Looks like a compound of 2 coincident prismatorhombated hexadecachora (proh), and indeed all but the Grünbaumian elements coincide by pairs. It occurs in different types: type A uses doubly wound hexagonal prisms (2trip), whereas type B uses 2 coincident triangular prisms (trip) each.

Incidence matrix according to Dynkin symbol

x3β3x4x   (type A)

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  1  0  0  0  0 |  1  1  1 0  0
both( . . x . ) |   2 |   * 192   *   *   * |  1  0  1  0  1  0  0  0 |  1  1  0 1  0
both( . . . x ) |   2 |   *   * 192   *   * |  0  1  1  0  0  0  1  1 |  1  0  1 1  1
sefa( x3β . . ) |   2 |   *   *   * 192   * |  0  0  0  1  0  1  1  0 |  0  1  1 0  1
sefa( . β3x . ) |   2 |   *   *   *   * 192 |  0  0  0  0  1  1  0  1 |  0  1  0 1  1
----------------+-----+---------------------+-------------------------+--------------
both( x . x . ) |   4 |   2   2   0   0   0 | 96  *  *  *  *  *  *  * |  1  1  0 0  0
both( x . . x ) |   4 |   2   0   2   0   0 |  * 96  *  *  *  *  *  * |  1  0  1 0  0
both( . . x4x ) |   8 |   0   4   4   0   0 |  *  * 48  *  *  *  *  * |  1  0  0 1  0
      x3β . .   ♦   6 |   3   0   0   3   0 |  *  *  * 64  *  *  *  * |  0  1  1 0  0
      . β3x .   ♦   6 |   0   3   0   0   3 |  *  *  *  * 64  *  *  * |  0  1  0 1  0
sefa( x3β3x . ) |   4 |   0   0   0   2   2 |  *  *  *  *  * 96  *  * |  0  1  0 0  1
sefa( x3β 2 x ) |   4 |   0   0   2   2   0 |  *  *  *  *  *  * 96  * |  0  0  1 0  1
sefa( . β3x4x ) |   8 |   0   0   4   0   4 |  *  *  *  *  *  *  * 48 |  0  0  0 1  1
----------------+-----+---------------------+-------------------------+--------------
both( x . x4x ) ♦  16 |   8   8   8   0   0 |  4  4  2  0  0  0  0  0 | 24  *  * *  *
      x3β3x .   ♦  24 |  12  12   0  12  12 |  6  0  0  4  4  6  0  0 |  * 16  * *  *
      x3β 2 x   ♦  12 |   6   0   6   6   0 |  0  3  0  2  0  0  3  0 |  *  * 32 *  *
      . β3x4x   ♦  48 |   0  24  24   0  24 |  0  0  6  0  8  0  0  6 |  *  *  * 8  *
both( x3β3x4x ) ♦  16 |   0   0   8   8   8 |  0  0  0  0  0  4  4  2 |  *  *  * * 24

starting figure: x3x3x4x

β3β3x4x   (type B)

both( . . . . ) | 384 |   1   1   2   1 |  1   1  1   2   2  1 |  1  1 1  2
----------------+-----+-----------------+----------------------+-----------
both( . . x . ) |   2 | 192   *   *   * |  1   0  1   1   0  0 |  1  0 1  1
both( . . . x ) |   2 |   * 192   *   * |  1   0  0   0   2  1 |  0  1 1  2
sefa( s3s . . ) |   2 |   *   * 384   * |  0   1  0   1   1  0 |  1  1 0  1
sefa( . β3x . ) |   2 |   *   *   * 192 |  0   0  1   1   0  1 |  1  0 1  1
----------------+-----+-----------------+----------------------+-----------
both( . . x4x ) |   8 |   4   4   0   0 | 48   *  *   *   *  * |  0  0 1  1
both( s3s . . ) ♦   3 |   0   0   3   0 |  * 128  *   *   *  * |  1  1 0  0
      . β3x .   ♦   6 |   3   0   0   3 |  *   * 64   *   *  * |  1  0 1  0
sefa( β3β3x . ) |   4 |   1   0   2   1 |  *   *  * 192   *  * |  1  0 0  1
sefa( s3s 2 x ) |   4 |   0   2   2   0 |  *   *  *   * 192  * |  0  1 0  1
sefa( . β3x4x ) |   8 |   0   4   0   4 |  *   *  *   *   * 48 |  0  0 1  1
----------------+-----+-----------------+----------------------+-----------
      β3β3x .   ♦  24 |  12   0  24  12 |  0   8  4  12   0  0 | 16  * *  *
both( s3s 2 x ) ♦   6 |   0   3   6   0 |  0   2  0   0   3  0 |  * 64 *  *
      . β3x4x   ♦  48 |  24  24   0  24 |  6   0  8   0   0  6 |  *  * 8  *
sefa( β3β3x4x ) ♦  16 |   4   8   8   4 |  1   0  0   4   4  1 |  *  * * 48

starting figure: x3x3x4x