Acronym	...
Name	Shephard's 3-generalised tesseract, complex polychoron x3-4-o2-3-o2-3-o2, γ34
	©
Vertex figure	tet
Coordinates	(ε3n, ε3m, ε3k, ε3l)   for any 1≤n,m,k,l≤3, where ε3=exp(2πi/3)
Dual	x2-3-o2-3-o2-4-o3
Face vector	81, 108, 54, 12
Confer	more general: xp-4-o2-3-o2-3-o2   real space embedding: triquip general polytopal classes: complex polytopes
External links

The (complex) faces are x₃-4-o₂, the (complex) cells are x₃-4-o₂-3-o₂, and the vertex figure here is just x₂-3-o₂-3-o₂, i.e. nothing but the real space tet.

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. In fact, these complex polychora simply re-use the even-dimensional elements of all the triangles, triddips, and trittips from this real quadprism only.

Incidence matrix according to Dynkin symbol

x3-4-o2-3-o2-3-o2

.    .    .    .  | 81 ♦   4 |  6 |  4
-----------------+----+-----+----+---
x3   .    .    .  |  3 | 108 |  3 |  3
-----------------+----+-----+----+---
x3-4-o2   .    .  ♦  9 |   6 | 54 |  2
-----------------+----+-----+----+---
x3-4-o2-3-o2   .  ♦ 27 |  27 |  9 | 12

x3   x3-4-o2-3-o2

.    .    .    .  | 81 ♦  1  3 |  3  3 | 3 1
-----------------+----+-------+-------+----
x3   .    .    .  |  3 | 27  * |  3  0 | 3 0
.    x3   .    .  |  3 |  * 81 |  1  2 | 2 1
-----------------+----+-------+-------+----
x3   x3   .    .  ♦  9 |  3  3 | 27  * | 2 0
.    x3-4-o2   .  ♦  9 |  0  6 |  * 27 | 1 1
-----------------+----+-------+-------+----
x3   x3-4-o2   .  ♦ 27 |  9 18 |  6  3 | 9 *
.    x3-4-o2-3-o2 ♦ 27 |  0 27 |  0  9 | * 3

x3-4-o2   x3-4-o2

.    .    .    .  | 81 ♦  2  2 | 1  4 1 | 2 2
-----------------+----+-------+--------+----
x3   .    .    .  |  3 | 54  * | 1  2 0 | 2 1
.    .    x3   .  |  3 |  * 54 | 0  2 1 | 1 2
-----------------+----+-------+--------+----
x3-4-o2   .    .  ♦  9 |  6  0 | 9  * * | 2 0
x3   .    x3   .  ♦  9 |  3  3 | * 36 * | 1 1
.    .    x3-4-o2 ♦  9 |  0  6 | *  * 9 | 0 2
-----------------+----+-------+--------+----
x3-4-o2   x3   .  ♦ 27 | 18  9 | 3  6 0 | 6 *
x3   .    x3-4-o2 ♦ 27 |  9 18 | 0  6 3 | * 6

x3   x3   x3-4-o2

.    .    .    .  | 81 ♦  1  1  2 | 1  2  2 1 | 2 1 1
-----------------+----+----------+-----------+-------
x3   .    .    .  |  3 | 27  *  * | 1  2  0 0 | 2 1 0
.    x3   .    .  |  3 |  * 27  * | 1  0  2 0 | 2 0 1
.    .    x3   .  |  3 |  *  * 54 | 0  1  1 1 | 1 1 1
-----------------+----+----------+-----------+-------
x3   x3   .    .  ♦  9 |  3  3  0 | 9  *  * * | 2 0 0
x3   .    x3   .  ♦  9 |  3  0  3 | * 18  * * | 1 1 0
.    x3   x3   .  ♦  9 |  0  3  3 | *  * 18 * | 1 0 1
.    .    x3-4-o2 ♦  9 |  0  0  6 | *  *  * 9 | 0 1 1
-----------------+----+----------+-----------+-------
x3   x3   x3   .  ♦ 27 |  9  9  9 | 3  3  3 0 | 6 * *
x3   .    x3-4-o2 ♦ 27 |  9  0 18 | 0  6  0 3 | * 3 *
.    x3   x3-4-o2 ♦ 27 |  0  9 18 | 0  0  6 3 | * * 3

x3   x3   x3   x3 

.    .    .    .  | 81 ♦  1  1  1  1 | 1 1 1 1 1 1 | 1 1 1 1
-----------------+----+-------------+-------------+--------
x3   .    .    .  |  3 | 27  *  *  * | 1 1 1 0 0 0 | 1 1 1 0
.    x3   .    .  |  3 |  * 27  *  * | 1 0 0 1 1 0 | 1 1 0 1
.    .    x3   .  |  3 |  *  * 27  * | 0 1 0 1 0 1 | 1 0 1 1
.    .    .    x3 |  3 |  *  *  * 27 | 0 0 1 0 1 1 | 0 1 1 1
-----------------+----+-------------+-------------+--------
x3   x3   .    .  ♦  9 |  3  3  0  0 | 9 * * * * * | 1 1 0 0
x3   .    x3   .  ♦  9 |  3  0  3  0 | * 9 * * * * | 1 0 1 0
x3   .    .    x3 ♦  9 |  3  0  0  3 | * * 9 * * * | 0 1 1 0
.    x3   x3   .  ♦  9 |  0  3  3  0 | * * * 9 * * | 1 0 0 1
.    x3   .    x3 ♦  9 |  0  3  0  3 | * * * * 9 * | 0 1 0 1
.    .    x3   x3 ♦  9 |  0  0  3  3 | * * * * * 9 | 0 0 1 1
-----------------+----+-------------+-------------+--------
x3   x3   x3   .  ♦ 27 |  3  3  3  0 | 3 3 0 3 0 0 | 3 * * *
x3   x3   .    x3 ♦ 27 |  3  3  0  3 | 3 0 3 0 3 0 | * 3 * *
x3   .    x3   x3 ♦ 27 |  3  0  3  3 | 0 3 3 0 0 3 | * * 3 *
.    x3   x3   x3 ♦ 27 |  0  3  3  3 | 0 0 0 3 3 3 | * * * 3