Looks like a compound of an icosidodecahedron (id), a small icosihemidodecahedron (seihid), and a small dodecahemidodecahedron (sidhid), and indeed vertices coincide by 4, edges coincide by three, {10} coincide by pairs.

Incidence matrix according to Dynkin symbol

x3/2x5/2x5*a

.   .   .    | 120 |  1  1  1 |  1  1  1
-------------+-----+----------+---------
x   .   .    |   2 | 60  *  * |  1  1  0
.   x   .    |   2 |  * 60  * |  1  0  1
.   .   x    |   2 |  *  * 60 |  0  1  1
-------------+-----+----------+---------
x3/2x   .    |   6 |  3  3  0 | 20  *  *
x   .   x5*a |  10 |  5  0  5 |  * 12  *
.   x5/2x    |  10 |  0  5  5 |  *  * 12