Truncated square antiprism

The truncated square antiprism one in an infinite series of truncated antiprisms, constructed as a truncated square antiprism. It has 18 faces, 2 octagons, 8 hexagons, and 8 squares.

Gyroelongated triamond square bicupola
If the hexagons are folded, it can be constructed by regular polygons. Or each folded hexagon can be replaced by two triamonds, adding 8 edges (56), and 4 faces (32). This form is called a gyroelongated triamond square bicupola.
 * Gyroelongated_triamond_square_bicupola.png

Snub square antiprism
Although it can't be made by all regular planar faces, its alternation is the Johnson solid, the snub square antiprism.
 * Snub_square_antiprism_colored.png