Acronym	hocucup
Name	hollow cuboid cupoliprism, six tepe blend, "coord-planes squares star" atop "coord-planes squares star"
Circumradius	sqrt(5/8) = 0.790569
General of army	cope
Face vector	24, 72, 72, 18
Confer	related segmentochora: "coord-axes edge star" || cube   "thah-squares star" || cube   "coord-planes square star" || cube   general polytopal classes: scaliform
External links

Idea: take 2 parallel 1/q-sized cuboctahedra as pseudo bases and consider from those the 3 diametral (then unit sized) squares each. From these inscribed squares their edges are to be taken only. Thereby 2 of those each describe a unit cross into the (smaller) cuboctahedral square faces. Then connect each (hollow) unit square of one cuboctahedral pseudo base with any of these parallel crosses of the other cuboctahedral pseudo base, and vice versa, by means of 6+6=12 tutrips. The remainder then would be filled by 6 stellated octahedra, each thereby connecting pairs of alike crosses on either cuboctahedron in the sense of a digonal antiprism for each of its tetrahedral components.

Even though it is inscribed into cope its name was chosen to contain cubic instead, as it does not contain cells within octahedral positions.

This polychoron comes in 2 types, considering the pairs of tets each as a compound facet (type A), or considering them as merely corealmic, separate facets (type B).

Incidence matrix

(Type A)
pseudo 1/q-co || pseudo 1/q-co   → height = 1/sqrt(2) = 0.707107

24 |  2  4 |  4  6 |  4 2
---+-------+-------+-----
 2 | 24  * |  2  2 |  3 1  hollow base square's edges
 2 |  * 48 |  1  2 |  2 1  lacing edges
---+-------+-------+-----
 4 |  2  2 | 24  * |  2 0
 3 |  1  2 |  * 48 |  1 1
---+-------+-------+-----
 8 |  6  8 |  4  4 | 12 *  tutrip
 8 |  4  8 |  0  8 |  * 6  so

(Type B)
pseudo 1/q-co || pseudo 1/q-co   → height = 1/sqrt(2) = 0.707107

24 |  2  4 |  4  6 |  4  2
---+-------+-------+------
 2 | 24  * |  2  2 |  3  1  hollow base square's edges
 2 |  * 48 |  1  2 |  2  1  lacing edges
---+-------+-------+------
 4 |  2  2 | 24  * |  2  0
 3 |  1  2 |  * 48 |  1  1
---+-------+-------+------
 8 |  6  8 |  4  4 | 12  *  tutrip
 4 |  2  4 |  0  4 |  * 12  tet