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

International Tables for Crystallography (2006). Vol. C, ch. 9.8, pp. 919-920


T. Janssen,a A. Janner,a A. Looijenga-Vosb and P. M. de Wolffc

aInstitute 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

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(3 + 1)-Dimensional point groups and arithmetic crystal classes

The four-dimensional point group Ks has external part KE, which belongs to a three-dimensional system. Depending on the Bravais class of the four-dimensional lattice left invariant by Ks, 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.

SystemPoint groupExternal Bravais classArithmetic crystal class(es)
Triclinic 1 (1, 1) [{\bar 1}]P 1P(αβγ)
[{\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)