空間群

在數學和物理學中，空間群（space group）是空間中（通常是三維空間）一種形態的空間對稱群。在三維空間中有219種不同的類型，或230種不同的手性類型。對超過三維的空間中的空間群也有研究，它們有時被稱作比貝爾巴赫群，並且是離散的緊群，具有歐氏空間的等距同構。它是由俄國結晶學家費多洛夫和德國結晶學家薛弗利斯（Artur Moritz Schoenflies，1853-1928）於1890至1891年間各自獨立地先後推導得出來的。

在晶體學中，空間群也被稱為費奧多羅夫群，是對晶體對稱型的一種描述。三維空間群的權威參考文獻是《國際晶體學表》。空間群可以分為兩類：一類稱為簡單空間群或稱點空間群；一類稱為複雜空間群或稱非點空間群。其中73種為簡單空間群，餘下的157種為複雜空間群。

三維的空間群

主條目：空間群列表

#	晶系 （空間群數量） 布拉維晶格	點群					空間群 (國際短符號)
		國際標記法	熊夫利標記法	軌形	考克斯特符號	點群階
1	三斜晶系 (2)	1	C1	11	[ ]+	1	P1
2		1	Ci	1×	[2+,2+]	2	P1
3–5	單斜晶系 (13)	2	C2	22	[2]+	2	P2, P21 C2
6–9		m	Cs	*11	[ ]	2	Pm, Pc Cm, Cc
10–15		2/m	C2h	2*	[2,2+]	4	P2/m, P21/m C2/m, P2/c, P21/c C2/c
16–24	正交晶系 (59)	222	D2	222	[2,2]+	4	P222, P2221, P21212, P212121, C2221, C222, F222, I222, I212121
25–46		mm2	C2v	*22	[2]	4	Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2, Pmn21, Pba2, Pna21, Pnn2 Cmm2, Cmc21, Ccc2, Amm2, Aem2, Ama2, Aea2 Fmm2, Fdd2 Imm2, Iba2, Ima2
47–74		mmm	D2h	*222	[2,2]	8	Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce Fmmm, Fddd Immm, Ibam, Ibca, Imma
75–80	四方晶系 (68)	4	C4	44	[4]+	4	P4, P41, P42, P43, I4, I41
81–82		4	S4	2×	[2+,4+]	4	P4, I4
83–88		4/m	C4h	4*	[2,4+]	8	P4/m, P42/m, P4/n, P42/n I4/m, I41/a
89–98		422	D4	224	[2,4]+	8	P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212 I422, I4122
99–110		4mm	C4v	*44	[4]	8	P4mm, P4bm, P42cm, P42nm, P4cc, P4nc, P42mc, P42bc I4mm, I4cm, I41md, I41cd
111–122		42m	D2d	2*2	[2+,4]	8	P42m, P42c, P421m, P421c, P4m2, P4c2, P4b2, P4n2 I4m2, I4c2, I42m, I42d
123–142		4/mmm	D4h	*224	[2,4]	16	P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P42/mmc, P42/mcm, P42/nbc, P42/nnm, P42/mbc, P42/mnm, P42/nmc, P42/ncm I4/mmm, I4/mcm, I41/amd, I41/acd
143–146	三方晶系 (25)	3	C3	33	[3]+	3	P3, P31, P32 R3
147–148		3	S6	3×	[2+,6+]	6	P3, R3
149–155		32	D3	223	[2,3]+	6	P312, P321, P3112, P3121, P3212, P3221 R32
156–161		3m	C3v	*33	[3]	6	P3m1, P31m, P3c1, P31c R3m, R3c
162–167		3m	D3d	2*3	[2+,6]	12	P31m, P31c, P3m1, P3c1 R3m, R3c
168–173	六方晶系 (27)	6	C6	66	[6]+	6	P6, P61, P65, P62, P64, P63
174		6	C3h	3*	[2,3+]	6	P6
175–176		6/m	C6h	6*	[2,6+]	12	P6/m, P63/m
177–182		622	D6	226	[2,6]+	12	P622, P6122, P6522, P6222, P6422, P6322
183–186		6mm	C6v	*66	[6]	12	P6mm, P6cc, P63cm, P63mc
187–190		6m2	D3h	*223	[2,3]	12	P6m2, P6c2, P62m, P62c
191–194		6/mmm	D6h	*226	[2,6]	24	P6/mmm, P6/mcc, P63/mcm, P63/mmc
195–199	立方晶系 (36)	23	T	332	[3,3]+	12	P23, F23, I23 P213, I213
200–206		m3	Th	3*2	[3+,4]	24	Pm3, Pn3, Fm3, Fd3, Im3, Pa3, Ia3
207–214		432	O	432	[3,4]+	24	P432, P4232 F432, F4132 I432 P4332, P4132, I4132
215–220		43m	Td	*332	[3,3]	24	P43m, F43m, I43m P43n, F43c, I43d
221–230		m3m	Oh	*432	[3,4]	48	Pm3m, Pn3n, Pm3n, Pn3m Fm3m, Fm3c, Fd3m, Fd3c Im3m, Ia3d

注: e 面是雙滑移面，是在兩個不同方向的滑移，存在於七個正交群，五個四方群和五個立方群中，都具有含有中心的晶格，官方的符號為e

參考資料

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外部連結

• International Union of Crystallography （頁面存檔備份，存於互聯網檔案館）
• Point Groups and Bravais Lattices
• [1] Bilbao Crystallographic Server
• Space Group Info (old)
• Space Group Info (new)
• Crystal Lattice Structures: Index by Space Group
• Full list of 230 crystallographic space groups
• Interactive 3D visualization of all 230 crystallographic space groups （頁面存檔備份，存於互聯網檔案館）
• Huson, Daniel H., The Fibrifold Notation and Classification for 3D Space Groups (PDF), 1999 
• The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions) （頁面存檔備份，存於互聯網檔案館）
• The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions) （頁面存檔備份，存於互聯網檔案館）