Looks like a compound of a prismatointercepted prismatotesseractihexadecachoron (pinpith) plus 48 additional squares coincident to those which are diametral to the octahedra (oct).

Incidence matrix according to Dynkin symbol

o3β3o4x

both( . . . . ) | 96 |  2   4   4 |  1  2  2  4  4  4 |  1  2 2  4
----------------+----+------------+-------------------+-----------
both( . . . x ) |  2 | 96   *   * |  1  0  0  0  2  2 |  0  1 2  2
sefa( o3β . . ) |  2 |  * 192   * |  0  1  0  1  1  0 |  1  1 0  1
sefa( . β3o . ) |  2 |  *   * 192 |  0  0  1  1  0  1 |  1  0 1  1
----------------+----+------------+-------------------+-----------
both( . . o4x ) |  4 |  4   0   0 | 24  *  *  *  *  * |  0  0 2  0
      o3β . .   ♦  3 |  0   3   0 |  * 64  *  *  *  * |  1  1 0  0
      . β3o .   ♦  3 |  0   0   3 |  *  * 64  *  *  * |  1  0 1  0
sefa( o3β3o . ) |  4 |  0   2   2 |  *  *  * 96  *  * |  1  0 0  1
sefa( o3β 2 x ) |  4 |  2   2   0 |  *  *  *  * 96  * |  0  1 0  1
sefa( . β3o4x ) |  8 |  4   0   4 |  *  *  *  *  * 48 |  0  0 1  1
----------------+----+------------+-------------------+-----------
      o3β3o .   ♦  6 |  0  12  12 |  0  4  4  6  0  0 | 16  * *  *
      o3β 2 x   ♦  6 |  3   6   0 |  0  2  0  0  3  0 |  * 32 *  *
      . β3o4x   ♦ 24 | 24   0  24 |  6  0  8  0  0  6 |  *  * 8  *
sefa( o3β3o4x ) ♦ 16 |  8   8   8 |  0  0  0  4  4  2 |  *  * * 24

starting figure: o3x3o4x