Acronym	...
Name	Shephard's 5-generalised octahedron, complex polyhedron x2-3-o2-4-o5, β53
	©
Circumradius	1/sqrt(2) = 0.707107
Vertex figure	x2-4-o5
Coordinates	(ε5n, 0, 0)/sqrt(2)   & all permutations, each for any 1≤n≤5, where ε5=exp(2πi/5)
Dual	x5-4-o2-3-o2
Face vector	15, 75, 125
Confer	more general: x2-3-o2-4-op   general polytopal classes: complex polytopes
External links

This complex polyhedron is somewhat special in so far as both its edges as well as its faces remain mere real polytopes. The vertex figure however, becomes the complex polygon x₂-4-o₅. Thence it not only is the formal dual of the complex polytope x₅-4-o₂-3-o₂, but also its real space embedding is still 6-dimensional and is the dual of that there given one, i.e. the tegum poduct of 3 (fully orthogonal) pentagons. In fact, the to be chosen triangles therefrom are just those lacing triangles, which have one vertex on each of those pentagons. For the provided metrical properties the overall scaling of that embedding is to be chosen such that those lacing edges become unity.

The incidence matrix thence is simply the same as for Shephard's 5-generalised cube, just rotated 180° around

Incidence matrix according to Dynkin symbol

x2-3-o2-4-o5

.    .    .  | 15 ♦ 10 |  25
-------------+----+----+-----
x2   .    .  |  2 | 75 |   5
-------------+----+----+-----
x2-3-o2   .  |  3 |  3 | 125