Looks like a compound of 2 tetrahedra (tet) with vertices identified, edges and faces both coincide by pairs.

Incidence matrix according to Dynkin symbol

x3/2o3o3*a   (type A)

.   . .    | 4 |  6 | 3 3
-----------+---+----+----
x   . .    | 2 | 12 | 1 1
-----------+---+----+----
x3/2o .    | 3 |  3 | 4 *
x   . o3*a | 3 |  3 | * 4

o3/2o3x3*a   (type A)

.   . .    | 4 |  6 | 3 3
-----------+---+----+----
.   . x    | 2 | 12 | 1 1
-----------+---+----+----
.   o3x    | 3 |  3 | 4 *
o   . x3*a | 3 |  3 | * 4

x3/2o3/2o3/2*a   (type A)

.   .   .      | 4 |  6 | 3 3
---------------+---+----+----
x   .   .      | 2 | 12 | 1 1
---------------+---+----+----
x3/2o   .      | 3 |  3 | 4 *
x   .   o3/2*a | 3 |  3 | * 4

β3o3o   (type A)

both( . . . ) | 4 |  6 | 3 3
--------------+---+----+----
sefa( β3o . ) | 2 | 12 | 1 1
--------------+---+----+----
      β3o .   ♦ 3 |  3 | 4 *
sefa( β3o3o ) | 3 |  3 | * 4

β3/2o3o   (type A)

both( .   . . ) | 4 |  6 | 3 3
----------------+---+----+----
sefa( β3/2o . ) | 2 | 12 | 1 1
----------------+---+----+----
      β3/2o .   ♦ 3 |  3 | 4 *
sefa( β3/2o3o ) | 3 |  3 | * 4

o3/2o3β   (type A)

both( .   . . ) | 4 |  6 | 3 3
----------------+---+----+----
sefa( .   o3β ) | 2 | 12 | 1 1
----------------+---+----+----
      .   o3β   ♦ 3 |  3 | 4 *
sefa( o3/2o3β ) | 3 |  3 | * 4

β3/2o3/2o   (type A)

both( .   .   . ) | 4 |  6 | 3 3
------------------+---+----+----
sefa( β3/2o   . ) | 2 | 12 | 1 1
------------------+---+----+----
      β3/2o   .   ♦ 3 |  3 | 4 *
sefa( β3/2o3/2o ) | 3 |  3 | * 4

β3/2x3o   (type A)

third( .   . . ) | 4 |  6 | 3 3
-----------------+---+----+----
both(  .   x . ) | 2 | 12 | 1 1
-----------------+---+----+----
both(  .   x3o ) | 3 |  3 | 4 *
       β3/2x .   ♦ 3 |  3 | * 4
       
(Here sefa( β3/2x3o ) degenerates to a point,
thus 3 points each from x3/2x3o will be identified.)

x3/2β3o   (type B)

third( .   . . ) | 4 | 3 3 | 3 3
-----------------+---+-----+----
both(  x   . . ) | 2 | 6 * | 2 0
sefa(  .   β3o ) | 2 | * 6 | 0 2
-----------------+---+-----+----
       x3/2β .   ♦ 3 | 3 0 | 4 *
       .   β3o   ♦ 3 | 0 3 | * 4
       
(Here the tetragons sefa( x3/2β3o ) degenerate to a line,
thus their remaining edges, sefa( β3o ), will be identified,
alike pairwise their vertices.)