Looks like a compound of 2 coincident small rhombated tesseracts (srit) with 64 further triangular prisms (trip) which increase the so far double coincidences to quadruple coincidences.

Incidence matrix according to Dynkin symbol

o3β3β4x

both( . . . . ) | 192 |  1   2   4  1 |  1   2  1   1  2   4 |  1  1  2  3
----------------+-----+---------------+----------------------+------------
both( . . . x ) |   2 | 96   *   *  * |  0   0  1   0  2   2 |  0  1  2  2
sefa( o3β . . ) |   2 |  * 192   *  * |  1   0  0   1  1   0 |  1  1  0  1
sefa( . s3s . ) |   2 |  *   * 384  * |  0   1  0   1  0   1 |  1  0  1  1
sefa( . . s4x ) |   2 |  *   *   * 96 |  0   0  1   0  0   2 |  0  0  2  1
----------------+-----+---------------+----------------------+------------
      o3β . .   ♦   3 |  0   3   0  0 | 64   *  *   *  *   * |  1  1  0  0
both( . s3s . ) ♦   3 |  0   0   3  0 |  * 128  *   *  *   * |  1  0  1  0
both( . . s4x ) ♦   4 |  2   0   0  2 |  *   * 48   *  *   * |  0  0  2  0
sefa( o3β3β . ) |   3 |  0   1   2  0 |  *   *  * 192  *   * |  1  0  0  1
sefa( o3β 2 x ) |   4 |  2   2   0  0 |  *   *  *   * 96   * |  0  1  0  1
sefa( . s3s4x ) |   4 |  1   0   2  1 |  *   *  *   *  * 192 |  0  0  1  1
----------------+-----+---------------+----------------------+------------
      o3β3β .   ♦  12 |  0  12  24  0 |  4   8  0  12  0   0 | 16  *  *  *
      o3β 2 x   ♦   6 |  3   6   0  0 |  2   0  0   0  3   0 |  * 32  *  *
both( . s3s4x ) ♦  24 | 12   0  24 12 |  0   8  6   0  0  12 |  *  * 16  *
sefa( o3β3β4x ) ♦   6 |  2   2   4  1 |  0   0  0   2  1   2 |  *  *  * 96

starting figure: o3x3x4x