Snub polyhedron

Class	Number and properties
Polyhedron
Platonic solids	(5, convex, regular)
Archimedean solids	(13, convex, uniform)
Kepler-Poinsot polyhedra	(4, regular, non-convex)
Uniform polyhedra	(75, uniform)
Prismatoid: prisms, antiprisms etc.	(4 infinite uniform classes)
Polyhedra tilings	(11 regular, in the plane)
Quasi-regular polyhedra	(8)
Johnson solids	(92, convex, non-uniform)
Pyramids and Bipyramids	(infinite)
Stellations	Stellations
Polyhedral compounds	(5 regular)
Deltahedra	(Deltahedra, equalatial triangle faces)
Snub polyhedra	(12 uniform, not mirror image)
Zonohedron	(Zonohedra, faces have 180°symmetry)
Dual polyhedron
Self-dual polyhedron	(infinite)
Catalan solid	(13, Archimedean dual)

A snub polyhedron is a polyhedron obtained by adding extra triangles around each vertex.

Chiral snub polyhedra do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. Their symmetry groups are all point groups and are one of:

O - chiral octahedral symmetry;the rotation group of the cube and octahedron; order 24.
I - chiral icosahedral symmetry; the rotation group of the icosahedron and the dodecahedron; order 60.

For example, the snub cube:

Snub polyhedra have Wythoff symbol | p q r and by extension, vertex configuration 3.p.3.q.3.r.

List of snub polyhedra

There are 12 uniform snub polyhedra, not including the icosahedron as a snub tetrahedron, the great icosahedron as a retrosnub tetrahedron and the great disnub dirhombidodecahedron, also known as Skilling's figure.

Snub polyhedron	Original polyhedron	Symmetry group
Icosahedron	Octahedron	I_h
Great icosahedron	Octahedron	I_h
Snub cube	Cuboctahedron	O
Snub dodecahedron	Icosidodecahedron	I
Great snub dodecicosidodecahedron	Great dodecicosidodecahedron	I
Snub icosidodecadodecahedron	Icosidodecadodecahedron	I
Snub dodecadodecahedron	Dodecadodecahedron	I
Inverted snub dodecadodecahedron	Dodecadodecahedron	I
Great inverted snub icosidodecahedron	Great icosidodecahedron	I
Great retrosnub icosidodecahedron	Great icosidodecahedron	I
Great snub icosidodecahedron	Great icosidodecahedron	I
Small snub icosicosidodecahedron	Small icosicosidodecahedron	I_h
Small retrosnub icosicosidodecahedron	Small icosicosidodecahedron	I_h
Great dirhombicosidodecahedron		I_h
Great disnub dirhombidodecahedron	Great dirhombicosidodecahedron	I_h

Notes:

The icosahedron, snub cube and snub dodecahedron are the only three convex ones. They are obtained by snubification of the octahedron, cuboctahedron and the icosidodecahedron - the three convex quasiregular polyhedra.
The only snub polyhedron with the chiral octahedral group of symmetries is the snub cube.
Only the icosahedron and the great icosahedron are also regular polyhedra. They are also deltahedra.
Only the first two and the last four also have reflective symmetries.

There is also the infinite set of antiprisms. They are formed from dihedra, degenerate regular polyhedra. Those up to hexagonal are listed below.

Snub polyhedron	Original polyhedron	Symmetry group
Tetrahedron	Digonal dihedron	T_d
Octahedron	Trigonal dihedron	O_h
Square antiprism	Tetragonal dihedron	D_4d
Pentagonal antiprism	Pentagonal dihedron	D_5d
Pentagrammic antiprism	Pentagrammic dihedron	D_5d
Pentagrammic crossed-antiprism	Retrogade pentagrammic dihedron	D_5d
Hexagonal antiprism	Hexagonal dihedron	D_6d

Notes:

Two of these polyhedra may be constructed from the first two snub polyhedra in the list starting with the icosahedron: the pentagonal antiprism is a parabidiminished icosahedron and a pentagrammic crossed-antiprism is a parabidiminished great icosahedron, also known as a parabireplenished great icosahedron.

Two Johnson solids are snub polyhedra: the snub disphenoid and the snub square antiprism. Neither is chiral.

Snub polyhedron	Image	Original polyhedron	Image	Symmetry group
Snub disphenoid		Disphenoid		D_2d
Snub square antiprism		Square antiprism		D_4d

Notes: