Acronym	...
Name	Shephard's 3-generalised cube, complex polyhedron x3-4-o2-3-o2, γ33
	©
Circumradius	1
Vertex figure	trig
Coordinates	(ε3n, ε3m, ε3k)/sqrt(3)   for any 1≤n,m,k≤3, where ε3=exp(2πi/3)
Dual	x2-3-o2-4-o3
Face vector	27, 27, 9
Confer	more general: xp-4-o2-3-o2   real space embedding: trittip general polytopal classes: complex polytopes
External links

The (complex) edges then are x₃-4-o₂, which were nothing but the set of triangles of the respective triddips each. 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

x3-4-o2-3-o2

.    .    .  | 27 |  3 | 3
-------------+----+----+--
x3   .    .  |  3 | 27 | 2
-------------+----+----+--
x3-4-o2   .  ♦  9 |  6 | 9

x3   x3-4-o2

.    .    .  | 27 | 1  2 | 2 1
-------------+----+------+----
x3   .    .  |  3 | 9  * | 2 0
.    x3   .  |  3 | * 18 | 1 1
-------------+----+------+----
x3   x3   .  ♦  9 | 3  3 | 6 *
.    x3-4-o2 ♦  9 | 0  6 | * 3

x3   x3   x3

.    .    .  | 27 | 1 1 1 | 1 1 1
-------------+----+-------+------
x3   .    .  |  3 | 9 * * | 1 1 0
.    x3   .  |  3 | * 9 * | 1 0 1
.    .    x3 |  3 | * * 9 | 0 1 1
-------------+----+-------+------
x3   x3   .  ♦  9 | 3 3 0 | 3 * *
x3   .    x3 ♦  9 | 3 0 3 | * 3 *
.    x3   x3 ♦  9 | 0 3 3 | * * 3