Looks like a compound of 2 coincident hexacosihecatonicosachora (xhi), and indeed all but the Grünbaumian elements coincide by pairs. It comes in 2 types depending on which cells are Grünbaumian double-covers: type A uses the former truncated tetrahedra (tut) as Grünbaumian double-covered cells, while type B uses the former truncated icosahedra (ti) as such.

Incidence matrix according to Dynkin symbol

β3x3x5o   (type A)

both( . . . . ) | 7200 |    1    2    1 |    2    1    1    2 |   1   2   1
----------------+------+----------------+---------------------+------------
both( . x . . ) |    2 | 3600    *    * |    2    0    1    0 |   1   2   0
both( . . x . ) |    2 |    * 7200    * |    1    1    0    1 |   1   1   1
sefa( β3x . . ) |    2 |    *    * 3600 |    0    0    1    2 |   0   2   1
----------------+------+----------------+---------------------+------------
both( . x3x . ) |    6 |    3    3    0 | 2400    *    *    * |   1   1   0
both( . . x5o ) |    5 |    0    5    0 |    * 1440    *    * |   1   0   1
      β3x . .   ♦    6 |    3    0    3 |    *    * 1200    * |   0   2   0
sefa( β3x3x . ) |    6 |    0    3    3 |    *    *    * 2400 |   0   1   1
----------------+------+----------------+---------------------+------------
both( . x3x5o ) ♦   60 |   30   60    0 |   20   12    0    0 | 120   *   *
      β3x3x .   ♦   24 |   12   12   12 |    4    0    4    4 |   * 600   *
sefa( β3x3x5o ) ♦   60 |    0   60   30 |    0   12    0   20 |   *   * 120

starting figure: x3x3x5o

o3x3x5β   (type B)

both( . . . . ) | 7200 |    2    1    1 |    1    2   1    2 |   1   2   1
----------------+------+----------------+--------------------+------------
both( . x . . ) |    2 | 7200    *    * |    1    1   0    1 |   1   1   1
both( . . x . ) |    2 |    * 3600    * |    0    2   1    0 |   1   2   0
sefa( . . x5β ) |    2 |    *    * 7200 |    0    0   1    2 |   0   2   1
----------------+------+----------------+--------------------+------------
both( o3x . . ) |    3 |    3    0    0 | 2400    *   *    * |   1   0   1
both( . x3x . ) |    6 |    3    3    0 |    * 2400   *    * |   1   1   0
      . . x5β   ♦   10 |    0    5    5 |    *    * 720    * |   0   2   0
sefa( . x3x5β ) |    6 |    3    0    3 |    *    *   * 2400 |   0   1   1
----------------+------+----------------+--------------------+------------
both( o3x3x . ) ♦   12 |   12    6    0 |    4    4   0    0 | 600   *   *
      . x3x5β   ♦  120 |   60   60   60 |    0   20  12   20 |   * 120   *
sefa( o3x3x5β ) ♦   12 |   12    0    6 |    4    0   0    4 |   *   * 600

starting figure: o3x3x5x