The Diminished Rhombicosidodecahedron
The diminished rhombicosidodecahedron is the 76th Johnson solid, or J76 for short. It has 52 faces (1 decagon, 11 pentagons, 25 squares, and 15 triangles), 105 edges, and 55 vertices. It is obtained by cutting off a pentagonal cupola segment from the rhombicosidodecahedron.
Cutting off a second pentagonal cupola from the opposite side to the decagonal face produces the parabidiminished rhombicosidodecahedron (J80). Cutting off a cupola that isn't directly opposite the decagon yields the metabidiminished rhombicosidodecahedron (J81).
Projections
Here are some views of the diminished rhombicosidodecahedron from various angles:
Projection  Description 

Top view. 

Front view. 

Side view. 
Coordinates
The Cartesian coordinates of the diminished rhombicosidodecahedron with edge length 2 are:
 (0, ±(φ+2), ±φ^{2})
 (±φ^{2}, 0, −(φ+2))
 (−φ^{2}, 0, (φ+2))
 (±(φ+2), ±φ^{2}, 0)
 (±1, ±1, ±φ^{3})
 (±1, ±φ^{3}, ±1)
 (±φ^{3}, ±1, −1)
 (−φ^{3}, ±1, 1)
 (±φ, ±φ^{2}, ±2φ)
 (±φ^{2}, ±2φ, ±φ)
 (±2φ, ±φ, −φ^{2})
 (−2φ, ±φ, φ^{2})
where φ=(1+√5)/2 is the Golden Ratio.