Cubic symmetry

Cubic symmetry
Rank	3
Space	Spherical
Order	48
Info
Coxeter diagram
Elements
Axes	3 × BC2×A1, 4 × (G2×A1)/2, 6 × K3
Related polytopes
Omnitruncate	Great rhombicuboctahedron

Cubic symmetry, also known as octahedral symmetry and notated B₃ or BC₃, is a 3D spherical Coxeter group. It is the symmetry group of the cube and octahedron.

Subgroups

• B₃/2 (maximal)
• B₃+ (maximal)
• A₃ (maximal)
• A₃+
• (G₂×A₁)/2 (maximal)
• (G₂+×A₁)/2
• B₂×A₁ (maximal)
• (B₂×A₁)/2
• (B₂×A₁)+
• B₂×I
• B₂+×A₁
• (B₂+×A₁)/2
• B₂+×I
• (A₂×A₁)+
• A₂×I
• A₂+×I
• K₃
• K₃+
• K₂×I
• K₂+×A₁
• K₂+×I
• ±(I×I×I)
• A₁×I×I
• I×I×I

Convex polytopes with B₃ symmetry

• Cube (regular)/Octahedron (regular)
• Cuboctahedron (isogonal)/Rhombic dodecahedron (isotopic)
• Truncated cube (isogonal)/Triakis octahedron (isotopic)
• Truncated octahedron (isogonal)/Tetrakis hexahedron (isotopic)
• Small rhombicuboctahedron (isogonal)/Deltoidal icositetrahedron (isotopic)
• Great rhombicuboctahedron (isogonal)/Disdyakis dodecahedron (isotopic)

Wythoffians with B₃ symmetry

o4o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Cube	cube	{4,3}	x4o3o
Truncated cube	tic	t{4,3}	x4x3o
Cuboctahedron	co	r{4,3}	o4x3o
Truncated octahedron	toe	t{3,4}	o4x3x
Octahedron	oct	{3,4}	o4o3x
Small rhombicuboctahedron	sirco	rr{4,3}	x4o3x
Great rhombicuboctahedron	girco	tr{4,3}	x4x3x
Snub cube	snic	sr{4,3}	s4s3s

o4/3o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Cube	cube	{4/3,3}	x4/3o3o ()
Quasitruncated hexahedron	quith	t{4/3,3}	x4/3x3o ()
Cuboctahedron	co	r{3,4/3}	o4/3x3o ()
Truncated octahedron	toe	t{3,4/3}	o4/3x3x ()
Octahedron	oct	{3,4/3}	o4/3o3x ()
Quasirhombicuboctahedron	querco	rr{3,4/3}	x4/3o3x ()
Quasitruncated cuboctahedron	quitco	tr{3,4/3}	x4/3x3x ()
(degenerate, oct+6(4))			o4/3o3ß ()
Icosahedron	ike	s{3,4/3}	o4/3s3s ()

o4/3o3o4*a truncations

Name	OBSA	CD diagram	Picture
(degenerate, double cover of cube)		x4/3o3o4*a ()
Great cubicuboctahedron	gocco	x4/3x3o4*a ()
(degenerate, oct+6(4))		o4/3x3o4*a ()
(degenerate, double cover of cho)		o4/3x3x4*a ()
(degenerate, oct+6(4))		o4/3o3x4*a ()
Small cubicuboctahedron	socco	x4/3o3x4*a ()
Cuboctatruncated cuboctahedron	cotco	x4/3x3x4*a ()