Acronym	...
Name	Shephard's 3-generalised cuboctahedron, complex polyhedron o3-4-x2-3-o2
	©
Vertex figure	x3   x2
Coordinates	(ε3n, ε3m, 0)   & all permutations, each for any 1≤n,m≤3, where ε3=exp(2πi/3)
Face vector	27, 81, 36
Confer	more general: op-4-x2-3-o2   general polytopal classes: complex polytopes
External links

Applying rectification onto either of x₃-4-o₂-3-o₂ or x₂-3-o₂-4-o₃ in the respectively other direction each, re-uses the former edge counts for new vertex count, and uses the duals of the faces each (somewhere further out) as the two face types in here. Alternatively those additional face types could be obtained as the respective vertex figures of the pre-images each. Edges themselves in here are 2-fold, as can be read from the symbol directly, i.e. are down to real ones again.

Incidence matrix according to Dynkin symbol

o3-4-x2-3-o2

.    .    .  | 27 ♦  6 | 2  3
-------------+----+----+-----
.    x2   .  |  2 | 81 | 1  1
-------------+----+----+-----
o3-4-x2   .  ♦  6 |  9 | 9  *
.    x2-3-o2 |  3 |  3 | * 27