Looks like a compound of 3 pentachora (pen) with (in type A) every triple of vertices identified, edges and faces coincide by 3, cells by pairs. One of the coincident cells each is a Grünbaumian double cover of the tetrahedron (2tet) itself.

Further there also is type B, the then non-uniform mere pyramid on 3tet, where the tip is a single vertex by definition, while the base vertices have triples of coincident vertices and edges. The lacing edges and faces therefore are coincident by three as well. Base faces however are coincident pairs of a triangle and a {6/2} each and thence induced the lacing cells will be coincident pairs of a tet and a 2tet (in fact a {6/2}-pyramid) each. The bottom cell clearly is a single 3tet.

Incidence matrix according to Dynkin symbol

β3o3o3o   (type A)

both( . . . . ) | 5 ♦ 12 |  6 12 | 4 4
----------------+---+----+-------+----
sefa( β3o . . ) | 2 | 30 |  1  2 | 2 1
----------------+---+----+-------+----
      β3o . .   ♦ 3 |  3 | 10  * | 2 0
sefa( β3o3o . ) | 3 |  3 |  * 20 | 1 1
----------------+---+----+-------+----
      β3o3o .   ♦ 4 | 12 |  4  4 | 5 *
sefa( β3o3o3o ) ♦ 4 |  6 |  0  4 | * 5

starting figure: x3o3o3o

ox3/2ox3oo&#x   (type B)   → height = sqrt(5/8) = 0.790569

o.3/2o.3o.    | 1  * ♦ 12 0  0 | 6 12 0 0 | 4 4 0
.o3/2.o3.o    | * 12 ♦  1 1  2 | 1  2 2 1 | 2 1 1
--------------+------+---------+----------+------
oo3/2oo3oo&#x | 1  1 | 12 *  * | 1  2 0 0 | 2 1 0
.x   .. ..    | 0  2 |  * 6  * | 1  0 2 0 | 2 0 1
..   .x ..    | 0  2 |  * * 12 | 0  1 1 1 | 1 1 1
--------------+------+---------+----------+------
ox   .. ..&#x | 1  2 |  2 1  0 | 6  * * * | 2 0 0
..   ox ..&#x | 1  2 |  2 0  1 | * 12 * * | 1 1 0
.x3/2.x ..    | 0  6 |  0 3  3 | *  * 4 * | 1 0 1
..   .x3.o    | 0  3 |  0 0  3 | *  * * 4 | 0 1 1
--------------+------+---------+----------+------
ox3/2ox ..&#x ♦ 1  6 |  6 3  3 | 3  3 1 0 | 4 * *  {6/2}-pyramid
..   ox3oo&#x ♦ 1  3 |  3 0  3 | 0  3 0 1 | * 4 *
.x3/2.x3.o    ♦ 0 12 |  0 6 12 | 0  0 4 4 | * * 1