International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.5, p. 829

Table 3.5.2.1 

E. Koch,a W. Fischera and U. Müllerb

Table 3.5.2.1| top | pdf |
Euclidean normalizers of the plane groups

For the restrictions of the cell metric of the two oblique plane groups see text and Fig. 3.5.2.3[link].

Plane group [{\cal G}]Euclidean normalizer [{\cal N}\!_{{\cal E}}({\cal G})]Additional generators of [{\cal N}\!_{{\cal E}}({\cal G})]Index of [{\cal G}] in [{\cal N}\!_{{\cal E}}({\cal G})]
No.Hermann–Mauguin symbolCell metricSymbolBasis vectorsTranslationsTwofold rotationFurther generators
1 p1 General [p^{2}2] [\varepsilon_{1}{{\bf a}},\varepsilon_{2}{{\bf b}}] r, 0; 0, s [-x, {-y}]   [\infty^2\cdot 2\cdot 1]
    [a \,\lt\, b,\ \gamma =90^{\circ}] [p^{2}2mm] [\varepsilon_{1}{{\bf a}}, \varepsilon_{2}{{\bf b}}] r, 0; 0, s [-x, {-y}] [-x,\; y] [\infty^{2}\cdot 2\cdot 2]
    [2\cos\gamma = -a/b], [90^{\circ} \,\lt\, \gamma \,\lt\, 120^{\circ}] [c^{2}2mm] [\varepsilon_{1}{{\bf a}},\varepsilon_{2}({\textstyle{1 \over 2}}{{\bf a}}+{{\bf b}})] r, 0; 0, s [-x, {-y}] [x-y, {-y}] [\infty^{2}\cdot 2\cdot 2]
    [a = b], [90^{\circ} \,\lt\, \gamma \,\lt\, 120^{\circ}] [c^{2}2mm] [\varepsilon_{1}({{\bf a}-{\bf b}}), \varepsilon_{2}({{\bf a}}+{{\bf b}})] r, 0; 0, s [-x, {-y}] y, x [\infty^{2}\cdot 2\cdot 2]
    [a = b, \gamma = 90^{\circ}] [p^{2}4mm] [\varepsilon {{\bf a}}, \varepsilon{{\bf b}}] r, 0; 0, s [-x, {-y}] [-x,\; y\hbox{; } y,\; x] [\infty^{2}\cdot 2\cdot 4]
    [a = b, \gamma = 120^{\circ}] [p^{2}6mm] [\varepsilon {{\bf a}}, \varepsilon{{\bf b}}] r, 0; 0, s [-x, {-y}] [y,\; x\hbox{; } x,\; x-y] [\infty^{2}\cdot 2\cdot 6]
2 p2 General p2 [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf b}}] [{\textstyle{1 \over 2}},0;\ 0,{\textstyle{1 \over 2}}]     [4\cdot 1\cdot 1]
    [a \,\lt\, b, \gamma = 90^{\circ}] p2mm [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf b}}] [{\textstyle{1 \over 2}},0;\ 0,{\textstyle{1 \over 2}}]   [-x,\; y] [4\cdot 1\cdot 2]
    [2\cos\gamma = -a/b], [90^{\circ} \,\lt\, \gamma \,\lt\, 120^{\circ}] c2mm [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf a}}+{{\bf b}}] [{\textstyle{1 \over 2}},0;\ 0,{\textstyle{1 \over 2}}]   [x-y, -y] [4\cdot 1\cdot 2]
    [a = b], [90^{\circ} \,\lt\, \gamma \,\lt\, 120^{\circ}] c2mm [{\textstyle{1 \over 2}}({{\bf a}-{\bf b}}), {\textstyle{1 \over 2}}({{\bf a}}+{{\bf b}})] [{\textstyle{1 \over 2}},0;\ 0,{\textstyle{1 \over 2}}]   [y,\; x] [4\cdot 1\cdot 2]
    [a = b, \gamma = 90^{\circ}] p4mm [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf b}}] [{\textstyle{1 \over 2}},0;\ 0,{\textstyle{1 \over 2}}]   [-x,\; y\hbox{; } y,\; x] [4\cdot 1\cdot 4]
    [a = b, \gamma = 120^{\circ}] p6mm [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf b}}] [{\textstyle{1 \over 2}},0;\ 0,{\textstyle{1 \over 2}}]   [y,\; x\hbox{; } x,\; x-y] [4\cdot 1\cdot 6]
3 p1m1   [p^{1}2mm] [{\textstyle{1 \over 2}}{{\bf a}}, \varepsilon{{\bf b}}] [{\textstyle{1 \over 2}},0;\ 0,s] [-x, {-y}]   [(2\cdot\infty)\cdot 2\cdot 1]
4 p1g1   [p^{1}2mm] [{\textstyle{1 \over 2}}{{\bf a}}, {\varepsilon{{\bf b}}}] [{\textstyle{1 \over 2}},0;\ 0,s] [-x, {-y}]   [(2\cdot\infty)\cdot 2\cdot 1]
5 c1m1   [p^{1}2mm] [{\textstyle{1 \over 2}}{{\bf a}}, \varepsilon{{\bf b}}] [0,s] [-x, {-y}]   [\infty\cdot 2\cdot 1]
6 p2mm [a\neq b] p2mm [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf b}}] [{\textstyle{1 \over 2}},0;\ 0,{\textstyle{1 \over 2}}]     [4\cdot 1\cdot 1]
    [a = b] p4mm [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf b}}] [{\textstyle{1 \over 2}},0;\ 0,{\textstyle{1 \over 2}}]   y, x [4\cdot 1\cdot 2]
7 p2mg   p2mm [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf b}}] [{\textstyle{1 \over 2}},0;\ 0,{\textstyle{1 \over 2}}]     [4\cdot 1\cdot 1]
8 p2gg [a\neq b] p2mm [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf b}}] [{\textstyle{1 \over 2}},0;\ 0,{\textstyle{1 \over 2}}]     [4\cdot 1\cdot 1]
    [a = b] p4mm [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf b}}] [{\textstyle{1 \over 2}},0;\ 0,{\textstyle{1 \over 2}}]   y, x [4\cdot 1\cdot 2]
9 c2mm [a\neq b] p2mm [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf b}}] [{\textstyle{1 \over 2}},0]     [2\cdot 1\cdot 1]
    [a = b] p4mm [{\textstyle{1 \over 2}}{{\bf a}}, {\textstyle{1 \over 2}}{{\bf b}}] [{\textstyle{1 \over 2}},0]   y, x [2\cdot 1\cdot 2]
10 p4   p4mm [{\textstyle{1 \over 2}}({{\bf a}}-{{\bf b}}), {\textstyle{1 \over 2}}({{\bf a}}+{{\bf b}})] [{\textstyle{1 \over 2}},{\textstyle{1 \over 2}}]   y, x [2\cdot 1\cdot 2]
11 p4mm   p4mm [{\textstyle{1 \over 2}}({{\bf a}}-{{\bf b}}), {\textstyle{1 \over 2}}({{\bf a}}+{{\bf b}})] [{\textstyle{1 \over 2}},{\textstyle{1 \over 2}}]     [2\cdot 1\cdot 1]
12 p4gm   p4mm [{\textstyle{1 \over 2}}({{\bf a}}-{{\bf b}}), {\textstyle{1 \over 2}}({{\bf a}}+{{\bf b}})] [{\textstyle{1 \over 2}},{\textstyle{1 \over 2}}]     [2\cdot 1\cdot 1]
13 p3   p6mm [{\textstyle{1 \over 3}}(2{{\bf a}}+{{\bf b}})], [{\textstyle{1 \over 3}}(-{{\bf a}}+{{\bf b}})] [{\textstyle{2 \over 3}},{\textstyle{1 \over 3}}] [-x, {-y}] y, x [3\cdot 2\cdot 2]
14 p3m1   p6mm [{\textstyle{1 \over 3}}(2{{\bf a}}+{{\bf b}})], [{\textstyle{1 \over 3}}(-{{\bf a}}+{{\bf b}})] [{\textstyle{2 \over 3}},{\textstyle{1 \over 3}}] [-x, {-y}]   [3\cdot 2\cdot 1]
15 p31m   p6mm [{{\bf a}}, {{\bf b}}]   [-x, {-y}]   [1\cdot 2\cdot 1]
16 p6   p6mm [{{\bf a}}, {{\bf b}}]     y, x [1\cdot 1\cdot 2]
17 p6mm   p6mm [{{\bf a}}, {{\bf b}}]       [1\cdot 1\cdot 1]