The Elongated Square Cupola
The elongated square cupola is the 18th Johnson solid (J19). It has 20 vertices and 18 faces (4 equilateral triangles, 13 squares, and 1 octagon).
The elongated square cupola can be constructed by attaching an octagonal prism to a square cupola (J4), lengthening it; hence the name elongated.
It can also be considered as a diminishing of the rhombicuboctahedron: the result of cutting off a square cupola from the latter.
Projections
Here are some views of the elongated square cupola from various angles:
Projection  Envelope  Description 

Regular octagon  Top view. 

Hexagon  Front view. 

Hexagon  22.5° side view. 
Coordinates
The Cartesian coordinates of the elongated square cupola with edge length 2 are:
 (±1, ±1, 1+√2)
 (±1, ±(1+√2), ±1)
 (±(1+√2), ±1, ±1)
These coordinates are obtained by deleting 4 vertices from the rhombicuboctahedron.