This is an edge-faceting of (the Grünbaumian double cover of) the rectified hexacosachoron (rox). The 720 pentagonal prisms (pip) are the vertex figures of the rectified hexacosachoron, and their faces are facets of its former cells which thus get an inner structure.

Incidence matrix according to Dynkin symbol

o3β3o5o

both( . . . . ) | 720 |   10   20 |    5   10   20   10 |   5   2  10
----------------+-----+-----------+---------------------+------------
sefa( o3β . . ) |   2 | 3600    * |    1    0    2    0 |   2   0   1
sefa( . β3o . ) |   2 |    * 7200 |    0    1    1    1 |   1   1   1
----------------+-----+-----------+---------------------+------------
      o3β . .   ♦   3 |    3    0 | 1200    *    *    * |   2   0   0
      . β3o .   ♦   3 |    0    3 |    * 2400    *    * |   1   1   0
sefa( o3β3o . ) |   4 |    2    2 |    *    * 3600    * |   1   0   1
sefa( . β3o5o ) |   5 |    0    5 |    *    *    * 1440 |   0   1   1
----------------+-----+-----------+---------------------+------------
      o3β3o .   ♦   6 |   12   12 |    4    4    6    0 | 600   *   *
      . β3o5o   ♦  12 |    0   60 |    0   20    0   12 |   * 120   *
sefa( o3β3o5o ) ♦  10 |    5   10 |    0    0    5    2 |   *   * 720

starting figure: o3x3o5o