This polyhedron is identified with the indexed name U78 as a uniform polyhedron.
It is a special case of a right prism with a pentagram as base, which in general has rectangular non-base faces.
NOTE: The pentagram face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered interior or exterior depending on how interior is defined. One definition of interior is the set of points that have a ray that crosses the boundary an odd number of times to escape the perimeter.
In either case, it is best to show the pentagram boundary line to distinguish it from a concave decagon.
An alternative representation with hollow centers to the pentagrams.
The pentagrammic dipyramid is the dual to the Pentagrammic prism