Looks like the compound of 2 small rhombated icositetrachora (srico), and indeed all but the Grünbaumian elements coincide by pairs. It occurs in different types: either it incorporates Grünbaumian doubly wrapped small rhombicuboctahedra (2srico) - type A, or it uses Grünbaumian doubly wrapped cuboctahedra (2co) and Grünbaumian doubly wrapped triangular prisms (2trip) - type B.

Incidence matrix according to Dynkin symbol

x3β4x3o   (type A)

both( . . . . ) | 576 |   1   2   1   2 |   2   1  1   2   2   1 |  1  2  1  1
----------------+-----+-----------------+------------------------+------------
both( x . . . ) |   2 | 288   *   *   * |   2   0  1   0   0   0 |  1  2  0  0
both( . . x . ) |   2 |   * 576   *   * |   1   1  0   1   0   0 |  1  1  1  0
sefa( x3β . . ) |   2 |   *   * 288   * |   0   0  1   0   2   0 |  0  2  0  1
sefa( . s4x . ) |   2 |   *   *   * 576 |   0   0  0   1   1   1 |  0  1  1  1
----------------+-----+-----------------+------------------------+------------
both( x . x . ) |   4 |   2   2   0   0 | 288   *  *   *   *   * |  1  1  0  0
both( . . x3o ) |   3 |   0   3   0   0 |   * 192  *   *   *   * |  1  0  1  0
      x3β . .   ♦   6 |   3   0   3   0 |   *   * 96   *   *   * |  0  2  0  0
both( . s4x . ) ♦   4 |   0   2   0   2 |   *   *  * 288   *   * |  0  1  1  0
sefa( x3β4x . ) |   4 |   0   0   2   2 |   *   *  *   * 288   * |  0  1  0  1
sefa( . s4x3o ) |   3 |   0   0   0   3 |   *   *  *   *   * 192 |  0  0  1  1
----------------+-----+-----------------+------------------------+------------
both( x . x3o ) ♦   6 |   3   6   0   0 |   3   2  0   0   0   0 | 96  *  *  *
      x3β4x .   ♦  48 |  24  24  24  24 |  12   0  8  12  12   0 |  * 24  *  *
both( . s4x3o ) ♦  12 |   0  12   0  12 |   0   4  0   6   0   4 |  *  * 48  *
sefa( x3β4x3o ) ♦   6 |   0   0   3   6 |   0   0  0   0   3   2 |  *  *  * 96

starting figure: x3x4x3o

β3x4o3x   (type B)

both( . . . . ) | 576 |   2   2   2 |   1   2   1   2   1   2 |  1  1  2  1
----------------+-----+-------------+-------------------------+------------
both( . x . . ) |   2 | 576   *   * |   1   1   0   1   0   0 |  1  1  1  0
both( . . . x ) |   2 |   * 576   * |   0   1   1   0   0   1 |  1  0  1  1
sefa( β3x . . ) |   2 |   *   * 576 |   0   0   0   1   1   1 |  0  1  1  1
----------------+-----+-------------+-------------------------+------------
both( . x4o . ) |   4 |   4   0   0 | 144   *   *   *   *   * |  1  1  0  0
both( . x . x ) |   4 |   2   2   0 |   * 288   *   *   *   * |  1  0  1  0
both( . . o3x ) |   3 |   0   3   0 |   *   * 192   *   *   * |  1  0  0  1
      β3x . .   ♦   6 |   3   0   3 |   *   *   * 192   *   * |  0  1  1  0
sefa( β3x4o . ) |   4 |   0   0   4 |   *   *   *   * 144   * |  0  1  0  1
sefa( β3x . x ) |   4 |   0   2   2 |   *   *   *   *   * 288 |  0  0  1  1
----------------+-----+-------------+-------------------------+------------
both( . x4o3x ) ♦  24 |  24  24   0 |   6  12   8   0   0   0 | 24  *  *  *
      β3x4o .   ♦  24 |  24   0  24 |   6   0   0   8   6   0 |  * 24  *  *
      β3x . x   ♦  12 |   6   6   6 |   0   3   0   2   0   3 |  *  * 96  *
sefa( β3x4o3x ) ♦  24 |   0  24  24 |   0   0   8   0   6  12 |  *  *  * 24

starting figure: x3x4o3x