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This polyhedron can be obtained from snic, if the 4 equatorial squares get reduced into their shallower diagonals each, thereby replacing the to be omitted vertices by their vertex figure pentagon.

Incidence matrix

4 * * * | 2 1 0 0 0 0 | 1 2 0 0  top polar
* 4 * * | 0 1 1 1 0 0 | 0 2 1 0  upper tropal
* * 4 * | 0 0 1 1 1 0 | 0 1 2 0  lower tropal
* * * 4 | 0 0 0 0 1 2 | 0 0 2 1  bottom polar
--------+-------------+--------
2 0 0 0 | 4 * * * * * | 1 1 0 0  x
1 1 0 0 | * 4 * * * * | 0 2 0 0  x
0 1 1 0 | * * 4 * * * | 0 1 1 0  x (zig)
0 1 1 0 | * * * 4 * * | 0 1 1 0  q (zag)
0 0 1 1 | * * * * 4 * | 0 0 2 0  x
0 0 0 2 | * * * * * 4 | 0 0 1 1  x
--------+-------------+--------
4 0 0 0 | 4 0 0 0 0 0 | 1 * * *  x4o oct-verf
2 2 1 0 | 1 2 1 1 0 0 | * 4 * *  (xxxxq) snic-verf
0 1 2 2 | 0 0 1 1 2 1 | * * 4 *  (xxxxq) snic-verf
0 0 0 4 | 0 0 0 0 0 4 | * * * 1  x4o oct-verf

or
8 * | 2 1 0 0 | 1 2  polar
* 8 | 0 1 1 1 | 0 3  tropal
----+---------+----
2 0 | 8 * * * | 1 1  x
1 1 | * 8 * * | 0 2  x
0 2 | * * 4 * | 0 2  x (zig)
0 2 | * * * 4 | 0 2  q (zag)
----+---------+----
4 0 | 4 0 0 0 | 2 *  x4o oct-verf
2 3 | 1 2 1 1 | * 8  (xxxxq) snic-verf