Table 9.8.3.3

Janssen, T.; Janner, A.; Looijenga-Vos, A.; Wolff, P. M. de

doi:10.1107/97809553602060000624

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. 9.8, pp. 919-920

Table 9.8.3.3

T. Janssen,^a A. Janner,^a A. Looijenga-Vos^b and P. M. de Wolff^c ^‡

^a Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands,^bRoland Holstlaan 908, NL-2624 JK Delft, The Netherlands, and ^cMeermanstraat 126, 2614 AM, Delft, The Netherlands

Table 9.8.3.3 | top | pdf |
(3 + 1)-Dimensional point groups and arithmetic crystal classes

The four-dimensional point group K_s has external part K_E, which belongs to a three-dimensional system. Depending on the Bravais class of the four-dimensional lattice left invariant by K_s, this point group gives rise to an integral 4 × 4 matrix group Γ(K) which belongs to one of the arithmetic crystal classes given in the last column.

System	Point group		External Bravais class	Arithmetic crystal class(es)
System	K_E	K_s	External Bravais class	Arithmetic crystal class(es)
Triclinic	1	(1, 1)	$[{\bar 1}]$ P	1P(αβγ)
Triclinic	$[{\bar 1}]$	( $[{\bar 1},{\bar 1}]$ )	$[{\bar 1}]$ P	$[{\bar 1}]$ P(αβγ)
Monoclinic	2	( $[2, {\bar 1}]$ )	2/mP	2P(αβ0), 2P(αβ $[{{1}\over{2}}]$ )
			2/mB	2B(αβ0)
		(2, 1)	2/mP	2P(00γ), 2P( $[{{1}\over{2}}]$ 0γ)
			2/mB	2B(00γ), 2B(0 $[{{1}\over{2}}]$ γ)
	m	(m, 1)	2/mP	mP(αβ0), mP(αβ $[{{1}\over{2}}]$ )
			2/mB	mB(αβ0)
		(m, $[{\bar 1}]$ )	2/mP	mP(00γ), mP( $[{{1}\over{2}}]$ 0γ)
			2/mB	mB(00γ), mB(0 $[{{1}\over{2}}]$ γ)
	2/m	(2/m, $[{\bar 1}1]$ )	2/mP	2/mP(αβ0), 2/mP(αβ $[{{1}\over{2}}]$ )
			2/mB	2/mB(αβ0)
		(2/m, $[1{\bar 1}]$ )	2/mP	2/mP(00γ), 2/mP( $[{{1}\over{2}}]$ 0γ)
			2/mB	2/mB(00γ), 2/mB(0 $[{{1}\over{2}}]$ γ)
Orthorhombic	222	(222, $[{\bar 1}{\bar 1}1]$ )	mmmP	222P(00γ), 222P(0 $[{{1}\over{2}}]$ γ), 222P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			mmmI	222I(00γ)
			mmmF	222F(00γ), 222F(10γ)
			mmmC	222C(00γ), 222C(10γ)
		(222, $[1{\bar 1}{\bar 1}]$ )	mmmC	222C(α00), 222C(α0 $[{{1}\over{2}}]$ )
	mm2	(mm2, 111)	mmmP	mm2P(00γ), mm2P(0 $[{{1}\over{2}}]$ γ), mm2P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			mmmI	mm2I(00γ)
			mmmF	mm2F(00γ), mm2F(10γ)
			mmmC	mm2C(00γ), mm2C(10γ)
		(2mm, 111)	mmmC	2mmC(α00), 2mmC(α0 $[{{1}\over{2}}]$ )
		(2mm, $[{\bar 1}1{\bar 1}]$ )	mmmP	2mmP(00γ), 2mmP(0 $[{{1}\over{2}}]$ γ), 2mmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			mmmI	2mmI(00γ)
			mmmF	2mmF(00γ), 2mmF(10γ)
			mmmC	2mmC(00γ), 2mmC(10γ)
		(mm2, $[{\bar 1}1{\bar 1}]$ )	mmmC	mm2C(α00), mm2C(α0 $[{{1}\over{2}}]$ )
		(m2m, $[1{\bar 1}{\bar 1}]$ )	mmmP	m2mP(0 $[{{1}\over{2}}]$ γ)
		(m2m, $[{\bar 1}{\bar 1}1]$ )	mmmC	m2mC(α00), m2mC(α0 $[{{1}\over{2}}]$ )
	mmm	(mmm, $[11{\bar 1}]$ )	mmmP	mmmP(00γ), mmmP(0 $[{{1}\over{2}}]$ γ), mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			mmmI	mmmI(00γ)
			mmmF	mmmF(00γ), mmmF(10γ)
			mmmC	mmmC(00γ), mmmC(10γ)
		(mmm, $[{\bar 1}11]$ )	mmmC	mmmC(α00), mmmC(α0 $[{{1}\over{2}}]$ )
Tetragonal	4	(4, 1)	4/mmmP	4P(00γ), 4P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	4I(00γ)
	$[{\bar 4}]$	( $[{\bar 4}]$ , $[{\bar 1}]$ )	4/mmmP	$[{\bar 4}]$ P(00γ), $[{\bar 4}]$ P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	$[{\bar 4}]$ I(00γ)
	4/m	(4/m, $[1{\bar 1}]$ )	4/mmmP	4/mP(00γ), 4/mP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	4/mI(00γ)
	422	(422, $[1{\bar 1}{\bar 1}]$ )	4/mmmP	422P(00γ), 422P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	422I(00γ)
	4mm	(4mm, 111)	4/mmmP	4mmP(00γ), 4mmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	4mmI(00γ)
	$[{\bar 4}]$ 2m	( $[{\bar 4}]$ 2m, $[{\bar 1}{\bar 1}1]$ )	4/mmmP	$[{\bar 4}]$ 2mP(00γ), $[{\bar 4}]$ 2mP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	$[{\bar 4}]$ 2mI(00γ)
	$[{\bar 4}]$ m2	( $[{\bar 4}]$ m2, $[{\bar 1}1{\bar 1}]$ )	4/mmmP	$[{\bar 4}]$ m2P(00γ), $[{\bar 4}]$ m2P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	$[{\bar 4}]$ m2I(00γ)
	4/mmm	(4/mmm, $[1{\bar 1}11]$ )	4/mmmP	4/mmmP(00γ), 4/mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	4/mmmI(00γ)
Trigonal	3	(3, 1)	$[{\bar 3}]$ mR	3R(00γ)
			6/mmmP	3P(00γ), 3P( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
	$[{\bar 3}]$	( $[{\bar 3}]$ , $[{\bar 1}]$ )	$[{\bar 3}]$ mR	$[{\bar 3}]$ R(00γ)
			6/mmmP	$[{\bar 3}]$ P(00γ), $[{\bar 3}]$ P( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
	32	(32, $[1{\bar 1}]$ )	$[{\bar 3}]$ mR	32R(00γ)
			6/mmmP	312P(00γ), 312P( $[{{1}\over{3}}{{1}\over{3}}]$ γ), 321P(00γ)
	3m	(3m, 11)	$[{\bar 3}]$ mR	3mR(00γ)
			6/mmmP	3m1P(00γ), 31mP(00γ), 31mP( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
	$[{\bar 3}]$ m	( $[{\bar 3}m]$ , $[{\bar 1}1]$ )	$[{\bar 3}]$ mR	$[{\bar 3}]$ mR(00γ)
			6/mmmP	$[{\bar 3}]$ 1mP(00γ), $[{\bar 3}]$ 1mP( $[{{1}\over{3}}{{1}\over{3}}]$ γ), $[{\bar 3}]$ m1P(00γ)
Hexagonal	6	(6, 1)	6/mmmP	6P(00γ)
	$[{\bar 6}]$	( $[{\bar 6}]$ , $[{\bar 1}]$ )	6/mmmP	$[{\bar 6}]$ P(00γ)
	6/m	(6/m, $[1{\bar 1}]$ )	6/mmmP	6/mP(00γ)
	622	(622, $[1{\bar 1}{\bar 1}]$ )	6/mmmP	622P(00γ)
	6mm	(6mm, 111)	6/mmmP	6mmP(00γ)
	$[{\bar 6}]$ m2	( $[{\bar 6}m2]$ , $[{\bar 1}1{\bar 1}]$ )	6/mmmP	$[{\bar 6}]$ m2P(00γ)
	$[{\bar 6}2m]$	( $[{\bar 6}2m]$ , $[{\bar 1}{\bar 1}1]$ )	6/mmmP	$[{\bar 6}]$ 2mP(00γ)
	6/mmm	(6/mmm, $[1{\bar 1}11]$ )	6/mmmP	6/mmmP(00γ)