Looks like a compound of an icositetrachoron (ico) with the compound of 3 vertex-inscribed tesseracts (tes, i.e. the great icositetrachoron, gico). But the squares all are doubled up to completely coincident pairs. Even so, the cubes are not doubled up.

Incidence matrix according to Dynkin symbol

β3o4o3o

both( . . . . ) | 24 |  24 | 12  24 |  6  8
----------------+----+-----+--------+------
sefa( β3o . . ) |  2 | 288 |  1   2 |  2  1
----------------+----+-----+--------+------
      β3o . .   ♦  3 |   3 | 96   * |  2  0
sefa( β3o4o . ) |  4 |   4 |  * 144 |  1  1
----------------+----+-----+--------+------
      β3o4o .   ♦  6 |  24 |  8   6 | 24  *
sefa( β3o4o3o ) ♦  8 |  12 |  0   6 |  * 24

o3β3o4o

both( . . . . ) | 24 |  8  16 |  4  8 16  8 |  4 2  8
----------------+----+--------+-------------+--------
sefa( o3β . . ) |  2 | 96   * |  1  0  2  0 |  2 0  1
sefa( . β3o . ) |  2 |  * 192 |  0  1  1  1 |  1 1  1
----------------+----+--------+-------------+--------
      o3β . .   ♦  3 |  3   0 | 32  *  *  * |  2 0  0
      . β3o .   ♦  3 |  0   3 |  * 64  *  * |  1 1  0
sefa( o3β3o . ) |  4 |  2   2 |  *  * 96  * |  1 0  1
sefa( . β3o4o ) |  4 |  0   4 |  *  *  * 48 |  0 1  1
----------------+----+--------+-------------+--------
      o3β3o .   ♦  6 | 12  12 |  4  4  6  0 | 16 *  *
      . β3o4o   ♦  6 |  0  24 |  0  8  0  6 |  * 8  *
sefa( o3β3o4o ) ♦  8 |  4   8 |  0  0  4  2 |  * * 24

o3β3o *b3o

both( . . .    . ) | 24 |  8  8  8 |  4  4  4  8  8  8 | 2 2 2  8
-------------------+----+----------+-------------------+---------
sefa( o3β .    . ) |  2 | 96  *  * |  1  0  0  1  1  0 | 1 1 0  1
sefa( . β3o    . ) |  2 |  * 96  * |  0  1  0  1  0  1 | 1 0 1  1
sefa( . β . *b3o ) |  2 |  *  * 96 |  0  0  1  0  1  1 | 0 1 1  1
-------------------+----+----------+-------------------+---------
      o3β .    .   ♦  3 |  3  0  0 | 32  *  *  *  *  * | 1 1 0  0
      . β3o    .   ♦  3 |  0  3  0 |  * 32  *  *  *  * | 1 0 1  0
      . β . *b3o   ♦  3 |  0  0  3 |  *  * 32  *  *  * | 0 1 1  0
sefa( o3β3o    . ) |  4 |  2  2  0 |  *  *  * 48  *  * | 1 0 0  1
sefa( o3β . *b3o ) |  4 |  2  0  2 |  *  *  *  * 48  * | 0 1 0  1
sefa( . β3o *b3o ) |  4 |  0  2  2 |  *  *  *  *  * 48 | 0 0 1  1
-------------------+----+----------+-------------------+---------
      o3β3o    .   ♦  6 | 12 12  0 |  4  4  0  6  0  0 | 8 * *  *
      o3β . *b3o   ♦  6 | 12  0 12 |  4  0  4  0  6  0 | * 8 *  *
      . β3o *b3o   ♦  6 |  0 12 12 |  0  4  4  0  0  6 | * * 8  *
sefa( o3β3o *b3o ) ♦  8 |  4  4  4 |  0  0  0  2  2  2 | * * * 24