Describe a $3$-dimensional solid whose symmetry group is isomorphic to $D_5$ Describe a $3$-dimensional solid whose symmetry group is isomorphic to $D_5$.
Would a pentagonal frustum satisfy this?  I can't think of any other rotations or reflections that could exist for this shape, or any other shape that would satisfy this.
 A: Yes, a (regular) pentagonal frustum would satisfy this.  Since the two pentagonal faces of the frustum do not have the same size, any element of the symmetric group must preserve these faces.  Hence the symmetric group is, "at most", the symmetric group of a pentagon, which is $D_5$.  Since all five rotations and five reflections preserve the frustum, we are done.
Another choice of solid with symmetric group $D_5$ would be a regular pentagonal pyramid.
A: Consider a solid which is a pentagonal prism, but with the pentagonal faces twisted by an angle that is not a multiple of 36° (e.g. 7°); the square faces are replaced with two triangles such that the overall solid is convex. This is similar to the construction of the Schönhardt polyhedron and is chiral; the symmetry group of this solid is $D_5$, generated by


*

*a rotation through 72° on an axis through the pentagonal faces' centres

*a rotation of 180° that swaps the two pentagonal faces


Note that "reflect through middle plane and rotate through 36°" is not a symmetry, as it changes the handedness.
