International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A. ch. 13.2, pp. 843-844

Table 13.2.2.1 

Y. Billieta‡ and E. F. Bertautb§

a Département de Chimie, Faculté des Sciences et Techniques, Université de Bretagne Occidentale, Brest, France, and bLaboratoire de Cristallographie, CNRS, Grenoble, France

Table 13.2.2.1 | top | pdf |
Three-dimensional derivative lattices of indices 2 to 7

The entry for each derivative lattice starts with a running number which is followed, between parentheses, by the appropriate basis-vector relations. It should be noted that the seven derivative lattices of index 2 are also recorded in the space-group table of P1 (No. 1) under Maximal isomorphic subgroups of lowest index but in the slightly different sequence 1, 2, 3, 6, 5, 4, 7.

Index 2 [1(2{\bf a},{\bf b},{\bf c})]; [2({\bf a},2{\bf b},{\bf c})]; [3({\bf a},{\bf b},2{\bf c})]; [4(2{\bf a},{\bf b} + {\bf a},{\bf c})]; [5(2{\bf a},{\bf b},{\bf c} + {\bf a})]; [6({\bf a}, 2{\bf b}, {\bf c} + {\bf b})]; [7(2{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]
Index 3 [1(3{\bf a}, {\bf b}, {\bf c})]; [2({\bf a}, 3{\bf b}, {\bf c})]; [3({\bf a}, {\bf b}, 3{\bf c})]; [4(3{\bf a}, {\bf b} + {\bf a}, {\bf c})]; [5(3{\bf a}, {\bf b}, {\bf c} + {\bf a})]; [6({\bf a}, 3{\bf b}, {\bf c} + {\bf b})]; [7(3{\bf a}, {\bf b} - {\bf a}, {\bf c})]; [8(3{\bf a}, {\bf b}, {\bf c} - {\bf a})]; [9({\bf a}, 3{\bf b}, {\bf c} - {\bf b})]; [10(3{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]; [11(3{\bf a}, {\bf b} + {\bf a},{\bf c} - {\bf a})]; [12(3{\bf a}, {\bf b} - {\bf a},{\bf c} + {\bf a})]; [13(3{\bf a}, {\bf b} - {\bf a},{\bf c} - {\bf a})]
Index 4 [1(4{\bf a}, {\bf b}, {\bf c})]; [2({\bf a}, 4{\bf b}, {\bf c})]; [3({\bf a}, {\bf b}, 4{\bf c})]; [4(4{\bf a}, {\bf b} + {\bf a}, {\bf c})]; [5(4{\bf a}, {\bf b}, {\bf c} + {\bf a})]; [6({\bf a}, 4{\bf b}, {\bf c} + {\bf b})]; [7(4{\bf a}, {\bf b} - {\bf a}, {\bf c})]; [8(4{\bf a}, {\bf b}, {\bf c} - {\bf a})]; [9({\bf a}, 4{\bf b}, {\bf c} - {\bf b})]; [10(4{\bf a}, {\bf b} + 2{\bf a}, {\bf c})]; [11(4{\bf a}, {\bf b}, {\bf c} + 2{\bf a})]; [12(2{\bf a}, 2{\bf b} + {\bf a}, {\bf c})]; [13({\bf a}, 4{\bf b}, {\bf c} + 2{\bf b})]; [14(2{\bf a}, {\bf b}, 2{\bf c} + {\bf a})]; [15({\bf a}, 2{\bf b}, 2{\bf c} + {\bf b})]; [16(4{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]; [17(4{\bf a}, {\bf b} + {\bf a},{\bf c} - {\bf a})]; [18(4{\bf a}, {\bf b} - {\bf a},{\bf c} + {\bf a})]; [19(4{\bf a}, {\bf b} - {\bf a},{\bf c} - {\bf a})]; [20(4{\bf a}, {\bf b} + {\bf a},{\bf c} + 2{\bf a})]; [21(4{\bf a}, {\bf b} + 2{\bf a},{\bf c} + {\bf a})]; [22(2{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf b})]; [23(4{\bf a}, {\bf b} - {\bf a},{\bf c} + 2{\bf a})]; [24(4{\bf a}, {\bf b} + 2{\bf a},{\bf c} - {\bf a})]; [25(2{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf a} + {\bf b})]; [26(4{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 2{\bf a})]; [27(2{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf a})]; [28(2{\bf a}, {\bf b} + {\bf a}, 2{\bf c} + {\bf a})]; [29(2{\bf a}, 2{\bf b}, {\bf c})]; [30(2{\bf a}, {\bf b}, 2{\bf c})]; [31({\bf a}, 2{\bf b}, 2{\bf c})]; [32(2{\bf a}, 2{\bf b}, {\bf c} + {\bf a})]; [33(2{\bf a}, 2{\bf b}, {\bf c} + {\bf b})]; [34(2{\bf a}, {\bf b} + {\bf a}, 2{\bf c})]; [35(2{\bf a}, 2{\bf b}, {\bf c} + {\bf a} + {\bf b})]
Index 5 [1(5{\bf a}, {\bf b}, {\bf c})]; [2({\bf a}, 5{\bf b}, {\bf c})]; [3({\bf a}, {\bf b}, 5{\bf c})]; [4(5{\bf a}, {\bf b} + {\bf a},{\bf c})]; [5(5{\bf a}, {\bf b}, {\bf c} + {\bf a})]; [6({\bf a}, 5{\bf b}, {\bf c} + {\bf b})]; [7(5{\bf a}, {\bf b} - {\bf a}, {\bf c})]; [8(5{\bf a}, {\bf b}, {\bf c} - {\bf a})]; [9({\bf a}, 5{\bf b}, {\bf c} - {\bf b})]; [10(5{\bf a}, {\bf b} + 2{\bf a}, {\bf c})]; [11(5{\bf a}, {\bf b}, {\bf c} - 2{\bf a})]; [12({\bf a}, 5{\bf b}, {\bf c} + 2{\bf b})]; [13(5{\bf a}, {\bf b} - 2{\bf a}, {\bf c})]; [14(5{\bf a}, {\bf b}, {\bf c} + 2{\bf a})]; [15({\bf a}, 5{\bf b}, {\bf c} - 2{\bf b})]; [16(5{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]; [17(5{\bf a}, {\bf b} + {\bf a},{\bf c} - {\bf a})]; [18(5{\bf a}, {\bf b} - {\bf a},{\bf c} + {\bf a})]; [19(5{\bf a}, {\bf b} - {\bf a},{\bf c} - {\bf a})]; [20(5{\bf a}, {\bf b} + {\bf a},{\bf c} + 2{\bf a})]; [21(5{\bf a}, {\bf b} + {\bf a},{\bf c} - 2{\bf a})]; [22(5{\bf a}, {\bf b} + 2{\bf a},{\bf c} + {\bf a})]; [23(5{\bf a}, {\bf b} - 2{\bf a},{\bf c} + {\bf a})]; [24(5{\bf a}, {\bf b} + 2{\bf a},{\bf c} - 2{\bf a})]; [25(5{\bf a}, {\bf b} - 2{\bf a},{\bf c} + 2{\bf a})]; [26(5{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 2{\bf a})]; [27(5{\bf a}, {\bf b} - 2{\bf a},{\bf c} - {\bf a})]; [28(5{\bf a}, {\bf b} - {\bf a},{\bf c} - 2{\bf a})]; [29(5{\bf a}, {\bf b} - 2{\bf a},{\bf c} - 2{\bf a})]; [30(5{\bf a}, {\bf b} + 2{\bf a},{\bf c} - {\bf a})]; [31(5{\bf a}, {\bf b} - {\bf a},{\bf c} + 2{\bf a})]
Index 6 [1(6{\bf a}, {\bf b}, {\bf c})]; [2({\bf a}, 6{\bf b}, {\bf c})]; [3({\bf a}, {\bf b}, 6{\bf c})]; [4(6{\bf a}, {\bf b} + {\bf a}, {\bf c})]; [5(6{\bf a}, {\bf b}, {\bf c} + {\bf a})]; [6({\bf a}, 6{\bf b}, {\bf c} + {\bf b})]; [7(6{\bf a}, {\bf b} - {\bf a}, {\bf c})]; [8(6{\bf a}, {\bf b}, {\bf c} - {\bf a})]; [9({\bf a}, 6{\bf b}, {\bf c} - {\bf b})]; [10(6{\bf a}, {\bf b} + 2{\bf a}, {\bf c})]; [11(6{\bf a}, {\bf b}, {\bf c} + 2{\bf a})]; [12({\bf a}, 6{\bf b}, {\bf c} + 2{\bf b})]; [13(3{\bf a}, 2{\bf b} + {\bf a}, {\bf c})]; [14({\bf a}, 3{\bf b}, 2{\bf c} + {\bf b})]; [15(3{\bf a}, {\bf b}, 2{\bf c} + {\bf a})]; [16(6{\bf a}, {\bf b} - 2{\bf a}, {\bf c})]; [17(6{\bf a}, {\bf b}, {\bf c} - 2{\bf a})]; [18({\bf a}, 6{\bf b}, {\bf c} - 2{\bf b})]; [19(3{\bf a}, 2{\bf b} - {\bf a}, {\bf c})]; [20({\bf a}, 3{\bf b}, 2{\bf c} - {\bf b})]; [21(3{\bf a}, {\bf b}, 2{\bf c} - {\bf a})]; [22(6{\bf a}, {\bf b} + 3{\bf a}, {\bf c})]; [23(6{\bf a}, {\bf b}, {\bf c} + 3{\bf a})]; [24({\bf a}, 6{\bf b}, {\bf c} + 3{\bf b})]; [25(2{\bf a}, 3{\bf b} + {\bf a}, {\bf c})]; [26({\bf a}, 2{\bf b}, 3{\bf c} + {\bf b})]; [27(2{\bf a}, {\bf b}, 3{\bf c} + {\bf a})]; [28(6{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]; [29(6{\bf a}, {\bf b} + {\bf a},{\bf c} - {\bf a})]; [30(6{\bf a}, {\bf b} - {\bf a},{\bf c} + {\bf a})]; [31(6{\bf a}, {\bf b} - {\bf a},{\bf c} - {\bf a})]; [32(6{\bf a}, {\bf b} + {\bf a},{\bf c} + 2{\bf a})]; [33(6{\bf a}, {\bf b} + {\bf a},{\bf c} - 2{\bf a})]; [34(3{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf b})]; [35(3{\bf a}, 2{\bf b} - {\bf a},{\bf c} + {\bf b})]; [36(6{\bf a}, {\bf b} + 2{\bf a},{\bf c} + {\bf a})]; [37(6{\bf a}, {\bf b} - 2{\bf a},{\bf c} + {\bf a})]; [38(6{\bf a}, {\bf b} - {\bf a},{\bf c} + 2{\bf a})]; [39(6{\bf a}, {\bf b} + 2{\bf a},{\bf c} - {\bf a})]; [40(3{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf a} + {\bf b})]; [41(6{\bf a}, {\bf b} - {\bf a},{\bf c} - 2{\bf a})]; [42(6{\bf a}, {\bf b} - 2{\bf a},{\bf c} - {\bf a})]; [43(3{\bf a}, 2{\bf b} - {\bf a},{\bf c} - {\bf a} + {\bf b})]; [44(6{\bf a}, {\bf b} + {\bf a},{\bf c} + 3{\bf a})]; [45(6{\bf a}, {\bf b} + 3{\bf a},{\bf c} + {\bf a})]; [46(2{\bf a}, 3{\bf b} + {\bf a},{\bf c} + {\bf b})]; [47(6{\bf a}, {\bf b} - {\bf a},{\bf c} + 3{\bf a})]; [48(6{\bf a}, {\bf b} + 3{\bf a},{\bf c} - {\bf a})]; [49(2{\bf a}, 3{\bf b} + {\bf a},{\bf c} - {\bf b})]; [50(6{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 2{\bf a})]; [51(3{\bf a}, 2{\bf b} + {\bf a},{\bf c} - {\bf a})]; [52(3{\bf a}, {\bf b} - {\bf a}, 2{\bf c} + {\bf a})]; [53(6{\bf a}, {\bf b} - 2{\bf a},{\bf c} - 2{\bf a})]; [54(3{\bf a}, 2{\bf b} - {\bf a},{\bf c} - {\bf a})]; [55(3{\bf a}, {\bf b} - {\bf a}, 2{\bf c} - {\bf a})]; [56(6{\bf a}, {\bf b} + 2{\bf a},{\bf c} - 2{\bf a})]; [57(3{\bf a}, 2{\bf b} - {\bf a},{\bf c} + {\bf a})]; [58(3{\bf a}, {\bf b} + {\bf a}, 2{\bf c} + {\bf a})]; [59(6{\bf a}, {\bf b} - 2{\bf a},{\bf c} + 2{\bf a})]; [60(3{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf a})]; [61(3{\bf a}, {\bf b} + {\bf a}, 2{\bf c} - {\bf a})]; [62(6{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 3{\bf a})]; [63(2{\bf a}, 3{\bf b} + {\bf a},{\bf c} + {\bf a} - {\bf b})]; [64(3{\bf a}, 2{\bf b}, {\bf c} - {\bf a} + {\bf b})]; [65(6{\bf a}, {\bf b} + 3{\bf a},{\bf c} + 2{\bf a})]; [66(3{\bf a}, 2{\bf b} + {\bf a},{\bf c} - {\bf a} + {\bf b})]; [67(2{\bf a}, 3{\bf b}, {\bf c} + {\bf a} - {\bf b})]; [68(6{\bf a}, {\bf b} - 2{\bf a},{\bf c} + 3{\bf a})]; [69(2{\bf a}, 3{\bf b} + {\bf a},{\bf c} + {\bf b} + {\bf a})]; [70(3{\bf a}, 2{\bf b}, {\bf c} + {\bf a} + {\bf b})]; [71(6{\bf a}, {\bf b} + 3{\bf a},{\bf c} - 2{\bf a})]; [72(3{\bf a}, 2{\bf b} - {\bf a},{\bf c} + {\bf a} + {\bf b})]; [73(2{\bf a}, 3{\bf b}, {\bf c} + {\bf a} + {\bf b})]; [74(6{\bf a}, {\bf b} + 3{\bf a},{\bf c} + 3{\bf a})]; [75(2{\bf a}, 3{\bf b} + {\bf a},{\bf c} + {\bf a})]; [76(2{\bf a}, {\bf b} + {\bf a}, 3{\bf c} + {\bf a})]; [77(3{\bf a}, 2{\bf b}, {\bf c})]; [78(3{\bf a}, {\bf b}, 2{\bf c})]; [79(2{\bf a}, 3{\bf b}, {\bf c})]; [80(2{\bf a}, {\bf b}, 3{\bf c})]; [81({\bf a}, 3{\bf b}, 2{\bf c})]; [82({\bf a}, 2{\bf b}, 3{\bf c})]; [83(3{\bf a}, 2{\bf b}, {\bf c} + {\bf a})]; [84(3{\bf a}, {\bf b} + {\bf a}, 2{\bf c})]; [85(2{\bf a}, 3{\bf b}, {\bf c} + {\bf b})]; [86(3{\bf a}, 2{\bf b}, {\bf c} - {\bf a})]; [87(3{\bf a}, {\bf b} - {\bf a}, 2{\bf c})]; [88(2{\bf a}, 3{\bf b}, {\bf c} - {\bf b})]; [89(3{\bf a}, 2{\bf b}, {\bf c} + {\bf b})]; [90(2{\bf a}, 3{\bf b}, {\bf c} + {\bf a})]; [91(2{\bf a}, {\bf a} + {\bf b}, 3{\bf c})]
Index 7 [1(7{\bf a}, {\bf b}, {\bf c})]; [2({\bf a}, 7{\bf b}, {\bf c})]; [3({\bf a}, {\bf b}, 7{\bf c})]; [4(7{\bf a}, {\bf b} + {\bf a}, {\bf c})]; [5(7{\bf a}, {\bf b}, {\bf c} + {\bf b})]; [6({\bf a}, 7{\bf b}, {\bf c} + {\bf b})]; [7(7{\bf a}, {\bf b} - {\bf a}, {\bf c})]; [8(7{\bf a}, {\bf b}, {\bf c} - {\bf a})]; [9({\bf a}, 7{\bf b}, {\bf c} - {\bf b})]; [10(7{\bf a}, {\bf b} + 2{\bf a}, {\bf c})]; [11(7{\bf a}, {\bf b}, {\bf c} - 3{\bf a})]; [12({\bf a}, 7{\bf b}, {\bf c} + 2{\bf b})]; [13(7{\bf a}, {\bf b} - 3{\bf a}, {\bf c})]; [14(7{\bf a}, {\bf b}, {\bf c} + 2{\bf a})]; [15({\bf a}, 7{\bf b}, {\bf c} - 3{\bf b})]; [16(7{\bf a}, {\bf b} - 2{\bf a}, {\bf c})]; [17(7{\bf a}, {\bf b}, {\bf c} + 3{\bf a})]; [18({\bf a}, 7{\bf b}, {\bf c} - 2{\bf b})]; [19(7{\bf a}, {\bf b} + 3{\bf a}, {\bf c})]; [20(7{\bf a}, {\bf b}, {\bf c} - 2{\bf a})]; [21({\bf a}, 7{\bf b}, {\bf c} + 3{\bf b})]; [22(7{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]; [23(7{\bf a}, {\bf b} + {\bf a},{\bf c} - {\bf a})]; [24(7{\bf a}, {\bf b} - {\bf a},{\bf c} + {\bf a})]; [25(7{\bf a}, {\bf b} - {\bf a},{\bf c} - {\bf a})]; [26(7{\bf a}, {\bf b} + {\bf a},{\bf c} + 2{\bf a})]; [27(7{\bf a}, {\bf b} + {\bf a},{\bf c} - 2{\bf a})]; [28(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} + {\bf a})]; [29(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} + {\bf a})]; [30(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} - 3{\bf a})]; [31(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} + 3{\bf a})]; [32(7{\bf a}, {\bf b} + {\bf a},{\bf c} + 3{\bf a})]; [33(7{\bf a}, {\bf b} + {\bf a},{\bf c} - 3{\bf a})]; [34(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} + {\bf a})]; [35(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} + {\bf a})]; [36(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} - 2{\bf a})]; [37(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} + 2{\bf a})]; [38(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 2{\bf a})]; [39(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} - {\bf a})]; [40(7{\bf a}, {\bf b} - {\bf a},{\bf c} - 3{\bf a})]; [41(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} - 2{\bf a})]; [42(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} - {\bf a})]; [43(7{\bf a}, {\bf b} - {\bf a},{\bf c} + 3{\bf a})]; [44(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 3{\bf a})]; [45(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} + 2{\bf a})]; [46(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} - 3{\bf a})]; [47(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} + 2{\bf a})]; [48(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} - 3{\bf a})]; [49(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} - 2{\bf a})]; [50(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} + 3{\bf a})]; [51(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} - 2{\bf a})]; [52(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} + 3{\bf a})]; [53(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} - {\bf a})]; [54(7{\bf a}, {\bf b} - {\bf a},{\bf c} - 2{\bf a})]; [55(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} - 3{\bf a})]; [56(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} - {\bf a})]; [57(7{\bf a}, {\bf b} - {\bf a},{\bf c} + 2{\bf a})]