Library of Crystallographic Prototypes

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & z_{1} \, \mathbf{a}_{3} & = & z_{1}c \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Al I} \\ \mathbf{B}_{2} & = & \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Al I} \\ \mathbf{B}_{3} & = & \frac{1}{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \frac{1}{2}b \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(2b\right) & \mbox{Al II} \\ \mathbf{B}_{4} & = & \frac{1}{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(2b\right) & \mbox{Al II} \\ \mathbf{B}_{5} & = & \frac{1}{2} \, \mathbf{a}_{1} + z_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + z_{3}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{Al III} \\ \mathbf{B}_{6} & = & \frac{1}{2} \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{Al III} \\ \mathbf{B}_{7} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2}b \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(2d\right) & \mbox{Al IV} \\ \mathbf{B}_{8} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(2d\right) & \mbox{Al IV} \\ \mathbf{B}_{9} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + y_{5}b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al V} \\ \mathbf{B}_{10} & = & -x_{5} \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}}-y_{5}b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al V} \\ \mathbf{B}_{11} & = & x_{5} \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}}-y_{5}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al V} \\ \mathbf{B}_{12} & = & -x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + y_{5}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al V} \\ \mathbf{B}_{13} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + y_{6}b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VI} \\ \mathbf{B}_{14} & = & -x_{6} \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}}-y_{6}b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VI} \\ \mathbf{B}_{15} & = & x_{6} \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}}-y_{6}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VI} \\ \mathbf{B}_{16} & = & -x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}} + y_{6}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VI} \\ \mathbf{B}_{17} & = & x_{7} \, \mathbf{a}_{1} + y_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}} + y_{7}b \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VII} \\ \mathbf{B}_{18} & = & -x_{7} \, \mathbf{a}_{1}-y_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}}-y_{7}b \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VII} \\ \mathbf{B}_{19} & = & x_{7} \, \mathbf{a}_{1}-y_{7} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}}-y_{7}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VII} \\ \mathbf{B}_{20} & = & -x_{7} \, \mathbf{a}_{1} + y_{7} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}} + y_{7}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VII} \\ \mathbf{B}_{21} & = & x_{8} \, \mathbf{a}_{1} + y_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}} + y_{8}b \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VIII} \\ \mathbf{B}_{22} & = & -x_{8} \, \mathbf{a}_{1}-y_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}}-y_{8}b \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VIII} \\ \mathbf{B}_{23} & = & x_{8} \, \mathbf{a}_{1}-y_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}}-y_{8}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VIII} \\ \mathbf{B}_{24} & = & -x_{8} \, \mathbf{a}_{1} + y_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}} + y_{8}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Al VIII} \\ \mathbf{B}_{25} & = & x_{9} \, \mathbf{a}_{1} + y_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & x_{9}a \, \mathbf{\hat{x}} + y_{9}b \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca I} \\ \mathbf{B}_{26} & = & -x_{9} \, \mathbf{a}_{1}-y_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & -x_{9}a \, \mathbf{\hat{x}}-y_{9}b \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca I} \\ \mathbf{B}_{27} & = & x_{9} \, \mathbf{a}_{1}-y_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & x_{9}a \, \mathbf{\hat{x}}-y_{9}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca I} \\ \mathbf{B}_{28} & = & -x_{9} \, \mathbf{a}_{1} + y_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & -x_{9}a \, \mathbf{\hat{x}} + y_{9}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca I} \\ \mathbf{B}_{29} & = & x_{10} \, \mathbf{a}_{1} + y_{10} \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & x_{10}a \, \mathbf{\hat{x}} + y_{10}b \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca II} \\ \mathbf{B}_{30} & = & -x_{10} \, \mathbf{a}_{1}-y_{10} \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & -x_{10}a \, \mathbf{\hat{x}}-y_{10}b \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca II} \\ \mathbf{B}_{31} & = & x_{10} \, \mathbf{a}_{1}-y_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & x_{10}a \, \mathbf{\hat{x}}-y_{10}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca II} \\ \mathbf{B}_{32} & = & -x_{10} \, \mathbf{a}_{1} + y_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & -x_{10}a \, \mathbf{\hat{x}} + y_{10}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca II} \\ \mathbf{B}_{33} & = & x_{11} \, \mathbf{a}_{1} + y_{11} \, \mathbf{a}_{2} + z_{11} \, \mathbf{a}_{3} & = & x_{11}a \, \mathbf{\hat{x}} + y_{11}b \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca III} \\ \mathbf{B}_{34} & = & -x_{11} \, \mathbf{a}_{1}-y_{11} \, \mathbf{a}_{2} + z_{11} \, \mathbf{a}_{3} & = & -x_{11}a \, \mathbf{\hat{x}}-y_{11}b \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca III} \\ \mathbf{B}_{35} & = & x_{11} \, \mathbf{a}_{1}-y_{11} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{11}\right) \, \mathbf{a}_{3} & = & x_{11}a \, \mathbf{\hat{x}}-y_{11}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca III} \\ \mathbf{B}_{36} & = & -x_{11} \, \mathbf{a}_{1} + y_{11} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{11}\right) \, \mathbf{a}_{3} & = & -x_{11}a \, \mathbf{\hat{x}} + y_{11}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca III} \\ \mathbf{B}_{37} & = & x_{12} \, \mathbf{a}_{1} + y_{12} \, \mathbf{a}_{2} + z_{12} \, \mathbf{a}_{3} & = & x_{12}a \, \mathbf{\hat{x}} + y_{12}b \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca IV} \\ \mathbf{B}_{38} & = & -x_{12} \, \mathbf{a}_{1}-y_{12} \, \mathbf{a}_{2} + z_{12} \, \mathbf{a}_{3} & = & -x_{12}a \, \mathbf{\hat{x}}-y_{12}b \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca IV} \\ \mathbf{B}_{39} & = & x_{12} \, \mathbf{a}_{1}-y_{12} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{12}\right) \, \mathbf{a}_{3} & = & x_{12}a \, \mathbf{\hat{x}}-y_{12}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca IV} \\ \mathbf{B}_{40} & = & -x_{12} \, \mathbf{a}_{1} + y_{12} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{12}\right) \, \mathbf{a}_{3} & = & -x_{12}a \, \mathbf{\hat{x}} + y_{12}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca IV} \\ \mathbf{B}_{41} & = & x_{13} \, \mathbf{a}_{1} + y_{13} \, \mathbf{a}_{2} + z_{13} \, \mathbf{a}_{3} & = & x_{13}a \, \mathbf{\hat{x}} + y_{13}b \, \mathbf{\hat{y}} + z_{13}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O I} \\ \mathbf{B}_{42} & = & -x_{13} \, \mathbf{a}_{1}-y_{13} \, \mathbf{a}_{2} + z_{13} \, \mathbf{a}_{3} & = & -x_{13}a \, \mathbf{\hat{x}}-y_{13}b \, \mathbf{\hat{y}} + z_{13}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O I} \\ \mathbf{B}_{43} & = & x_{13} \, \mathbf{a}_{1}-y_{13} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{13}\right) \, \mathbf{a}_{3} & = & x_{13}a \, \mathbf{\hat{x}}-y_{13}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O I} \\ \mathbf{B}_{44} & = & -x_{13} \, \mathbf{a}_{1} + y_{13} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{13}\right) \, \mathbf{a}_{3} & = & -x_{13}a \, \mathbf{\hat{x}} + y_{13}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O I} \\ \mathbf{B}_{45} & = & x_{14} \, \mathbf{a}_{1} + y_{14} \, \mathbf{a}_{2} + z_{14} \, \mathbf{a}_{3} & = & x_{14}a \, \mathbf{\hat{x}} + y_{14}b \, \mathbf{\hat{y}} + z_{14}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O II} \\ \mathbf{B}_{46} & = & -x_{14} \, \mathbf{a}_{1}-y_{14} \, \mathbf{a}_{2} + z_{14} \, \mathbf{a}_{3} & = & -x_{14}a \, \mathbf{\hat{x}}-y_{14}b \, \mathbf{\hat{y}} + z_{14}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O II} \\ \mathbf{B}_{47} & = & x_{14} \, \mathbf{a}_{1}-y_{14} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{14}\right) \, \mathbf{a}_{3} & = & x_{14}a \, \mathbf{\hat{x}}-y_{14}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O II} \\ \mathbf{B}_{48} & = & -x_{14} \, \mathbf{a}_{1} + y_{14} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{14}\right) \, \mathbf{a}_{3} & = & -x_{14}a \, \mathbf{\hat{x}} + y_{14}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O II} \\ \mathbf{B}_{49} & = & x_{15} \, \mathbf{a}_{1} + y_{15} \, \mathbf{a}_{2} + z_{15} \, \mathbf{a}_{3} & = & x_{15}a \, \mathbf{\hat{x}} + y_{15}b \, \mathbf{\hat{y}} + z_{15}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O III} \\ \mathbf{B}_{50} & = & -x_{15} \, \mathbf{a}_{1}-y_{15} \, \mathbf{a}_{2} + z_{15} \, \mathbf{a}_{3} & = & -x_{15}a \, \mathbf{\hat{x}}-y_{15}b \, \mathbf{\hat{y}} + z_{15}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O III} \\ \mathbf{B}_{51} & = & x_{15} \, \mathbf{a}_{1}-y_{15} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{15}\right) \, \mathbf{a}_{3} & = & x_{15}a \, \mathbf{\hat{x}}-y_{15}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O III} \\ \mathbf{B}_{52} & = & -x_{15} \, \mathbf{a}_{1} + y_{15} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{15}\right) \, \mathbf{a}_{3} & = & -x_{15}a \, \mathbf{\hat{x}} + y_{15}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O III} \\ \mathbf{B}_{53} & = & x_{16} \, \mathbf{a}_{1} + y_{16} \, \mathbf{a}_{2} + z_{16} \, \mathbf{a}_{3} & = & x_{16}a \, \mathbf{\hat{x}} + y_{16}b \, \mathbf{\hat{y}} + z_{16}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IV} \\ \mathbf{B}_{54} & = & -x_{16} \, \mathbf{a}_{1}-y_{16} \, \mathbf{a}_{2} + z_{16} \, \mathbf{a}_{3} & = & -x_{16}a \, \mathbf{\hat{x}}-y_{16}b \, \mathbf{\hat{y}} + z_{16}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IV} \\ \mathbf{B}_{55} & = & x_{16} \, \mathbf{a}_{1}-y_{16} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{16}\right) \, \mathbf{a}_{3} & = & x_{16}a \, \mathbf{\hat{x}}-y_{16}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{16}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IV} \\ \mathbf{B}_{56} & = & -x_{16} \, \mathbf{a}_{1} + y_{16} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{16}\right) \, \mathbf{a}_{3} & = & -x_{16}a \, \mathbf{\hat{x}} + y_{16}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{16}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IV} \\ \mathbf{B}_{57} & = & x_{17} \, \mathbf{a}_{1} + y_{17} \, \mathbf{a}_{2} + z_{17} \, \mathbf{a}_{3} & = & x_{17}a \, \mathbf{\hat{x}} + y_{17}b \, \mathbf{\hat{y}} + z_{17}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O V} \\ \mathbf{B}_{58} & = & -x_{17} \, \mathbf{a}_{1}-y_{17} \, \mathbf{a}_{2} + z_{17} \, \mathbf{a}_{3} & = & -x_{17}a \, \mathbf{\hat{x}}-y_{17}b \, \mathbf{\hat{y}} + z_{17}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O V} \\ \mathbf{B}_{59} & = & x_{17} \, \mathbf{a}_{1}-y_{17} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{17}\right) \, \mathbf{a}_{3} & = & x_{17}a \, \mathbf{\hat{x}}-y_{17}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{17}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O V} \\ \mathbf{B}_{60} & = & -x_{17} \, \mathbf{a}_{1} + y_{17} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{17}\right) \, \mathbf{a}_{3} & = & -x_{17}a \, \mathbf{\hat{x}} + y_{17}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{17}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O V} \\ \mathbf{B}_{61} & = & x_{18} \, \mathbf{a}_{1} + y_{18} \, \mathbf{a}_{2} + z_{18} \, \mathbf{a}_{3} & = & x_{18}a \, \mathbf{\hat{x}} + y_{18}b \, \mathbf{\hat{y}} + z_{18}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VI} \\ \mathbf{B}_{62} & = & -x_{18} \, \mathbf{a}_{1}-y_{18} \, \mathbf{a}_{2} + z_{18} \, \mathbf{a}_{3} & = & -x_{18}a \, \mathbf{\hat{x}}-y_{18}b \, \mathbf{\hat{y}} + z_{18}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VI} \\ \mathbf{B}_{63} & = & x_{18} \, \mathbf{a}_{1}-y_{18} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{18}\right) \, \mathbf{a}_{3} & = & x_{18}a \, \mathbf{\hat{x}}-y_{18}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{18}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VI} \\ \mathbf{B}_{64} & = & -x_{18} \, \mathbf{a}_{1} + y_{18} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{18}\right) \, \mathbf{a}_{3} & = & -x_{18}a \, \mathbf{\hat{x}} + y_{18}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{18}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VI} \\ \mathbf{B}_{65} & = & x_{19} \, \mathbf{a}_{1} + y_{19} \, \mathbf{a}_{2} + z_{19} \, \mathbf{a}_{3} & = & x_{19}a \, \mathbf{\hat{x}} + y_{19}b \, \mathbf{\hat{y}} + z_{19}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VII} \\ \mathbf{B}_{66} & = & -x_{19} \, \mathbf{a}_{1}-y_{19} \, \mathbf{a}_{2} + z_{19} \, \mathbf{a}_{3} & = & -x_{19}a \, \mathbf{\hat{x}}-y_{19}b \, \mathbf{\hat{y}} + z_{19}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VII} \\ \mathbf{B}_{67} & = & x_{19} \, \mathbf{a}_{1}-y_{19} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{19}\right) \, \mathbf{a}_{3} & = & x_{19}a \, \mathbf{\hat{x}}-y_{19}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{19}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VII} \\ \mathbf{B}_{68} & = & -x_{19} \, \mathbf{a}_{1} + y_{19} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{19}\right) \, \mathbf{a}_{3} & = & -x_{19}a \, \mathbf{\hat{x}} + y_{19}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{19}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VII} \\ \mathbf{B}_{69} & = & x_{20} \, \mathbf{a}_{1} + y_{20} \, \mathbf{a}_{2} + z_{20} \, \mathbf{a}_{3} & = & x_{20}a \, \mathbf{\hat{x}} + y_{20}b \, \mathbf{\hat{y}} + z_{20}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VIII} \\ \mathbf{B}_{70} & = & -x_{20} \, \mathbf{a}_{1}-y_{20} \, \mathbf{a}_{2} + z_{20} \, \mathbf{a}_{3} & = & -x_{20}a \, \mathbf{\hat{x}}-y_{20}b \, \mathbf{\hat{y}} + z_{20}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VIII} \\ \mathbf{B}_{71} & = & x_{20} \, \mathbf{a}_{1}-y_{20} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{20}\right) \, \mathbf{a}_{3} & = & x_{20}a \, \mathbf{\hat{x}}-y_{20}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{20}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VIII} \\ \mathbf{B}_{72} & = & -x_{20} \, \mathbf{a}_{1} + y_{20} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{20}\right) \, \mathbf{a}_{3} & = & -x_{20}a \, \mathbf{\hat{x}} + y_{20}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{20}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VIII} \\ \mathbf{B}_{73} & = & x_{21} \, \mathbf{a}_{1} + y_{21} \, \mathbf{a}_{2} + z_{21} \, \mathbf{a}_{3} & = & x_{21}a \, \mathbf{\hat{x}} + y_{21}b \, \mathbf{\hat{y}} + z_{21}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IX} \\ \mathbf{B}_{74} & = & -x_{21} \, \mathbf{a}_{1}-y_{21} \, \mathbf{a}_{2} + z_{21} \, \mathbf{a}_{3} & = & -x_{21}a \, \mathbf{\hat{x}}-y_{21}b \, \mathbf{\hat{y}} + z_{21}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IX} \\ \mathbf{B}_{75} & = & x_{21} \, \mathbf{a}_{1}-y_{21} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{21}\right) \, \mathbf{a}_{3} & = & x_{21}a \, \mathbf{\hat{x}}-y_{21}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{21}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IX} \\ \mathbf{B}_{76} & = & -x_{21} \, \mathbf{a}_{1} + y_{21} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{21}\right) \, \mathbf{a}_{3} & = & -x_{21}a \, \mathbf{\hat{x}} + y_{21}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{21}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IX} \\ \mathbf{B}_{77} & = & x_{22} \, \mathbf{a}_{1} + y_{22} \, \mathbf{a}_{2} + z_{22} \, \mathbf{a}_{3} & = & x_{22}a \, \mathbf{\hat{x}} + y_{22}b \, \mathbf{\hat{y}} + z_{22}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O X} \\ \mathbf{B}_{78} & = & -x_{22} \, \mathbf{a}_{1}-y_{22} \, \mathbf{a}_{2} + z_{22} \, \mathbf{a}_{3} & = & -x_{22}a \, \mathbf{\hat{x}}-y_{22}b \, \mathbf{\hat{y}} + z_{22}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O X} \\ \mathbf{B}_{79} & = & x_{22} \, \mathbf{a}_{1}-y_{22} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{22}\right) \, \mathbf{a}_{3} & = & x_{22}a \, \mathbf{\hat{x}}-y_{22}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{22}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O X} \\ \mathbf{B}_{80} & = & -x_{22} \, \mathbf{a}_{1} + y_{22} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{22}\right) \, \mathbf{a}_{3} & = & -x_{22}a \, \mathbf{\hat{x}} + y_{22}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{22}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O X} \\ \mathbf{B}_{81} & = & x_{23} \, \mathbf{a}_{1} + y_{23} \, \mathbf{a}_{2} + z_{23} \, \mathbf{a}_{3} & = & x_{23}a \, \mathbf{\hat{x}} + y_{23}b \, \mathbf{\hat{y}} + z_{23}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XI} \\ \mathbf{B}_{82} & = & -x_{23} \, \mathbf{a}_{1}-y_{23} \, \mathbf{a}_{2} + z_{23} \, \mathbf{a}_{3} & = & -x_{23}a \, \mathbf{\hat{x}}-y_{23}b \, \mathbf{\hat{y}} + z_{23}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XI} \\ \mathbf{B}_{83} & = & x_{23} \, \mathbf{a}_{1}-y_{23} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{23}\right) \, \mathbf{a}_{3} & = & x_{23}a \, \mathbf{\hat{x}}-y_{23}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{23}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XI} \\ \mathbf{B}_{84} & = & -x_{23} \, \mathbf{a}_{1} + y_{23} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{23}\right) \, \mathbf{a}_{3} & = & -x_{23}a \, \mathbf{\hat{x}} + y_{23}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{23}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XI} \\ \mathbf{B}_{85} & = & x_{24} \, \mathbf{a}_{1} + y_{24} \, \mathbf{a}_{2} + z_{24} \, \mathbf{a}_{3} & = & x_{24}a \, \mathbf{\hat{x}} + y_{24}b \, \mathbf{\hat{y}} + z_{24}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XII} \\ \mathbf{B}_{86} & = & -x_{24} \, \mathbf{a}_{1}-y_{24} \, \mathbf{a}_{2} + z_{24} \, \mathbf{a}_{3} & = & -x_{24}a \, \mathbf{\hat{x}}-y_{24}b \, \mathbf{\hat{y}} + z_{24}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XII} \\ \mathbf{B}_{87} & = & x_{24} \, \mathbf{a}_{1}-y_{24} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{24}\right) \, \mathbf{a}_{3} & = & x_{24}a \, \mathbf{\hat{x}}-y_{24}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{24}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XII} \\ \mathbf{B}_{88} & = & -x_{24} \, \mathbf{a}_{1} + y_{24} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{24}\right) \, \mathbf{a}_{3} & = & -x_{24}a \, \mathbf{\hat{x}} + y_{24}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{24}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XII} \\ \mathbf{B}_{89} & = & x_{25} \, \mathbf{a}_{1} + y_{25} \, \mathbf{a}_{2} + z_{25} \, \mathbf{a}_{3} & = & x_{25}a \, \mathbf{\hat{x}} + y_{25}b \, \mathbf{\hat{y}} + z_{25}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XIII} \\ \mathbf{B}_{90} & = & -x_{25} \, \mathbf{a}_{1}-y_{25} \, \mathbf{a}_{2} + z_{25} \, \mathbf{a}_{3} & = & -x_{25}a \, \mathbf{\hat{x}}-y_{25}b \, \mathbf{\hat{y}} + z_{25}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XIII} \\ \mathbf{B}_{91} & = & x_{25} \, \mathbf{a}_{1}-y_{25} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{25}\right) \, \mathbf{a}_{3} & = & x_{25}a \, \mathbf{\hat{x}}-y_{25}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{25}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XIII} \\ \mathbf{B}_{92} & = & -x_{25} \, \mathbf{a}_{1} + y_{25} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{25}\right) \, \mathbf{a}_{3} & = & -x_{25}a \, \mathbf{\hat{x}} + y_{25}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{25}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XIII} \\ \mathbf{B}_{93} & = & x_{26} \, \mathbf{a}_{1} + y_{26} \, \mathbf{a}_{2} + z_{26} \, \mathbf{a}_{3} & = & x_{26}a \, \mathbf{\hat{x}} + y_{26}b \, \mathbf{\hat{y}} + z_{26}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XIV} \\ \mathbf{B}_{94} & = & -x_{26} \, \mathbf{a}_{1}-y_{26} \, \mathbf{a}_{2} + z_{26} \, \mathbf{a}_{3} & = & -x_{26}a \, \mathbf{\hat{x}}-y_{26}b \, \mathbf{\hat{y}} + z_{26}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XIV} \\ \mathbf{B}_{95} & = & x_{26} \, \mathbf{a}_{1}-y_{26} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{26}\right) \, \mathbf{a}_{3} & = & x_{26}a \, \mathbf{\hat{x}}-y_{26}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{26}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XIV} \\ \mathbf{B}_{96} & = & -x_{26} \, \mathbf{a}_{1} + y_{26} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{26}\right) \, \mathbf{a}_{3} & = & -x_{26}a \, \mathbf{\hat{x}} + y_{26}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{26}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XIV} \\ \mathbf{B}_{97} & = & x_{27} \, \mathbf{a}_{1} + y_{27} \, \mathbf{a}_{2} + z_{27} \, \mathbf{a}_{3} & = & x_{27}a \, \mathbf{\hat{x}} + y_{27}b \, \mathbf{\hat{y}} + z_{27}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XV} \\ \mathbf{B}_{98} & = & -x_{27} \, \mathbf{a}_{1}-y_{27} \, \mathbf{a}_{2} + z_{27} \, \mathbf{a}_{3} & = & -x_{27}a \, \mathbf{\hat{x}}-y_{27}b \, \mathbf{\hat{y}} + z_{27}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XV} \\ \mathbf{B}_{99} & = & x_{27} \, \mathbf{a}_{1}-y_{27} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{27}\right) \, \mathbf{a}_{3} & = & x_{27}a \, \mathbf{\hat{x}}-y_{27}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{27}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XV} \\ \mathbf{B}_{100} & = & -x_{27} \, \mathbf{a}_{1} + y_{27} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{27}\right) \, \mathbf{a}_{3} & = & -x_{27}a \, \mathbf{\hat{x}} + y_{27}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{27}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XV} \\ \mathbf{B}_{101} & = & x_{28} \, \mathbf{a}_{1} + y_{28} \, \mathbf{a}_{2} + z_{28} \, \mathbf{a}_{3} & = & x_{28}a \, \mathbf{\hat{x}} + y_{28}b \, \mathbf{\hat{y}} + z_{28}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XVI} \\ \mathbf{B}_{102} & = & -x_{28} \, \mathbf{a}_{1}-y_{28} \, \mathbf{a}_{2} + z_{28} \, \mathbf{a}_{3} & = & -x_{28}a \, \mathbf{\hat{x}}-y_{28}b \, \mathbf{\hat{y}} + z_{28}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XVI} \\ \mathbf{B}_{103} & = & x_{28} \, \mathbf{a}_{1}-y_{28} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{28}\right) \, \mathbf{a}_{3} & = & x_{28}a \, \mathbf{\hat{x}}-y_{28}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{28}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XVI} \\ \mathbf{B}_{104} & = & -x_{28} \, \mathbf{a}_{1} + y_{28} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{28}\right) \, \mathbf{a}_{3} & = & -x_{28}a \, \mathbf{\hat{x}} + y_{28}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{28}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XVI} \\ \mathbf{B}_{105} & = & x_{29} \, \mathbf{a}_{1} + y_{29} \, \mathbf{a}_{2} + z_{29} \, \mathbf{a}_{3} & = & x_{29}a \, \mathbf{\hat{x}} + y_{29}b \, \mathbf{\hat{y}} + z_{29}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{S} \\ \mathbf{B}_{106} & = & -x_{29} \, \mathbf{a}_{1}-y_{29} \, \mathbf{a}_{2} + z_{29} \, \mathbf{a}_{3} & = & -x_{29}a \, \mathbf{\hat{x}}-y_{29}b \, \mathbf{\hat{y}} + z_{29}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{S} \\ \mathbf{B}_{107} & = & x_{29} \, \mathbf{a}_{1}-y_{29} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{29}\right) \, \mathbf{a}_{3} & = & x_{29}a \, \mathbf{\hat{x}}-y_{29}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{29}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{S} \\ \mathbf{B}_{108} & = & -x_{29} \, \mathbf{a}_{1} + y_{29} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{29}\right) \, \mathbf{a}_{3} & = & -x_{29}a \, \mathbf{\hat{x}} + y_{29}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{29}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{S} \\ \end{array} \]