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

International Tables for Crystallography (2015). Vol. A, ch. 1.4, p. 60

Table 1.4.3.2 

H. Wondratscheke

Table 1.4.3.2| top | pdf |
Generation of the space group [P6_122\equiv D_6^2 ] (178)

The entries in the second column designated by the numbers (1)–(12) correspond to the coordinate triplets of the general position of [P6_122].

 Coordinate tripletsSymmetry operations
[\ispecialfonts{\sfi g}_{1}] (1) [x, y, z]; Identity [\ispecialfonts\sfi I]
[\ispecialfonts\!\matrix{{\sfi g}_{2}\hfill\cr {\sfi g}_{3}\hfill\cr {\sfi g}_{4}\hfill\cr}] [\!\left.\matrix{t(100)\hfill\cr t(010)\hfill\cr t(001)\hfill\cr}\right\}] [\!\left\{\matrix{\cr \hbox{Generating translations}\hfill\cr \cr}\right.]
  The group [{\cal G}_{4} \equiv {\cal T}] of all translations of P6122 has been generated  
[\ispecialfonts{\sfi g}_{5}] [(2)\ \bar{y},x - y, z + {1 \over 3}\semi] Threefold screw rotation
[\ispecialfonts{\sfi g}_{5}^{2}] [(3)\ \bar{x} + y, \bar{x}, z + {2 \over 3}\semi] Threefold screw rotation
[\ispecialfonts{\sfi g}_{5}^{3} = t(001){:}] Now the space group [{\cal G}_{5} \equiv P3_{1}] has been generated  
[\ispecialfonts{\sfi g}_{6}] [(4)\ \bar{x},\bar{y}, z + {1 \over 2}\semi] Twofold screw rotation
[\ispecialfonts{\sfi g}_{6}* {\sfi g}_{5}] [(5)\ y,\bar{x} + y,z + {5 \over 6}\semi] Sixfold screw rotation
[\ispecialfonts{\sfi g}_{6}* {\sfi g}_{5}^{2}] [x - y,x,z + {7 \over 6} \sim (6)\ x - y,x, z + {1 \over 6}\semi] Sixfold screw rotation
[\ispecialfonts{\sfi g}_{6}^{2} = t(001){:}] Now the space group [{\cal G}_{6} \equiv P6_{1}] has been generated  
[\ispecialfonts{\sfi g}_{7}] [(7)\ y,x,\bar{z} + {1 \over 3}\semi] Twofold rotation, direction of axis [110]
[\ispecialfonts{\sfi g}_{7}*{\sfi g}_{5}] [(8)\ x - y,\bar{y},\bar{z}\semi] Twofold rotation, axis [100]
[\ispecialfonts{\sfi g}_{7}*{\sfi g}_{5}^{2}] [\bar{x},\bar{x} + y, \bar{z} - {1 \over 3} \sim (9)\ \bar{x},\bar{x} + y, \bar{z} + {2 \over 3}\semi] Twofold rotation, axis [010]
[\ispecialfonts{\sfi g}_{7}* {\sfi g}_{6}] [\bar{y},\bar{x},\bar{z} - {1 \over 6} \sim (10)\ \bar{y},\bar{x},\bar{z} + {5 \over 6}\semi] Twofold rotation, axis [[1\bar{1}0]]
[\ispecialfonts{\sfi g}_{7}* {\sfi g}_{6}*{\sfi g}_{5}] [\bar{x} + y,y, \bar{z} - {1 \over 2} \sim (11)\ \bar{x} + y, y,\bar{z} + {1 \over 2}\semi] Twofold rotation, axis [120]
[\ispecialfonts{\sfi g}_{7}* {\sfi g}_{6}* {\sfi g}_{5}^{2}] [x,x - y, \bar{z} - {5 \over 6} \sim (12)\ x,x - y,\bar{z} + {1 \over 6}\semi] Twofold rotation, axis [210]
[\ispecialfonts{\sfi g}_{7}^{2} = {\sfi I}] [{\cal G}_{7} \sim P6_{1}22]