There is a diagram of this geometric structure here (Fig.18 on p.198). It is a solid with 72 vertices and 140 triangular faces. Twelve of the vertices have five neighbours, and the remaining 60 vertices have six neighbours, just as you want. The author describes the solid as being obtained from a snub dodecahedron by a process of triangulation. He also mentions that it has been considered as a virus model (presumably with each vertex representing a capsomere).
Having said that, I have no idea how you would construct this object mathematically. But I hope that you may find that reference helpful.