Looks like a compound of 3 great ditrigonary icosidodecahedra (gidtid), and indeed vertices, edges, and faces all coincide by three.

Incidence matrix according to Dynkin symbol

o5/2β5β

both( .   . . ) | 60 |  2   4 |  1  2  3
----------------+----+--------+---------
sefa( o5/2β . ) |  2 | 60   * |  1  0  1
sefa( .   s5s ) |  2 |  * 120 |  0  1  1
----------------+----+--------+---------
      o5/2β .   ♦  5 |  5   0 | 12  *  *
both( .   s5s ) ♦  5 |  0   5 |  * 24  *
sefa( o5/2β5β ) |  3 |  1   2 |  *  * 60

starting figure: o5/2x5x