Looks like a compound of 3 icosahedra (ike) and 3 great dodecahedron (gad), and indeed vertices coincide by 3, edges coincide by 6, both face types coincide by 3 each.

It occurs in 2 types: either triangles and pentagon cycle in the same direction (type A), or in opposite direction (type B).

Incidence matrix according to Dynkin symbol

s5/4s5s5*a (type A)

demi( .   . .    ) | 60 |  2  2  2 |  1  1  1  3
-------------------+----+----------+------------
sefa( s5/4s .    ) |  2 | 60  *  * |  1  0  0  1
sefa( s   . s5*a ) |  2 |  * 60  * |  0  1  0  1
sefa( .   s5s    ) |  2 |  *  * 60 |  0  0  1  1
-------------------+----+----------+------------
s5/4s .        5 |  5  0  0 | 12  *  *  *
s   . s5*a   |  5 |  0  5  0 |  * 12  *  *
.   s5s      |  5 |  0  0  5 |  *  * 12  *
sefa( s5/4s5s5*a ) |  3 |  1  1  1 |  *  *  * 60

starting figure: x5/4x5x5*a

s5/4s5/4s5/4*a (type B)

demi( .   .   .      ) | 60 |  2  2  2 |  1  1  1  3
-----------------------+----+----------+------------
sefa( s5/4s   .      ) |  2 | 60  *  * |  1  0  0  1
sefa( s   .   s5/4*a ) |  2 |  * 60  * |  0  1  0  1
sefa( .   s5/4s      ) |  2 |  *  * 60 |  0  0  1  1
-----------------------+----+----------+------------
s5/4s   .          5 |  5  0  0 | 12  *  *  *
s   .   s5/4*a     5 |  0  5  0 |  * 12  *  *
.   s5/4s          5 |  0  0  5 |  *  * 12  *
sefa( s5/4s5/4s5/4*a ) |  3 |  1  1  1 |  *  *  * 60

starting figure: x5/4x5/4x5/4*a