There are two different compounds of great icosahedron and great stellated dodecahedron: one is a dual compound and a stellation of the great icosidodecahedron, the other is a stellation of the icosidodecahedron.
Dual compound
It can be seen as a polyhedron compound of a great icosahedron and great stellated dodecahedron. It is one of five compounds constructed from a Platonic solid or Kepler-Poinsot solid, and its dual. It is a stellation of the great icosidodecahedron.
It has icosahedral symmetry (Ih) and it has the same vertex arrangement as a great rhombic triacontahedron.
This can be seen as one of the two three-dimensional equivalents of the compound of two pentagrams ({ 10/4} "decagram"); this series continues into the fourth dimension as compounds of star 4-polytopes.
Stellation of the icosidodecahedron
This polyhedron is a stellation of the icosidodecahedron, and given as Wenninger model index 61. It has the same vertex arrangement as a rhombic triacontahedron, its convex hull.
The stellation facets for construction are:
See also
Compound of two tetrahedra
Compound of cube and octahedron
Compound of dodecahedron and icosahedron
Compound of small stellated dodecahedron and great dodecahedron
References
, p. 90. , pp. 51-53. Martyn Cundy and A. Rollett. "Great Icosahedron Plus Great Stellated Dodecahedron". §3.10.4 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 132-133, 1989.