Great Stellated Dodecahedron
The shape is essentially an icosahedron with a three-sided pointy pyramid on top of each face. There are 10 groups each consisting of 5 triangles which are coplanar.
Mathematically however, it is actually composed of twelve intersecting pentagrammic faces. At each outer vertex three faces meet. The inner vertices are not true vertices at all in the sense that they are points where edges end and meet.