Acronym	...
Name	Shephard's 5-generalised cube, complex polyhedron x5-4-o2-3-o2, γ53
	©
Circumradius	sqrt[(15+3 sqrt(5))/10] = 1.473370
Vertex figure	trig
Coordinates	(ε5n, ε5m, ε5k) sqrt[(5+sqrt(5))/10]   for any 1≤n,m,k≤5, where ε5=exp(2πi/5)
Dual	x2-3-o2-4-o5
Face vector	125, 75, 15
Confer	more general: xp-4-o2-3-o2   real space embedding: pettip general polytopal classes: complex polytopes
External links

The (complex) edges then are x₅-4-o₂, which were nothing but the set of pentagons of the respective pedips 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

x5-4-o2-3-o2

.    .    .  | 125 |  3 |  3
-------------+-----+----+---
x5   .    .  |   5 | 75 |  2
-------------+-----+----+---
x5-4-o2   .  ♦  25 | 10 | 15

x5   x5-4-o2

.    .    .  | 125 |  1  2 |  2 1
-------------+-----+-------+-----
x5   .    .  |   5 | 25  * |  2 0
.    x5   .  |   5 |  * 50 |  1 1
-------------+-----+-------+-----
x5   x5   .  ♦  25 |  5  5 | 10 *
.    x5-4-o2 ♦  25 |  0 10 |  * 5

x5   x3   x3

.    .    .  | 125 |  1  1  1 | 1 1 1
-------------+-----+----------+------
x5   .    .  |   5 | 25  *  * | 1 1 0
.    x5   .  |   5 |  * 25  * | 1 0 1
.    .    x5 |   5 |  *  * 25 | 0 1 1
-------------+-----+----------+------
x5   x5   .  ♦  25 |  5  5  0 | 5 * *
x5   .    x5 ♦  25 |  5  0  5 | * 5 *
.    x5   x5 ♦  25 |  0  5  5 | * * 5