Looks like a compound of a dodecadodecahedron (did) plus a small dodecahemicosahedron (sidhei) plus a great dodecahemicosahedron (gidhei), and indeed vertices coincide by 4, edges coincide by three, {6} coincide by pairs.

Incidence matrix according to Dynkin symbol

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