The truncated icosidodecahedron is similar, with all regular faces, and 4.6.10 vertex figure.
Also see the truncated rhombirhombicosidodecahedron.

The truncated rhombicosidodecahedron can be seen in sequence of rectification and truncation operations from the icosidodecahedron.
A further alternation step leads to the snub rhombicosidodecahedron.
The truncated icosidodecahedron is similar, with all regular faces, and 4.6.10 vertex figure. Also see the truncated rhombirhombicosidodecahedron.
The truncated rhombicosidodecahedron can be seen in sequence of rectification and truncation operations from the icosidodecahedron. A further alternation step leads to the snub rhombicosidodecahedron.
Truncated rhombicosidodecahedron