Looks like a compound of 3 small ditrigonal icosidodecahedra (sidtid), and indeed vertices, edges, and faces all coincide by three.

Incidence matrix according to Dynkin symbol

β5/2β5o

both( .   . . ) | 60 |   4  2 |  2  1  3
----------------+----+--------+---------
sefa( s5/2s . ) |  2 | 120  * |  1  0  1
sefa( .   β5o ) |  2 |   * 60 |  0  1  1
----------------+----+--------+---------
both( s5/2s . ) ♦  5 |   5  0 | 24  *  *
      .   β5o   ♦  5 |   0  5 |  * 12  *
sefa( β5/2β5o ) |  3 |   2  1 |  *  * 60