This interesting shape is a 5 sided die. When rolled, it can land either just on the points of the star or on points of the star and the tip of one of the pyramids. It's numbered in such a way that there are 5 possible outcomes, each appearing with equal probability.
It has a convex hull that's a lat-long polyisohedron, an order 3 polyisohedron. It consists of an overall star shape with 5 sided pyramids on each side.