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

Applying rectification onto either of x_p-4-o₂-3-o₂ or x₂-3-o₂-4-o_p 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

op-4-x2-3-o2

.    .    .  | 3p2 ♦ 2p  |  2 p 
------------+-----+-----+------
.    x2   .  |  2  | 3p3 |  1 1 
------------+-----+-----+------
op-4-x2   .  ♦ 2p  |  p2 | 3p * 
.    x2-3-o2 |  3  |  3  |  * p3