The Gyroelongated Triangular Cupola
The gyroelongated triangular cupola is the 22nd Johnson solid (J22). It has 15 vertices and 20 faces (16 equilateral triangles, 3 squares, and 1 hexagon).
The gyroelongated triangular cupola can be constructed by attaching a hexagonal antiprism to a triangular cupola (J3), thereby lengthening it. The gyro in the name refers to how the bottom hexagonal face is gyrated with respect to the hexagonal face of the constituent triangular cupola.
Adding a second cupola to the other side of the hexagonal antiprism produces the gyroelongated triangular bicupola (J44).
Projections
Here are some views of the gyroelongated triangular cupola from various angles:
Projection  Envelope  Description 

Regular dodecagon  Top view. 

Hexagon  Front view. 

Hexagon  Side view, with many coincident edges. 
Coordinates
The Cartesian coordinates of the gyroelongated triangular cupola with edge length 2 are:
 (±1, ±√3, √(√3−1))
 (±2, 0, √(√3−1))
 (±√3, ±1, −√(√3−1))
 (0, ±2, −√(√3−1))