Looks like a compound of 2 coincident cubes (cube), and indeed edges and faces both coincide by pairs, but vertices are identified (type A). Alternatively it might have 2 opposite Grünbaumian faces, while the other elements again coincide by pairs (types B).

Incidence matrix according to Dynkin symbol

o3/2o4x4*a   (type A)

.   . .    | 8 |  6 | 3 3
-----------+---+----+----
.   . x    | 2 | 24 | 1 1
-----------+---+----+----
.   o4x    | 4 |  4 | 6 *
o   . x4*a | 4 |  4 | * 6

x4/3o3o4*a   (type A)

.   . .    | 8 |  6 | 3 3
-----------+---+----+----
x   . .    | 2 | 24 | 1 1
-----------+---+----+----
x4/3o .    | 4 |  4 | 6 *
x   . o4*a | 4 |  4 | * 6

o4/3x4/3o3/2*a   (type A)

.   .   . | 8 |  6 | 3 3
----------+---+----+----
.   x   . | 2 | 24 | 1 1
----------+---+----+----
o4/3x   . | 4 |  4 | 6 *
.   x4/3o | 4 |  4 | * 6

x2β4x   (type B)

both( . . . ) | 16 | 1 1 1 | 1 1 1
--------------+----+-------+------
both( x . . ) |  2 | 8 * * | 0 1 1
both( . . x ) |  2 | * 8 * | 1 1 0
sefa( . β4x ) |  2 | * * 8 | 1 0 1
--------------+----+-------+------
      . β4x   ♦  8 | 0 4 4 | 2 * *
both( x . x ) |  4 | 2 2 0 | * 4 *
sefa( x2β4x ) |  4 | 2 0 2 | * * 4

starting figure: x x4x

β2β4x   (type B)

both( . . . ) | 16 | 1 1 1 | 1 2
--------------+----+-------+----
both( s2s   ) |  2 | 8 * * | 0 2
both( . . x ) |  2 | * 8 * | 1 1
sefa( . β4x ) |  2 | * * 8 | 1 1
--------------+----+-------+----
        β4x   ♦  8 | 0 4 4 | 2 *
sefa( β2β4x ) |  4 | 2 1 1 | * 8

starting figure: x x4x