Looks like a compound of 2 small dodecahemicosahedron (sidhei), and indeed vertices, edges and faces all coincide by pairs.

Incidence matrix according to Dynkin symbol

x5/3o5/2x3*a

.   .   .    | 60 |  2  2 |  1  2  1
-------------+----+-------+---------
x   .   .    |  2 | 60  * |  1  1  0
.   .   x    |  2 |  * 60 |  0  1  1
-------------+----+-------+---------
x5/3o   .    |  5 |  5  0 | 12  *  *
x   .   x3*a |  6 |  3  3 |  * 20  *
.   o5/2x    |  5 |  0  5 |  *  * 12