To make this polyhedron, I started with a snub dodecahedron. Next, I augmented all triangular faces of it with prisms, then took the convex hull of the result. Finally, I used Stella 4d‘s “try to make faces regular” function on the convex hull.

If you’d like to try the trial version of Stella for yourself, the website to visit is http://www.software3d.com/Stella.php.