-o2   .    .  |  3 |  3 | 108 |  3
-----------------+----+----+-----+---
x2-3-o2-3-o2   .  ♦  4 |  6 |   4 | 81