Looks like a compound of 3 small ditrigonary 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

starting figure: x5/2x5o

s5/2s5/2s5/2*a

demi( .   .   .      ) | 60 |  2  2  2 |  1  1  1  3
-----------------------+----+----------+------------
sefa( s5/2s   .      ) |  2 | 60  *  * |  1  0  0  1
sefa( s   .   s5/2*a ) |  2 |  * 60  * |  0  1  0  1
sefa( .   s5/2s      ) |  2 |  *  * 60 |  0  0  1  1
-----------------------+----+----------+------------
      s5/2s   .        ♦  5 |  5  0  0 | 12  *  *  *
      s   .   s5/2*a   ♦  5 |  0  5  0 |  * 12  *  *
      .   s5/2s        ♦  5 |  0  0  5 |  *  * 12  *
sefa( s5/2s5/2s5/2*a ) |  3 |  1  1  1 |  *  *  * 60

starting figure: x5/2x5/2x5/2*a