Table 9.7.1.2

Wilson, A. J. C.; Karen, V. L.; Mighell, A.

doi:10.1107/97809553602060000623

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 9.7, pp. 899-901

Table 9.7.1.2

A. J. C. Wilson,^a ^‡ V. L. Karen^b and A. Mighell^b

^a St John's College, Cambridge CB2 1TP, England, and ^bNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

Table 9.7.1.2 | top | pdf |
Space groups arranged by arithmetic crystal class and degree of symmorphism (Wilson, 1993d), as frequented by homomolecular structures with one molecule in the general position (in superscript numerals; according to Belsky, Zorkaya & Zorky, 1995)

(a) Triclinic, monoclinic and orthorhombic systems. The triclinic space groups are a special case, with `degree of symmorphism' undefined, and they are not assigned to any particular column. For *, † see Subsection 9.7.4.1.

Arithmetic crystal class	Fully symmorphic	Tending to symmorphism	Equally balanced	Tending to antimorphism	Fully antimorphic
1P	*P1†⁽⁹⁰⁾
$[\overline {1}P]$	*P $[{\overline 1}]$ †⁽¹⁷⁹⁶⁾
2P	*P2⁽⁰⁾	⋯	⋯	⋯	P2₁†⁽¹³²⁷⁾
2C	⋯	⋯	*C2 †⁽¹⁰⁹⁾	⋯	⋯
mP	*Pm⁽⁰⁾	⋯	⋯	⋯	Pc†⁽⁵⁸⁾
mC	⋯	⋯	*Cm⁽⁰⁾	⋯	Cc†⁽¹⁴⁴⁾
2/mP	*P2/m⁽⁰⁾	⋯	P2₁/m⁽⁰⁾	⋯	P2₁/c†⁽⁵⁹⁵¹⁾
2/mP			P2/c⁽¹¹⁾
2/mC	⋯	⋯	*C2/m⁽⁰⁾	C2/c †⁽⁵⁸⁷⁾	⋯
222P	*P222⁽⁰⁾	P222₁⁽⁰⁾	⋯	P2₁2₁2⁽³⁰⁾	P2₁2₁2₁†⁽²⁷⁹⁵⁾
222C	⋯	*C222⁽⁰⁾	⋯	C222₁†⁽¹¹⁾	⋯
222F	⋯	⋯	*F222⁽⁰⁾	⋯	⋯
222I	⋯	⋯	*I222⁽⁰⁾	⋯	⋯
222I			I2₁2₁2₁†⁽⁰⁾
mm2P	*Pmm2⁽⁰⁾	Pma2⁽⁰⁾	⋯	Pmc2₁⁽⁰⁾	Pca2₁⁽¹⁵³⁾
				Pcc2⁽¹⁾	Pna2₁†⁽³⁶⁷⁾
				Pnc2⁽¹⁾
				Pmn2₁⁽⁰⁾
				Pba2⁽¹⁾
				Pnn2⁽¹⁾
mm2C	⋯	*Cmm2⁽⁰⁾	⋯	Cmc2₁†⁽⁰⁾	⋯
mm2C				Ccc2⁽⁰⁾
2mmC	⋯	*Amm2⁽⁰⁾	⋯	Abm2⁽⁰⁾	⋯
				Ama2⁽⁰⁾
				Aba2† ⁽¹¹⁾
mm2F			*Fmm2⁽⁰⁾	Fdd2†⁽³⁵⁾
mm2I	⋯	⋯	*Imm2⁽⁰⁾	Iba2†⁽¹⁴⁾
mm2I				Ima2⁽⁰⁾
mmmP	*Pmmm⁽⁰⁾	Pccm ⁽⁰⁾	Pnnn ⁽⁰⁾	Pnna ⁽¹⁾	Pbca†⁽⁸²⁷⁾
		Pmma ⁽⁰⁾	Pban ⁽⁰⁾	Pcca ⁽³⁾
			Pmna ⁽⁰⁾	Pbam ⁽⁰⁾
			Pmmn ⁽⁰⁾	Pccn ⁽³⁷⁾
				Pbcm ⁽⁰⁾
				Pnnm ⁽⁰⁾
				Pbcn ⁽⁶⁰⁾
				Pnma ⁽⁰⁾
mmmC	⋯	*Cmmm⁽⁰⁾	Cmma ⁽⁰⁾	Cmcm ⁽⁰⁾	⋯
				Cmca ⁽⁰⁾
				Cccm ⁽⁰⁾
				Ccca†⁽⁰⁾
mmmF	⋯	⋯	*Fmmm⁽⁰⁾	Fddd†⁽²⁾	⋯
mmmI	⋯	⋯	*Immm⁽⁰⁾	Ibam ⁽⁰⁾
				Ibca⁽⁰⁾
				Imma ⁽⁰⁾

(b) Tetragonal space groups. For *, † see Subsection 9.7.4.1.

Arithmetic crystal class	Fully symmorphic	Tending to symmorphism	Equally balanced	Tending to antimorphism	Fully antimorphic
4P	*P4⁽⁰⁾	P4₂⁽¹⁾	⋯	⋯	P4_1,3†⁽⁴⁰⁾
4I	⋯	⋯	I4₁†⁽³⁾	*I4⁽³⁾	⋯
$[\overline {4}]$ P	*P $[\overline {4}]$ †⁽⁰⁾	⋯	⋯	⋯	⋯
$[\overline {4}]$ I	⋯	⋯	*I $[\overline {4}]$ †⁽⁷⁾	⋯	⋯
4/mP	*P4/m⁽⁰⁾	P4₂/m⁽⁰⁾ P4/n⁽¹⁾	P4₂/n†⁽²⁰⁾	⋯	⋯
4/mI	⋯	⋯	⋯	*I4/m⁽⁰⁾ I4₁/a†⁽²⁹⁾	⋯
422P	⋯	*P422⁽⁰⁾	P42₁2⁽⁰⁾	P4_1,32₁2†⁽⁴⁹⁾	⋯
422P		P4₂22⁽⁰⁾	P4_1,322⁽¹⁾	P4₂2₁2⁽¹⁾
422I	⋯	⋯	I4₁22†⁽⁰⁾	*I422⁽⁰⁾	⋯
4mmP	⋯	*P4mm⁽⁰⁾	P4bm⁽⁰⁾	P4₂cm⁽⁰⁾	⋯
				P4₂nm⁽⁰⁾
				P4cc⁽⁰⁾
				P4nc⁽⁰⁾
				P4₂mc⁽⁰⁾
				P4₂bc†⁽¹⁾
4mmI	⋯	⋯	⋯	*I4mm⁽⁰⁾	⋯
				I4cm⁽⁰⁾
				I4₁md⁽⁰⁾
				I4₁cd†⁽⁵⁾
$[\overline {4}]$ 2mP	⋯	*P $[\overline {4}]$ 2m⁽⁰⁾	P $[\overline {4}]$ 2c⁽⁰⁾	P $[\overline {4}]$ 2₁c†⁽¹²⁾	⋯
$[\overline {4}]$ 2mP			P $[\overline {4}]$ 2₁m⁽⁰⁾
$[\overline {4}]$ m2P	⋯	*P $[\overline {4}]$ m2⁽⁰⁾	P $[\overline {4}]$ c2⁽⁰⁾	⋯	⋯
			P $[\overline {4}]$ b2⁽⁰⁾
			P $[\overline {4}]$ n2⁽⁰⁾
$[\overline {4}]$ m2I	⋯	⋯	*I $[\overline {4}]$ m2⁽⁰⁾	I $[\overline {4}]$ c2†⁽⁰⁾	⋯
$[\overline {4}]$ 2mI	⋯	⋯	*I $[\overline {4}]$ 2m⁽⁰⁾	I $[\overline {4}]$ 2d†⁽⁰⁾	⋯
4/mmmP	⋯	P4/mmm*⁽⁰⁾	P4/mcc⁽⁰⁾	P4/nbm⁽⁰⁾	⋯
		P4₂/mmc⁽⁰⁾	P4/nmm⁽⁰⁾	P4/nnc⁽⁰⁾
		P4₂/mcm⁽⁰⁾		P4/mbm⁽⁰⁾
				P4/mnc⁽⁰⁾
				P4/ncc⁽⁰⁾
				P4₂/nbc⁽⁰⁾
				P4₂/nnm⁽⁰⁾
				P4₂/mbc⁽⁰⁾
				P4₂/mnm⁽⁰⁾
				P4₂/nmc⁽⁰⁾
				P4₂/ncm⁽⁰⁾
4/mmmI	⋯	I4/mmm*⁽⁰⁾	⋯	I4/mcm⁽⁰⁾
				I4₁/amd⁽⁰⁾
				I4₁/acd†⁽⁰⁾

Arithmetic crystal class	Fully symmorphic	Tending to symmorphism	Equally balanced	Tending to antimorphism	Fully antimorphic
3P	*P3⁽⁰⁾	⋯	⋯	⋯	P3_1,2†⁽³³⁾
3R	⋯	⋯	⋯	*R3†⁽¹¹⁾	⋯
$[\overline {3}]$ P	*P $[\overline {3}]$ †⁽¹⁾	⋯	⋯	⋯	⋯
$[\overline {3}]$ R	⋯	⋯	⋯	*R $[\overline {3}]$ †⁽³⁰⁾	⋯
312P 321P	⋯	P312⁽⁰⁾ P321⁽⁰⁾	⋯	P3_1,212†⁽⁰⁾ P3_1,221†⁽¹⁰⁾	⋯
32R	⋯	⋯	⋯	*R32†⁽⁰⁾	⋯
3m1P 31mP	⋯	P3m1⁽⁰⁾ P31m⁽⁰⁾	⋯	P3c1†⁽⁰⁾ P31c†⁽⁰⁾	⋯
3mR	⋯	⋯	⋯	*R3m⁽⁰⁾ R3c†⁽⁷⁾	⋯
$[\overline {3}]$ m1P $[\overline {3}]$ 1mP	⋯	P $[\overline {3}]$ m1⁽⁰⁾ P $[\overline {3}]$ 1m⁽⁰⁾	⋯	P $[\overline {3}]$ c1†⁽⁰⁾ P $[\overline {3}]$ 1c†⁽⁰⁾	⋯
$[\overline {3}]$ mR	⋯	⋯	⋯	*R $[\overline {3}]$ m⁽⁰⁾ R $[\overline {3}]$ c†⁽⁰⁾	$[\cdots ]$

(d) Hexagonal space groups. For *, † see Subsection 9.7.4.1.

Arithmetic crystal class	Fully symmorphic	Tending to symmorphism	Equally balanced	Tending to antimorphism	Fully antimorphic
6P	*P6⁽⁰⁾	⋯	P6_2,4⁽¹⁾ P6₃⁽⁰⁾	⋯	P6_1,5†⁽²²⁾
$[\overline {6}]$ P	*P $[\overline {6}]$ †⁽⁰⁾	⋯	⋯	⋯	⋯
6/mP	*P6/m⁽⁰⁾	⋯	P6₃/m†⁽⁰⁾	⋯	⋯
622P	⋯	*P622⁽⁰⁾ P6_2,422⁽⁰⁾	⋯	P6₃22⁽¹⁾ P6_1,522†⁽²⁾	⋯
6mmP	⋯	*P6mm⁽⁰⁾	⋯	P6cc⁽⁰⁾	⋯
6mmP				P6₃cm⁽⁰⁾ P6₃mc⁽⁰⁾
$[\overline {6}]$ m2P $[\overline {6}]$ 2mP	⋯	P $[\overline {6}]$ m2⁽⁰⁾ P $[\overline {6}]$ 2m⁽⁰⁾	⋯	P $[\overline {6}]$ c2†⁽⁰⁾ P $[\overline {6}]$ 2c†⁽⁰⁾
6/mmmP	⋯	P6/mmm*⁽⁰⁾	⋯	P6/mcc†⁽⁰⁾	⋯
6/mmmP				P6₃/mcm⁽⁰⁾ P6₃/mmc⁽⁰⁾

(e) Cubic space groups. For *, †, see Subsection 9.7.4.1. No examples with one molecule in general position were found, so the frequencies are omitted.

Arithmetic crystal class	Fully symmorphic	Tending to symmorphism	Equally balanced	Tending to antimorphism	Antimorphic except for 3
23P	⋯	*P23	⋯	⋯	P2₁3†
23F	⋯	⋯	*F23†	⋯	⋯
23I	⋯	⋯	*I23 I2₁3†	⋯	⋯
m $[\overline 3]$ P	⋯	*Pm $[\overline 3]$	Pn $[\overline 3]$	⋯	Pa $[\overline 3 ]$ †
m $[\overline 3]$ F	⋯	⋯	*Fm $[\overline 3]$	Fd $[\overline 3]$ †	⋯
m $[\overline 3]$ I	⋯	⋯	*Im $[\overline 3]$	Ia $[\overline 3]$ †	⋯
432P	⋯	*P432	⋯	P4₂32† P4_1,332†	⋯
432F	⋯	⋯	*F432	F4₁32†	⋯
432I	⋯	⋯	*I432	I4₁32†	⋯
$[\overline {4}]$ 3mP	⋯	*P $[\overline {4}]$ 3m	⋯	P $[\overline {4}]$ 3n†	⋯
$[\overline {4}]$ 3mF	⋯	⋯	*F $[\overline {4}]$ 3m	F $[\overline {4}]$ 3c†	⋯
$[\overline {4}]$ 3mI	⋯	⋯	*I $[\overline {4}]$ 3m	I $[\overline {4}]$ 3d†	⋯
m $[\overline {3}]$ mP	⋯	*Pm $[\overline {3}]$ m	Pm $[\overline {3}]$ n Pn $[\overline {3}]$ m	Pn $[\overline {3}]$ n†	⋯
m $[\overline {3}]$ mF	⋯	*Fm $[\overline {3}]$ m	⋯	Fm $[\overline {3}]$ c Fd $[\overline {3}]$ m	Fd $[\overline {3}]$ c†
$[m\overline {3}]$ mI	⋯	*Im $[\overline {3}]$ m	⋯	Ia $[\overline {3}]$ d†	⋯