Acronym	...
Name	Shephard's 4-generalised cube, complex polyhedron x4-4-o2-3-o2, γ43
	©
Circumradius	sqrt(3/2) = 1.224745
Vertex figure	trig
Coordinates	(in, im, ik)/sqrt(2)   for any 1≤n,m,k≤4
Dual	x2-3-o2-4-o4
Face vector	64, 48, 12
Confer	more general: xp-4-o2-3-o2   real space embedding: ax general polytopal classes: complex polytopes
External links

The (complex) edges then are x₄-4-o₂, which were nothing but the set of base-squares of the respective tess each, when considered as the (n,n)-duoprism with n=4. The vertex figure here is just x₂-3-o₂, i.e. nothing but the real space triangle.

The below various incidence representations are direct, next dimensional consequences from what was explained already at x_p-4-o₂ = x_p   x_p. I.e. the according cartesian or prism product applies for complex polytopes alike.

Incidence matrix according to Dynkin symbol

x4-4-o2-3-o2

.    .    .  | 64 |  3 |  3
-------------+----+----+---
x4   .    .  |  4 | 48 |  2
-------------+----+----+---
x4-4-o2   .  ♦ 16 |  8 | 12

x4   x4-4-o2

.    .    .  | 64 |  1  2 | 2 1
-------------+----+-------+----
x4   .    .  |  4 | 16  * | 2 0
.    x4   .  |  4 |  * 32 | 1 1
-------------+----+-------+----
x4   x4   .  ♦ 16 |  4  4 | 8 *
.    x4-4-o2 ♦ 16 |  0  8 | * 4

x4   x4   x4

.    .    .  | 64 |  1  1  1 | 1 1 1
-------------+----+----------+------
x4   .    .  |  4 | 16  *  * | 1 1 0
.    x4   .  |  4 |  * 16  * | 1 0 1
.    .    x4 |  4 |  *  * 16 | 0 1 1
-------------+----+----------+------
x4   x4   .  ♦ 16 |  4  4  0 | 4 * *
x4   .    x4 ♦ 16 |  4  0  4 | * 4 *
.    x4   x4 ♦ 16 |  0  4  4 | * * 4