Looks like a compound of 2 coincident prismatorhombated tesseracts (prit).

Incidence matrix according to Dynkin symbol

x3x3β4x

both( . . . . ) | 384 |   1   1   1   1   1 |  1  1  1  1  1  1  1  1 |  1  1  1 1  1
----------------+-----+---------------------+-------------------------+--------------
both( x . . . ) |   2 | 192   *   *   *   * |  1  1  0  0  0  1  1  0 |  1  1  1 0  1
both( . x . . ) |   2 |   * 192   *   *   * |  1  0  1  1  0  0  0  0 |  1  1  0 1  0
both( . . . x ) |   2 |   *   * 192   *   * |  0  1  1  0  1  0  0  0 |  1  0  1 1  0
sefa( . x3β . ) |   2 |   *   *   * 192   * |  0  0  0  1  0  1  0  1 |  0  1  0 1  1
sefa( . . s4x ) |   2 |   *   *   *   * 192 |  0  0  0  0  1  0  1  1 |  0  0  1 1  1
----------------+-----+---------------------+-------------------------+--------------
both( x3x . . ) |   6 |   3   3   0   0   0 | 64  *  *  *  *  *  *  * |  1  1  0 0  0
both( x . . x ) |   4 |   2   0   2   0   0 |  * 96  *  *  *  *  *  * |  1  0  1 0  0
both( . x . x ) |   4 |   0   2   2   0   0 |  *  * 96  *  *  *  *  * |  1  0  0 1  0
      . x3β .   ♦   6 |   0   3   0   3   0 |  *  *  * 64  *  *  *  * |  0  1  0 1  0
both( . . s4x ) ♦   4 |   0   0   2   0   2 |  *  *  *  * 96  *  *  * |  0  0  1 1  0
sefa( x3x3β . ) |   6 |   3   0   0   3   0 |  *  *  *  *  * 64  *  * |  0  1  0 0  1
sefa( x 2 s4x ) |   4 |   2   0   0   0   2 |  *  *  *  *  *  * 96  * |  0  0  1 0  1
sefa( . x3β4x ) |   4 |   0   0   0   2   2 |  *  *  *  *  *  *  * 96 |  0  0  0 1  1
----------------+-----+---------------------+-------------------------+--------------
both( x3x . x ) ♦  12 |   6   6   6   0   0 |  2  3  3  0  0  0  0  0 | 32  *  * *  *
      x3x3β .   ♦  24 |  12  12   0  12   0 |  4  0  0  4  0  4  0  0 |  * 16  * *  *
both( x 2 s4x ) ♦   8 |   4   0   4   0   4 |  0  2  0  0  2  0  2  0 |  *  * 48 *  *
      . x3β4x   ♦  48 |   0  24  24  24  24 |  0  0 12  8 12  0  0 12 |  *  *  * 8  *
sefa( x3x3β4x ) ♦  12 |   6   0   0   6   6 |  0  0  0  0  0  2  3  3 |  *  *  * * 32

starting figure: x3x3x4x

x3x3β4β

both( . . . . ) | 384 |   1   1   1   2 |  1  1  1  1   2   2 |  1  1 1  2
----------------+-----+-----------------+---------------------+-----------
both( x . . . ) |   2 | 192   *   *   * |  1  0  0  1   2   0 |  1  1 0  2
both( . x . . ) |   2 |   * 192   *   * |  1  1  0  0   0   1 |  1  0 1  1
sefa( . x3β . ) |   2 |   *   * 192   * |  0  1  0  1   0   1 |  1  0 1  1
sefa( . . s4s ) |   2 |   *   *   * 384 |  0  0  1  0   1   1 |  0  1 1  1
----------------+-----+-----------------+---------------------+-----------
both( x3x . . ) |   6 |   3   3   0   0 | 64  *  *  *   *   * |  1  0 0  1
      . x3β .   ♦   6 |   0   3   3   0 |  * 64  *  *   *   * |  1  0 1  0
both( . . s4s ) ♦   4 |   0   0   0   4 |  *  * 96  *   *   * |  0  1 1  0
sefa( x3x3β . ) |   6 |   3   0   3   0 |  *  *  * 64   *   * |  1  0 0  1
sefa( x 2 s4s ) |   4 |   2   0   0   2 |  *  *  *  * 192   * |  0  1 0  1
sefa( . x3β4β ) |   4 |   0   1   1   2 |  *  *  *  *   * 192 |  0  0 1  1
----------------+-----+-----------------+---------------------+-----------
      x3x3β .   ♦  24 |  12  12  12   0 |  4  4  0  4   0   0 | 16  * *  *
both( x 2 s4s ) ♦   8 |   4   0   0   8 |  0  0  2  0   4   0 |  * 48 *  *
      . x3β4β   ♦  48 |   0  24  24  48 |  0  8 12  0   0  24 |  *  * 8  *
sefa( x3x3β4β ) ♦  12 |   6   3   3   6 |  1  0  0  1   3   3 |  *  * * 64

starting figure: x3x3x4x