Looks like a compound of a great icosidodecahedron (gid) plus a great icosihemidodecahedron (geihid) plus a great dodecahemidodecahedron (gidhid), and indeed vertices coincide by 4, edges coincide by three, {10/3} coincide by pairs.

Incidence matrix according to Dynkin symbol

x5/4x3/2x5/3*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
---------------+-----+----------+---------
x5/4x   .      |  10 |  5  5  0 | 12  *  *
x   .   x5/3*a |  10 |  5  0  5 |  * 12  *
.   x3/2x      |   6 |  0  3  3 |  *  * 20