The Gyroelongated Square Cupola
The gyroelongated square cupola is the 23rd Johnson solid (J23). It has 20 vertices, 44 edges, and 26 faces (20 equilateral triangles, 5 squares, and 1 octagon).
The gyroelongated square cupola can be constructed by attaching an octagonal antiprism to a square cupola (J4), thereby lengthening it. The gyro in the name refers to how the bottom octagonal face is gyrated with respect to the octagonal face of the constituent square cupola.
Adding a second cupola to the other side of the octagonal antiprism produces the gyroelongated square bicupola (J45).
Projections
Here are some views of the gyroelongated square cupola from various angles:
Projection  Description 

Top view. 

Front view projection, parallel to two square faces. 

Side view projection, parallel to 4 triangular faces. 

11.25° side view, with rectangular antiprism image. 

22.5° side view. 
Coordinates
The Cartesian coordinates of the gyroelongated square cupola with edge length 2 are:
 (±1, ±1, √2+H)
 (±1, ±(1+√2), H)
 (±(1+√2), ±1, H)
 (0, ±√(4+2√2), −H)
 (±√(4+2√2), 0, −H)
 (±√(2+√2), ±√(2+√2), −H)
where H = √((1+√2)(√(1+1/√2)−1)), or approximately 0.860296.