Acronym	...
Name	complex polyhedron x3-3-o3-4-o2
Circumradius	1
Vertex figure	x3-4-o2
Coordinates	(ε3n, ε3m, ε3k)/sqrt(3)      & for any 1≤n,m,k≤3 (-ε3n, -ε3m, -ε3k)/sqrt(3)  & for any 1≤n,m,k≤3 (ε3n, 0, 0)                        & for any 1≤n≤3, all permutations, all changes of sign where: ε3=exp(2πi/3)=(-1+i sqrt(3))/2
Dual	x2-4-o3-3-o3
Face vector	72, 216, 54
Confer	real space embedding: mo general polytopal classes: complex polytopes
External links

When considered as a regular complex polyhedron, one derives its below provided incidence structure as follows. The vertex count is readily obtained either by its real space embedding (mo) or by the count of the above provided explicite coordinates (27+27+18). The in here to be used (complex) faces are x₃-3-o₃, which in turn were substructures of hex. Further the (complex) vertex figure here is x₃-4-o₂, thence the super-diagonal entries of the matrix are clear as well.

When considered as a rectified complex polyhedron, one derives its incidence structure as follows. That afore mentioned vertex figure could be viewed alternatively as x₃-2-x₃ instead, which then applies in here. Further one notes, that the pre-image for this rectification is just x₃-3-o₃-3-o₃. And because the rectification uses its vertices at the centers of the formers edges (up to some global scaling), it becomes clear that the new vertex number is the formers edge number. Finally one notes, that the faces of a rectified polyhedron consist out of the duals of the former's faces plus the former's vertex figures. Accordingly, the face number of the rectified polyhedron equals the sum of the former's face number plus the former's vertex number. Taking this all together on derives the incidence matrix of the second, rectified representation independently from the first, regular representation, i.e. directly from its non-rectified base.

Incidence matrix according to Dynkin symbol

x3-3-o3-4-o2

.    .    .  | 72 ♦   9 |  6
-------------+----+-----+---
x3   .    .  |  3 | 216 |  2
-------------+----+-----+---
x3-3-o3   .  ♦  8 |   8 | 54

o3-3-x3-3-o3

.    .    .  | 72 ♦   9 |  3  3
-------------+----+-----+------
.    x3   .  |  3 | 216 |  1  1
-------------+----+-----+------
o3-3-x3   .  ♦  8 |   8 | 27  *
.    x3-3-o3 ♦  8 |   8 |  * 27