This is a rhombicosidodecahedron, one of the Archimedean solids.
If one pentagonal cupola is removed from this polyhedron, the result is the diminished rhombicosidodecahedron, which is one of the Johnson solids (J76).
The next step is to take another J76, and attach it to the first one, so that their decagonal faces meet.
I’m calling the result the “double rhombicosidodecahedron.”
I did these manipulations of polyhedra and their images with a program called Stella 4d: Polyhedron Navigator. There’s a free trial download available, if you’d like to try the program for yourself, and it’s at this website.