空そら间群

在ざい数学すうがく和わ物理ぶつり学がく中なか，空そら间群（space group）是ぜ空そら间中（通常つうじょう是ぜ三さん维空间）一いち种形态的空そら间对称しょう群ぐん。在ざい三维空间中有219种不同ふどう的てき类型，或ある230种不同ふどう的てき手性てしょう类型。对超过三维的空间中的空间群也有研究，它们有ゆう时被称しょう作さく比ひ贝尔巴ともえ赫群ぐん，并且是ぜ离散的てき紧群ぐん，具有ぐゆう欧おう氏し空そら间的等とう距同构。它是由よし俄にわか国こく结晶学がく家か费多洛らく夫おっと和德わとく国こく结晶学がく家か薛弗利り斯（Artur Moritz Schoenflies，1853-1928）于1890至いたり1891年ねん间各自かくじ独立どくりつ地ち先さき后きさき推导得とく出来でき的てき。

在ざい晶あきら体からだ学がく中なか，空そら间群也被称しょう为费奥多た罗夫群ぐん，是ぜ对晶体たい对称型がた的てき一いち种描述じゅつ。三维空间群的权威参考文献是《国くに际晶体からだ学がく表ひょう》。空そら间群可か以分为两类：一类称为简单空间群或称点空间群；一类称为复杂空间群或称非点空间群。其中73种为简单空そら间群，余よ下か的てき157种为复杂空そら间群。

三维的空间群

主しゅ条目じょうもく：空そら间群列れつ表ひょう

#	晶あきら系けい （空そら间群数量すうりょう） 布ぬの拉ひしげ维晶格かく	点てん群ぐん					空そら间群 (国くに际短符号ふごう)
		國際こくさい標記ひょうき法ほう	熊夫くまお利り標記ひょうき法ほう	軌形	考こう克かつ斯特符號ふごう	點てん群ぐん階かい
1	三さん斜はす晶あきら系けい (2)	1	C1	11	[ ]+	1	P1
2		1	Ci	1×	[2+,2+]	2	P1
3–5	单斜晶あきら系けい (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	正せい交晶系けい (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	四方よも晶あきら系けい (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	三さん方ぽう晶あきら系けい (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	六方ろっぽう晶あきら系けい (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	立方りっぽう晶あきら系けい (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

注ちゅう: e 面めん是ぜ双そう滑すべり移うつり面めん，是ぜ在ざい两个不同ふどう方向ほうこう的てき滑すべり移うつり，存在そんざい于七なな个正交群，五个四方群和五个立方群中，都と具有ぐゆう含有がんゆう中心ちゅうしん的てき晶あきら格かく，官かん方かた的てき符号ふごう为e

参考さんこう资料

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• Bieberbach, Ludwig, Über die Bewegungsgruppen der Euklidischen Räume, Mathematische Annalen, 1911, 70 (3): 297–336, ISSN 0025-5831, doi:10.1007/BF01564500 
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• Brown, Harold; Bülow, Rolf; Neubüser, Joachim; Wondratschek, Hans; Zassenhaus, Hans, Crystallographic groups of four-dimensional space, New York: Wiley-Interscience [John Wiley & Sons], 1978, ISBN 978-0-471-03095-9, MR0484179 
• Burckhardt, Johann Jakob, Die Bewegungsgruppen der Kristallographie, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften 13, Verlag Birkhäuser, Basel, 1947, MR0020553 
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• Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P., On three-dimensional space groups, Beiträge zur Algebra und Geometrie. Contributions to Algebra and Geometry, 2001, 42 (2): 475–507 [2013-03-04], ISSN 0138-4821, MR1865535, （原始げんし内容ないよう存そん档于2021-04-18） 
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外部がいぶ链接

• International Union of Crystallography （页面存そん档备份，存そん于互联网档案あん馆）
• 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 （页面存そん档备份，存そん于互联网档案あん馆）
• 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) （页面存そん档备份，存そん于互联网档案あん馆）
• The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions) （页面存そん档备份，存そん于互联网档案あん馆）