Categorisation by Class: All

• Symmetry Home

• Crystal classes
• Unit cells
• Piezo/Pyroelectric classes

• Operations
• Dipole transitions
• Direct products
• Correlation tables
• Isomorphism
• Periodicity

• Stereographs
• 3D objects
• Molecules (non-java)
• Molecules (interactive)

• Links
• Live 3D
• References

• E-mail me

   

• Jon Goss
• AIMPRO

Operations	Order	Schönflies Symbol	International Symbol	Full Symmetry Symbol	Correlation Table	Irred. Rep. products	× i	Isomorph. with
Cubic
E, 4C3, 4C32, 3C2	12	T	23	23		⊗	Th
E, 8C3, 3C2, 3σv, i, 8S6	24	Th	m3			⊗	Th
E, 6C4, 8C3, 3C2, 6C2’	24	O	432	432		⊗	Oh	Td
E, 8C3, 3C2, 6S4, 6σd	24	Td	3m	3m		⊗	Oh	O
E, 8C3, 6C2, 6C4, 3C2’, i, 6S4, 8S6, 3σh, 6σd	48	Oh	m3m			⊗	Oh
Tetragonal
E, C4, C2, C43	4	C4	4	4		⊗	C4h	S4
E, S4, C2, S43	4	S4				⊗	C4h	C4
E, C4, C2, C43, i, S43, σh, S4	8	C4h	4/m			⊗	C4h
E, 2C4, C2, 2C2’, 2C2’’	8	D4	422	422		⊗	D4h	C4v, D2d
E, 2C4, C2, 2σv, 2σd	8	C4v	4mm	4mm		⊗	D4h	D4, D2d
E, 2S4, C2, 2C2’, 2σd	8	D2d(Vd)	2m	2m		⊗	D4h	D4, C4v
E, 2C4, C2, 2C2’, 2C2’’, i, 2S4, σh, 2σv, 2σd	16	D4h	4/mmm			⊗	D4h
Orthorhombic
E, C2, C2’, C2’’	4	D2(V)	222	222		⊗	D2h	C2v, C2h
E, C2, σv, σv’	4	C2v	mm2	mm2		⊗	D2h	D2, C2h
E, C2, C2’, C2’’, i, σ, σ’, σ’’	8	D2h(Vh)	mmm			⊗	D2h
Rhombic symmetry for defects in cubic crystals is often divided into two types: Type I: C2 coincides with the [110] direction and C2’ and C2’’ with [001] and [1-10] directions respectively (or, σv and σv’ coincide with the planes (1-10) and (001)). Also belonging to type I are centres for which C2 coincides with [001] and σv and σv’ with (110) and (1-10). Type II: C2 axis coincides with [001] and the axes C2’ and C2’’ with [100] and [010], (or, alternatively, σv and σv’ coincide with (010) and (100)).
Monoclinic
E, C2	2	C2	2	2		⊗	C2h	Cs, Ci
E, σh	2	Cs(C1h)	m	m		⊗	C2h	C2, Ci
E, C2, i, σh	4	C2h	2/m			⊗	C2h	D2, C2v
Monoclinic symmetry for defects in cubic crystals is often divided into two types: Type I: C2 coincides with <110> or σh with (110) Type II: C2 coincides with <100> or σh with (100)
Triclinic
E	1	C1	1	1		⊗	Ci
E, i	2	Ci(S2)				⊗	Ci	Cs, C2
Trigonal
E, C3, C32	3	C3	3	3		⊗	S6
E, C3, C32, i, S65, S6	6	S6(C3i)				⊗	S6	C6, C3h
E, 2C3, 3C2	6	D3	32	32		⊗	D3d	C3v
E, 2C3, 3σv	6	C3v	3m	3m		⊗	D3d	D3
E, 2C3, 3C2, i, 2S6, 3σd	12	D3d	m			⊗	D3d	C6v, D6, D3h
Hexagonal
E, C6, C3, C2, C32, C65	6	C6	6	6		⊗	C6h	S6, C3h
E, C3, C32, σh, S3, S32	6	C3h(S3)				⊗	C6h	S6, C6
E, C6, C3, C2, C32, C65, i, S32, S65, σh, S6, S3	12	C6h	6/m			⊗	C6h
E, 2C6, 2C3, C2, 3C2’, 3C2’’	12	D6	622	622		⊗	D6h	C6v, D3d, D3h
E, 2C6, 2C3, C2, 3σv, 3σd	12	C6v	6mm	6mm		⊗	D6h	D6, D3d, D3h
E, 2C3, 3C2, σh, 2S3, 3σv	12	D3h	m2	m2		⊗	D6h	D6, D3d, C6v
E, 2C6, 2C5, C2, 3C2’, 3C2’’, i, 2S3, 2S6, σh, 3σd, 3σv	24	D6h	6/mmm			⊗	D6h
Non-Crystallographic
E, 2C∞φ	∞	C∞	∞	-		-	C∞h
E, 2C∞φ, i, 2S∞φ	∞	C∞h	∞/m	-		-	C∞h
E, 2C∞φ, ∞σv	∞	C∞v	∞m	-		-	D∞h
E, 2C∞φ, ∞σv, i, 2S∞φ, ∞C2	∞	D∞h	∞/mm	-		-	D∞h
E, C5, C52, C53,  C54	5	C5	5	-		⊗	S10
E, S8, C4, S83, C2, S85, C43, S87	8	S8	-	-		⊗	C8h
E, 2C5, 2C52, 5C2	10	D5	-	-		⊗	D5d	C5v
E, 2C5, 2C52, 5σv	10	C5v	-	-		⊗	D5d	D5
E, C5, C52, C53, C54, σh, S5, S57, S53, S59	10	C5h	-	-		⊗	C10h
E, 2S8, 2C4, 2S83, C2, 4C2’, 4σd	16	D4d	-	-		⊗	D8h
E, 2C5, 2C52, 5C2, i, 2S10, 2S103, 5σd	20	D5d	-	-		⊗	D5d	D5h
E, 2C5, 2C52, 5C2, σh, 2S5, 2S52, 5σd	20	D5h	-	-		⊗	D10h	D5d
E, 2S12, 2C6, 2S4, 2C3, 2S125, C2, 6C2’, 6σd	24	D6d	-	-		⊗	D12h
E, 12C5, 12C52, 20C3, 15C2	60	I	-	-		⊗	Ih
E, 12C5, 12C52, 20C3, 15C2, i, 12S10, 12S103, 20S6, 15σ	120	Ih	-	-		⊗	Ih

This table lists point group symmetries along with their symmetry operations, the order of the group (i.e. the number of symmetry operations) and common notations.links to a correlation table, and ⊗ links to tables of products of irreducible representations. The group produced by combination with inversion is listed under "× i". This, in the case of crystolographic point groups, is the Laue class which corresponds to the symmetry of reciprocal space. Isomorphic groups are also listed where character tables are available.