I found this polyhedron by faceting the icosidodecahedron in such a way as to get twelve pentagons and twelve pentagrams as faces. Once I had done so, the result looked familiar, so I checked the uniform solids (a class of polyhedra I don’t yet know well), and found it there. Here is its dual, the medial rhombic triacontahedron.

I used Stella 4d to make these; you can try it for free at this website.