Looks like a compound of 4 icosahedra (ike) and a great dodecahedron (gad), and indeed vertices and edges coincide by five, {3} coincide by four.

Incidence matrix according to Dynkin symbol

s3/2s5s

demi( .   . . ) | 60 |  1  2  2 |  1  1  3
----------------+----+----------+---------
      s  2  s   |  2 | 30  *  * |  0  0  2
sefa( s3/2s . ) |  2 |  * 60  * |  1  0  1
sefa( .   s5s ) |  2 |  *  * 60 |  0  1  1
----------------+----+----------+---------
      s3/2s .   |  3 |  0  3  0 | 20  *  *
      .   s5s   |  5 |  0  0  5 |  * 12  *
sefa( s3/2s5s ) |  3 |  1  1  1 |  *  * 60

starting figure: x3/2x5x