The Gyroelongated Square Bipyramid
The gyroelongated square bipyramid is the 17th Johnson solid (J17). It has 10 vertices, 24 edges, and 16 faces (16 equilateral triangles).
The gyroelongated square bipyramid can be constructed by attaching two square pyramids to a square antiprism. Its name describes its relation to the octahedron regarded as a square bipyramid: inserting a square antiprism between the two constituent square pyramids to make the bipyramid longer and with the square pyramids gyrated with respect to each other.
Alternatively, it can also be regarded as the gyroelongated square pyramid (J10) augmented with another square pyramid, in gyrated orientation with respect to the first constituent square pyramid.
Projections
Here are some views of the gyroelongated square bipyramid from various angles:
Projection  Envelope  Description 

Regular octagon  Top view. 

Hexagon  Front view. 

Hexagon  22.5° side view. 
Coordinates
The Cartesian coordinates of the gyroelongated square bipyramid with edge length 2 are:
 (0, 0, ±(√2 + 1/∜2))
 (0, ±√2, 1/∜2)
 (±√2, 0, 1/∜2)
 (±1, ±1, −1/∜2)