Looks like a compound of 5 icosahedra (ike) and a great dodecahedron (gad), and indeed vertices coincide by five, edges by six, {3} coincide by five.

Incidence matrix according to Dynkin symbol

s3/2s3s5*a

demi( .   . .    ) | 60 |  2  2  2 |  1  1  1  3
-------------------+----+----------+------------
sefa( s3/2s .    ) |  2 | 60  *  * |  1  0  0  1
sefa( s   . s5*a ) |  2 |  * 60  * |  0  1  0  1
sefa( .   s3s    ) |  2 |  *  * 60 |  0  0  1  1
-------------------+----+----------+------------
      s3/2s .      |  3 |  3  0  0 | 20  *  *  *
      s   . s5*a   |  5 |  0  5  0 |  * 12  *  *
      .   s3s      |  3 |  0  0  3 |  *  * 20  *
sefa( s3/2s3s5*a ) |  3 |  1  1  1 |  *  *  * 60

starting figure: x3/2x3x5*a