This complex polyhedral lace prism has mutually dual complex polygons for bases. Accordingly the lacing faces here simply are pyramids only: vertex atop center of 3-edge. Each base had used a hex as their real space embeddings. Thus the here to be used embedding happens to result in the according alterprism.

Incidence matrix according to Dynkin symbol

x3-3-o3 || o3-3-x3

o3-3-o3   .    .  | 8 * | 3  3 0 | 1 3 1 0
.    .    o3-3-o3 | * 8 | 0  3 3 | 0 1 3 1
-----------------+-----+--------+--------
x3   .    .    .  | 3 0 | 8  * * | 1 1 0 0
o3-3-o3 || o3-3-o3 | 1 1 | * 24 * | 0 1 1 0
.    .    .    x3 | 0 3 | *  * 8 | 0 0 1 1
-----------------+-----+--------+--------
x3-3-o3   .    .  ♦ 8 0 | 8  0 0 | 1 * * *
x3   .  || o3   .  | 3 1 | 1  3 0 | * 8 * *
.    o3 || .    x3 | 1 3 | 0  3 1 | * * 8 *
.    .    o3-3-x3 ♦ 0 8 | 0  0 8 | * * * 1

or
o3-3-o3   .    .   & | 16 |  3  3 | 1  4
--------------------+----+-------+-----
x3   .    .    .   & |  3 | 16  * | 1  1
o3-3-o3 || o3-3-o3    |  2 |  * 24 | 0  2
--------------------+----+-------+-----
x3-3-o3   .    .   & ♦  8 |  8  0 | 2  *
x3   .  || o3   .   & |  4 |  1  3 | * 16