Vesuvianite (Ca10Al4(Mg,Fe)2Si9O34(OH)4, $S2_{3}$) Structure : A4B10C2D34E4F9_tP252_126_k_ce2k_f_h8k_k_d2k

Picture of Structure; Click for Big Picture
Prototype : Al4Ca10(Mg,Fe)2O34(OH)4Si9
AFLOW prototype label : A4B10C2D34E4F9_tP252_126_k_ce2k_f_h8k_k_d2k
Strukturbericht designation : $S2_{3}$
Pearson symbol : tP252
Space group number : 126
Space group symbol : $P4/nnc$
AFLOW prototype command : aflow --proto=A4B10C2D34E4F9_tP252_126_k_ce2k_f_h8k_k_d2k
--params=
$a$,$c/a$,$z_{3}$,$x_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$,$x_{10}$,$y_{10}$,$z_{10}$,$x_{11}$,$y_{11}$,$z_{11}$,$x_{12}$,$y_{12}$,$z_{12}$,$x_{13}$,$y_{13}$,$z_{13}$,$x_{14}$,$y_{14}$,$z_{14}$,$x_{15}$,$y_{15}$,$z_{15}$,$x_{16}$,$y_{16}$,$z_{16}$,$x_{17}$,$y_{17}$,$z_{17}$,$x_{18}$,$y_{18}$,$z_{18}$,$x_{19}$,$y_{19}$,$z_{19}$


  • Vesuvianite, also known as idocrase, comes in a variety of compositions and structures, see e.g., (Allen, 1992) and (Rucklidge, 1975) and references therein. For our example we use the original structure of (Warren, 1931), where the magnesium ($8f$) site is filled by a random (Mg,Fe) alloy. The positions of the hydrogen atoms in the OH ions were not determined, so we only give the positions of the oyxgen atoms (labeled as OH).

Simple Tetragonal primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & a \, \mathbf{\hat{x}} \\ \mathbf{a}_2 & = & a \, \mathbf{\hat{y}} \\ \mathbf{a}_3 & = & c \, \mathbf{\hat{z}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{3}{4}a \, \mathbf{\hat{y}} + \frac{3}{4}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{Ca I} \\ \mathbf{B}_{2} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{3}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{3}{4}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{Ca I} \\ \mathbf{B}_{3} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{3}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{Ca I} \\ \mathbf{B}_{4} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{3}{4}a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{Ca I} \\ \mathbf{B}_{5} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{3}{4}a \, \mathbf{\hat{y}} & \left(4d\right) & \mbox{Si I} \\ \mathbf{B}_{6} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} & = & \frac{3}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} & \left(4d\right) & \mbox{Si I} \\ \mathbf{B}_{7} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{3}{4}a \, \mathbf{\hat{y}} + \frac{1}{2}c \, \mathbf{\hat{z}} & \left(4d\right) & \mbox{Si I} \\ \mathbf{B}_{8} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{3}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{2}c \, \mathbf{\hat{z}} & \left(4d\right) & \mbox{Si I} \\ \mathbf{B}_{9} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca II} \\ \mathbf{B}_{10} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{3}\right) \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{3}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca II} \\ \mathbf{B}_{11} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & \frac{3}{4}a \, \mathbf{\hat{x}} + \frac{3}{4}a \, \mathbf{\hat{y}}-z_{3}c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca II} \\ \mathbf{B}_{12} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \frac{3}{4}a \, \mathbf{\hat{x}} + \frac{3}{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{Ca II} \\ \mathbf{B}_{13} & = & 0 \, \mathbf{a}_{1} + 0 \, \mathbf{a}_{2} + 0 \, \mathbf{a}_{3} & = & 0 \, \mathbf{\hat{x}} + 0 \, \mathbf{\hat{y}} + 0 \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{Mg} \\ \mathbf{B}_{14} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} & \left(8f\right) & \mbox{Mg} \\ \mathbf{B}_{15} & = & \frac{1}{2} \, \mathbf{a}_{1} & = & \frac{1}{2}a \, \mathbf{\hat{x}} & \left(8f\right) & \mbox{Mg} \\ \mathbf{B}_{16} & = & \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{2}a \, \mathbf{\hat{y}} & \left(8f\right) & \mbox{Mg} \\ \mathbf{B}_{17} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2}c \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{Mg} \\ \mathbf{B}_{18} & = & \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{2}c \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{Mg} \\ \mathbf{B}_{19} & = & \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}c \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{Mg} \\ \mathbf{B}_{20} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{2}c \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{Mg} \\ \mathbf{B}_{21} & = & x_{5} \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(8h\right) & \mbox{O I} \\ \mathbf{B}_{22} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(8h\right) & \mbox{O I} \\ \mathbf{B}_{23} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(8h\right) & \mbox{O I} \\ \mathbf{B}_{24} & = & x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(8h\right) & \mbox{O I} \\ \mathbf{B}_{25} & = & -x_{5} \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}} + \frac{3}{4}c \, \mathbf{\hat{z}} & \left(8h\right) & \mbox{O I} \\ \mathbf{B}_{26} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{y}} + \frac{3}{4}c \, \mathbf{\hat{z}} & \left(8h\right) & \mbox{O I} \\ \mathbf{B}_{27} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}} + \frac{3}{4}c \, \mathbf{\hat{z}} & \left(8h\right) & \mbox{O I} \\ \mathbf{B}_{28} & = & -x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{y}} + \frac{3}{4}c \, \mathbf{\hat{z}} & \left(8h\right) & \mbox{O I} \\ \mathbf{B}_{29} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + y_{6}a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{30} & = & \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{6}\right)a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{31} & = & \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{6}\right)a \, \mathbf{\hat{x}} + x_{6}a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{32} & = & y_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & y_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{6}\right)a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{33} & = & \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{6}\right)a \, \mathbf{\hat{x}} + y_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{6}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{34} & = & x_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{6}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{35} & = & y_{6} \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{3} & = & y_{6}a \, \mathbf{\hat{x}} + x_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{6}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{36} & = & \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{6}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{37} & = & -x_{6} \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}}-y_{6}a \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{6}\right)a \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{39} & = & \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{1}-x_{6} \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{6}\right)a \, \mathbf{\hat{x}}-x_{6}a \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{40} & = & -y_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & -y_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{41} & = & \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{x}}-y_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{42} & = & -x_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{43} & = & -y_{6} \, \mathbf{a}_{1}-x_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & -y_{6}a \, \mathbf{\hat{x}}-x_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{44} & = & \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ag} \\ \mathbf{B}_{45} & = & x_{7} \, \mathbf{a}_{1} + y_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}} + y_{7}a \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{46} & = & \left(\frac{1}{2} - x_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{7}\right) \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{7}\right)a \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{47} & = & \left(\frac{1}{2} - y_{7}\right) \, \mathbf{a}_{1} + x_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{7}\right)a \, \mathbf{\hat{x}} + x_{7}a \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{48} & = & y_{7} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{7}\right) \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & y_{7}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{7}\right)a \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{49} & = & \left(\frac{1}{2} - x_{7}\right) \, \mathbf{a}_{1} + y_{7} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{7}\right)a \, \mathbf{\hat{x}} + y_{7}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{7}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{50} & = & x_{7} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{7}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{51} & = & y_{7} \, \mathbf{a}_{1} + x_{7} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{3} & = & y_{7}a \, \mathbf{\hat{x}} + x_{7}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{7}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{52} & = & \left(\frac{1}{2} - y_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{7}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{53} & = & -x_{7} \, \mathbf{a}_{1}-y_{7} \, \mathbf{a}_{2}-z_{7} \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}}-y_{7}a \, \mathbf{\hat{y}}-z_{7}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{54} & = & \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{2}-z_{7} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{7}\right)a \, \mathbf{\hat{y}}-z_{7}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{55} & = & \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{1}-x_{7} \, \mathbf{a}_{2}-z_{7} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{7}\right)a \, \mathbf{\hat{x}}-x_{7}a \, \mathbf{\hat{y}}-z_{7}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{56} & = & -y_{7} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{2}-z_{7} \, \mathbf{a}_{3} & = & -y_{7}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{7}\right)a \, \mathbf{\hat{y}}-z_{7}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{57} & = & \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{1}-y_{7} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{7}\right)a \, \mathbf{\hat{x}}-y_{7}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{58} & = & -x_{7} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{59} & = & -y_{7} \, \mathbf{a}_{1}-x_{7} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & -y_{7}a \, \mathbf{\hat{x}}-x_{7}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{60} & = & \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca III} \\ \mathbf{B}_{61} & = & x_{8} \, \mathbf{a}_{1} + y_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}} + y_{8}a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{62} & = & \left(\frac{1}{2} - x_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{8}\right) \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{8}\right)a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{63} & = & \left(\frac{1}{2} - y_{8}\right) \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{8}\right)a \, \mathbf{\hat{x}} + x_{8}a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{64} & = & y_{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{8}\right) \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & y_{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{8}\right)a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{65} & = & \left(\frac{1}{2} - x_{8}\right) \, \mathbf{a}_{1} + y_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{8}\right)a \, \mathbf{\hat{x}} + y_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{66} & = & x_{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{67} & = & y_{8} \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & y_{8}a \, \mathbf{\hat{x}} + x_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{68} & = & \left(\frac{1}{2} - y_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{69} & = & -x_{8} \, \mathbf{a}_{1}-y_{8} \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}}-y_{8}a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{70} & = & \left(\frac{1}{2} +x_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{8}\right) \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{8}\right)a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{71} & = & \left(\frac{1}{2} +y_{8}\right) \, \mathbf{a}_{1}-x_{8} \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{8}\right)a \, \mathbf{\hat{x}}-x_{8}a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{72} & = & -y_{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{8}\right) \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & -y_{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{8}\right)a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{73} & = & \left(\frac{1}{2} +x_{8}\right) \, \mathbf{a}_{1}-y_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{8}\right)a \, \mathbf{\hat{x}}-y_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{74} & = & -x_{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{75} & = & -y_{8} \, \mathbf{a}_{1}-x_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & -y_{8}a \, \mathbf{\hat{x}}-x_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{76} & = & \left(\frac{1}{2} +y_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Ca IV} \\ \mathbf{B}_{77} & = & x_{9} \, \mathbf{a}_{1} + y_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & x_{9}a \, \mathbf{\hat{x}} + y_{9}a \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{78} & = & \left(\frac{1}{2} - x_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{9}\right) \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{9}\right)a \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{79} & = & \left(\frac{1}{2} - y_{9}\right) \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{9}\right)a \, \mathbf{\hat{x}} + x_{9}a \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{80} & = & y_{9} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{9}\right) \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & y_{9}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{9}\right)a \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{81} & = & \left(\frac{1}{2} - x_{9}\right) \, \mathbf{a}_{1} + y_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{9}\right)a \, \mathbf{\hat{x}} + y_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{9}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{82} & = & x_{9} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & x_{9}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{9}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{83} & = & y_{9} \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & y_{9}a \, \mathbf{\hat{x}} + x_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{9}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{84} & = & \left(\frac{1}{2} - y_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{9}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{85} & = & -x_{9} \, \mathbf{a}_{1}-y_{9} \, \mathbf{a}_{2}-z_{9} \, \mathbf{a}_{3} & = & -x_{9}a \, \mathbf{\hat{x}}-y_{9}a \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{86} & = & \left(\frac{1}{2} +x_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{9}\right) \, \mathbf{a}_{2}-z_{9} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{9}\right)a \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{87} & = & \left(\frac{1}{2} +y_{9}\right) \, \mathbf{a}_{1}-x_{9} \, \mathbf{a}_{2}-z_{9} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{9}\right)a \, \mathbf{\hat{x}}-x_{9}a \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{88} & = & -y_{9} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{9}\right) \, \mathbf{a}_{2}-z_{9} \, \mathbf{a}_{3} & = & -y_{9}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{9}\right)a \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{89} & = & \left(\frac{1}{2} +x_{9}\right) \, \mathbf{a}_{1}-y_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{9}\right)a \, \mathbf{\hat{x}}-y_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{90} & = & -x_{9} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & -x_{9}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{91} & = & -y_{9} \, \mathbf{a}_{1}-x_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & -y_{9}a \, \mathbf{\hat{x}}-x_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{92} & = & \left(\frac{1}{2} +y_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O II} \\ \mathbf{B}_{93} & = & x_{10} \, \mathbf{a}_{1} + y_{10} \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & x_{10}a \, \mathbf{\hat{x}} + y_{10}a \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{94} & = & \left(\frac{1}{2} - x_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{10}\right) \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{10}\right)a \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{95} & = & \left(\frac{1}{2} - y_{10}\right) \, \mathbf{a}_{1} + x_{10} \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{10}\right)a \, \mathbf{\hat{x}} + x_{10}a \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{96} & = & y_{10} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{10}\right) \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & y_{10}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{10}\right)a \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{97} & = & \left(\frac{1}{2} - x_{10}\right) \, \mathbf{a}_{1} + y_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{10}\right)a \, \mathbf{\hat{x}} + y_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{10}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{98} & = & x_{10} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & x_{10}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{10}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{10}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{99} & = & y_{10} \, \mathbf{a}_{1} + x_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & y_{10}a \, \mathbf{\hat{x}} + x_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{10}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{100} & = & \left(\frac{1}{2} - y_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{10}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{10}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{101} & = & -x_{10} \, \mathbf{a}_{1}-y_{10} \, \mathbf{a}_{2}-z_{10} \, \mathbf{a}_{3} & = & -x_{10}a \, \mathbf{\hat{x}}-y_{10}a \, \mathbf{\hat{y}}-z_{10}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{102} & = & \left(\frac{1}{2} +x_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{10}\right) \, \mathbf{a}_{2}-z_{10} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{10}\right)a \, \mathbf{\hat{y}}-z_{10}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{103} & = & \left(\frac{1}{2} +y_{10}\right) \, \mathbf{a}_{1}-x_{10} \, \mathbf{a}_{2}-z_{10} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{10}\right)a \, \mathbf{\hat{x}}-x_{10}a \, \mathbf{\hat{y}}-z_{10}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{104} & = & -y_{10} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{10}\right) \, \mathbf{a}_{2}-z_{10} \, \mathbf{a}_{3} & = & -y_{10}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{10}\right)a \, \mathbf{\hat{y}}-z_{10}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{105} & = & \left(\frac{1}{2} +x_{10}\right) \, \mathbf{a}_{1}-y_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{10}\right)a \, \mathbf{\hat{x}}-y_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{106} & = & -x_{10} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & -x_{10}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{10}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{107} & = & -y_{10} \, \mathbf{a}_{1}-x_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & -y_{10}a \, \mathbf{\hat{x}}-x_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{108} & = & \left(\frac{1}{2} +y_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{10}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O III} \\ \mathbf{B}_{109} & = & x_{11} \, \mathbf{a}_{1} + y_{11} \, \mathbf{a}_{2} + z_{11} \, \mathbf{a}_{3} & = & x_{11}a \, \mathbf{\hat{x}} + y_{11}a \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{110} & = & \left(\frac{1}{2} - x_{11}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{11}\right) \, \mathbf{a}_{2} + z_{11} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{11}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{11}\right)a \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{111} & = & \left(\frac{1}{2} - y_{11}\right) \, \mathbf{a}_{1} + x_{11} \, \mathbf{a}_{2} + z_{11} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{11}\right)a \, \mathbf{\hat{x}} + x_{11}a \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{112} & = & y_{11} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{11}\right) \, \mathbf{a}_{2} + z_{11} \, \mathbf{a}_{3} & = & y_{11}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{11}\right)a \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{113} & = & \left(\frac{1}{2} - x_{11}\right) \, \mathbf{a}_{1} + y_{11} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{11}\right)a \, \mathbf{\hat{x}} + y_{11}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{11}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{114} & = & x_{11} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{11}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{11}\right) \, \mathbf{a}_{3} & = & x_{11}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{11}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{11}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{115} & = & y_{11} \, \mathbf{a}_{1} + x_{11} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{11}\right) \, \mathbf{a}_{3} & = & y_{11}a \, \mathbf{\hat{x}} + x_{11}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{11}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{116} & = & \left(\frac{1}{2} - y_{11}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{11}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{11}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{11}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{11}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{117} & = & -x_{11} \, \mathbf{a}_{1}-y_{11} \, \mathbf{a}_{2}-z_{11} \, \mathbf{a}_{3} & = & -x_{11}a \, \mathbf{\hat{x}}-y_{11}a \, \mathbf{\hat{y}}-z_{11}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{118} & = & \left(\frac{1}{2} +x_{11}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{11}\right) \, \mathbf{a}_{2}-z_{11} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{11}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{11}\right)a \, \mathbf{\hat{y}}-z_{11}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{119} & = & \left(\frac{1}{2} +y_{11}\right) \, \mathbf{a}_{1}-x_{11} \, \mathbf{a}_{2}-z_{11} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{11}\right)a \, \mathbf{\hat{x}}-x_{11}a \, \mathbf{\hat{y}}-z_{11}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{120} & = & -y_{11} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{11}\right) \, \mathbf{a}_{2}-z_{11} \, \mathbf{a}_{3} & = & -y_{11}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{11}\right)a \, \mathbf{\hat{y}}-z_{11}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{121} & = & \left(\frac{1}{2} +x_{11}\right) \, \mathbf{a}_{1}-y_{11} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{11}\right)a \, \mathbf{\hat{x}}-y_{11}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{122} & = & -x_{11} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{11}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{11}\right) \, \mathbf{a}_{3} & = & -x_{11}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{11}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{123} & = & -y_{11} \, \mathbf{a}_{1}-x_{11} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{11}\right) \, \mathbf{a}_{3} & = & -y_{11}a \, \mathbf{\hat{x}}-x_{11}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{124} & = & \left(\frac{1}{2} +y_{11}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{11}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{11}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{11}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IV} \\ \mathbf{B}_{125} & = & x_{12} \, \mathbf{a}_{1} + y_{12} \, \mathbf{a}_{2} + z_{12} \, \mathbf{a}_{3} & = & x_{12}a \, \mathbf{\hat{x}} + y_{12}a \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{126} & = & \left(\frac{1}{2} - x_{12}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{12}\right) \, \mathbf{a}_{2} + z_{12} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{12}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{12}\right)a \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{127} & = & \left(\frac{1}{2} - y_{12}\right) \, \mathbf{a}_{1} + x_{12} \, \mathbf{a}_{2} + z_{12} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{12}\right)a \, \mathbf{\hat{x}} + x_{12}a \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{128} & = & y_{12} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{12}\right) \, \mathbf{a}_{2} + z_{12} \, \mathbf{a}_{3} & = & y_{12}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{12}\right)a \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{129} & = & \left(\frac{1}{2} - x_{12}\right) \, \mathbf{a}_{1} + y_{12} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{12}\right)a \, \mathbf{\hat{x}} + y_{12}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{12}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{130} & = & x_{12} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{12}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{12}\right) \, \mathbf{a}_{3} & = & x_{12}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{12}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{12}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{131} & = & y_{12} \, \mathbf{a}_{1} + x_{12} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{12}\right) \, \mathbf{a}_{3} & = & y_{12}a \, \mathbf{\hat{x}} + x_{12}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{12}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{132} & = & \left(\frac{1}{2} - y_{12}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{12}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{12}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{12}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{12}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{133} & = & -x_{12} \, \mathbf{a}_{1}-y_{12} \, \mathbf{a}_{2}-z_{12} \, \mathbf{a}_{3} & = & -x_{12}a \, \mathbf{\hat{x}}-y_{12}a \, \mathbf{\hat{y}}-z_{12}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{134} & = & \left(\frac{1}{2} +x_{12}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{12}\right) \, \mathbf{a}_{2}-z_{12} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{12}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{12}\right)a \, \mathbf{\hat{y}}-z_{12}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{135} & = & \left(\frac{1}{2} +y_{12}\right) \, \mathbf{a}_{1}-x_{12} \, \mathbf{a}_{2}-z_{12} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{12}\right)a \, \mathbf{\hat{x}}-x_{12}a \, \mathbf{\hat{y}}-z_{12}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{136} & = & -y_{12} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{12}\right) \, \mathbf{a}_{2}-z_{12} \, \mathbf{a}_{3} & = & -y_{12}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{12}\right)a \, \mathbf{\hat{y}}-z_{12}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{137} & = & \left(\frac{1}{2} +x_{12}\right) \, \mathbf{a}_{1}-y_{12} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{12}\right)a \, \mathbf{\hat{x}}-y_{12}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{138} & = & -x_{12} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{12}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{12}\right) \, \mathbf{a}_{3} & = & -x_{12}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{12}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{139} & = & -y_{12} \, \mathbf{a}_{1}-x_{12} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{12}\right) \, \mathbf{a}_{3} & = & -y_{12}a \, \mathbf{\hat{x}}-x_{12}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{140} & = & \left(\frac{1}{2} +y_{12}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{12}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{12}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{12}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O V} \\ \mathbf{B}_{141} & = & x_{13} \, \mathbf{a}_{1} + y_{13} \, \mathbf{a}_{2} + z_{13} \, \mathbf{a}_{3} & = & x_{13}a \, \mathbf{\hat{x}} + y_{13}a \, \mathbf{\hat{y}} + z_{13}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{142} & = & \left(\frac{1}{2} - x_{13}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{13}\right) \, \mathbf{a}_{2} + z_{13} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{13}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{13}\right)a \, \mathbf{\hat{y}} + z_{13}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{143} & = & \left(\frac{1}{2} - y_{13}\right) \, \mathbf{a}_{1} + x_{13} \, \mathbf{a}_{2} + z_{13} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{13}\right)a \, \mathbf{\hat{x}} + x_{13}a \, \mathbf{\hat{y}} + z_{13}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{144} & = & y_{13} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{13}\right) \, \mathbf{a}_{2} + z_{13} \, \mathbf{a}_{3} & = & y_{13}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{13}\right)a \, \mathbf{\hat{y}} + z_{13}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{145} & = & \left(\frac{1}{2} - x_{13}\right) \, \mathbf{a}_{1} + y_{13} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{13}\right)a \, \mathbf{\hat{x}} + y_{13}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{13}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{146} & = & x_{13} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{13}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{13}\right) \, \mathbf{a}_{3} & = & x_{13}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{13}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{13}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{147} & = & y_{13} \, \mathbf{a}_{1} + x_{13} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{13}\right) \, \mathbf{a}_{3} & = & y_{13}a \, \mathbf{\hat{x}} + x_{13}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{13}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{148} & = & \left(\frac{1}{2} - y_{13}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{13}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{13}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{13}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{13}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{149} & = & -x_{13} \, \mathbf{a}_{1}-y_{13} \, \mathbf{a}_{2}-z_{13} \, \mathbf{a}_{3} & = & -x_{13}a \, \mathbf{\hat{x}}-y_{13}a \, \mathbf{\hat{y}}-z_{13}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{150} & = & \left(\frac{1}{2} +x_{13}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{13}\right) \, \mathbf{a}_{2}-z_{13} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{13}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{13}\right)a \, \mathbf{\hat{y}}-z_{13}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{151} & = & \left(\frac{1}{2} +y_{13}\right) \, \mathbf{a}_{1}-x_{13} \, \mathbf{a}_{2}-z_{13} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{13}\right)a \, \mathbf{\hat{x}}-x_{13}a \, \mathbf{\hat{y}}-z_{13}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{152} & = & -y_{13} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{13}\right) \, \mathbf{a}_{2}-z_{13} \, \mathbf{a}_{3} & = & -y_{13}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{13}\right)a \, \mathbf{\hat{y}}-z_{13}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{153} & = & \left(\frac{1}{2} +x_{13}\right) \, \mathbf{a}_{1}-y_{13} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{13}\right)a \, \mathbf{\hat{x}}-y_{13}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{154} & = & -x_{13} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{13}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{13}\right) \, \mathbf{a}_{3} & = & -x_{13}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{13}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{155} & = & -y_{13} \, \mathbf{a}_{1}-x_{13} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{13}\right) \, \mathbf{a}_{3} & = & -y_{13}a \, \mathbf{\hat{x}}-x_{13}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{156} & = & \left(\frac{1}{2} +y_{13}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{13}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{13}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{13}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VI} \\ \mathbf{B}_{157} & = & x_{14} \, \mathbf{a}_{1} + y_{14} \, \mathbf{a}_{2} + z_{14} \, \mathbf{a}_{3} & = & x_{14}a \, \mathbf{\hat{x}} + y_{14}a \, \mathbf{\hat{y}} + z_{14}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{158} & = & \left(\frac{1}{2} - x_{14}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{14}\right) \, \mathbf{a}_{2} + z_{14} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{14}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{14}\right)a \, \mathbf{\hat{y}} + z_{14}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{159} & = & \left(\frac{1}{2} - y_{14}\right) \, \mathbf{a}_{1} + x_{14} \, \mathbf{a}_{2} + z_{14} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{14}\right)a \, \mathbf{\hat{x}} + x_{14}a \, \mathbf{\hat{y}} + z_{14}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{160} & = & y_{14} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{14}\right) \, \mathbf{a}_{2} + z_{14} \, \mathbf{a}_{3} & = & y_{14}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{14}\right)a \, \mathbf{\hat{y}} + z_{14}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{161} & = & \left(\frac{1}{2} - x_{14}\right) \, \mathbf{a}_{1} + y_{14} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{14}\right)a \, \mathbf{\hat{x}} + y_{14}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{14}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{162} & = & x_{14} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{14}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{14}\right) \, \mathbf{a}_{3} & = & x_{14}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{14}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{14}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{163} & = & y_{14} \, \mathbf{a}_{1} + x_{14} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{14}\right) \, \mathbf{a}_{3} & = & y_{14}a \, \mathbf{\hat{x}} + x_{14}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{14}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{164} & = & \left(\frac{1}{2} - y_{14}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{14}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{14}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{14}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{14}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{165} & = & -x_{14} \, \mathbf{a}_{1}-y_{14} \, \mathbf{a}_{2}-z_{14} \, \mathbf{a}_{3} & = & -x_{14}a \, \mathbf{\hat{x}}-y_{14}a \, \mathbf{\hat{y}}-z_{14}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{166} & = & \left(\frac{1}{2} +x_{14}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{14}\right) \, \mathbf{a}_{2}-z_{14} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{14}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{14}\right)a \, \mathbf{\hat{y}}-z_{14}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{167} & = & \left(\frac{1}{2} +y_{14}\right) \, \mathbf{a}_{1}-x_{14} \, \mathbf{a}_{2}-z_{14} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{14}\right)a \, \mathbf{\hat{x}}-x_{14}a \, \mathbf{\hat{y}}-z_{14}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{168} & = & -y_{14} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{14}\right) \, \mathbf{a}_{2}-z_{14} \, \mathbf{a}_{3} & = & -y_{14}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{14}\right)a \, \mathbf{\hat{y}}-z_{14}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{169} & = & \left(\frac{1}{2} +x_{14}\right) \, \mathbf{a}_{1}-y_{14} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{14}\right)a \, \mathbf{\hat{x}}-y_{14}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{170} & = & -x_{14} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{14}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{14}\right) \, \mathbf{a}_{3} & = & -x_{14}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{14}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{171} & = & -y_{14} \, \mathbf{a}_{1}-x_{14} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{14}\right) \, \mathbf{a}_{3} & = & -y_{14}a \, \mathbf{\hat{x}}-x_{14}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{172} & = & \left(\frac{1}{2} +y_{14}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{14}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{14}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{14}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VII} \\ \mathbf{B}_{173} & = & x_{15} \, \mathbf{a}_{1} + y_{15} \, \mathbf{a}_{2} + z_{15} \, \mathbf{a}_{3} & = & x_{15}a \, \mathbf{\hat{x}} + y_{15}a \, \mathbf{\hat{y}} + z_{15}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{174} & = & \left(\frac{1}{2} - x_{15}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{15}\right) \, \mathbf{a}_{2} + z_{15} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{15}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{15}\right)a \, \mathbf{\hat{y}} + z_{15}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{175} & = & \left(\frac{1}{2} - y_{15}\right) \, \mathbf{a}_{1} + x_{15} \, \mathbf{a}_{2} + z_{15} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{15}\right)a \, \mathbf{\hat{x}} + x_{15}a \, \mathbf{\hat{y}} + z_{15}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{176} & = & y_{15} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{15}\right) \, \mathbf{a}_{2} + z_{15} \, \mathbf{a}_{3} & = & y_{15}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{15}\right)a \, \mathbf{\hat{y}} + z_{15}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{177} & = & \left(\frac{1}{2} - x_{15}\right) \, \mathbf{a}_{1} + y_{15} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{15}\right)a \, \mathbf{\hat{x}} + y_{15}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{15}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{178} & = & x_{15} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{15}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{15}\right) \, \mathbf{a}_{3} & = & x_{15}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{15}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{15}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{179} & = & y_{15} \, \mathbf{a}_{1} + x_{15} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{15}\right) \, \mathbf{a}_{3} & = & y_{15}a \, \mathbf{\hat{x}} + x_{15}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{15}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{180} & = & \left(\frac{1}{2} - y_{15}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{15}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{15}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{15}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{15}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{181} & = & -x_{15} \, \mathbf{a}_{1}-y_{15} \, \mathbf{a}_{2}-z_{15} \, \mathbf{a}_{3} & = & -x_{15}a \, \mathbf{\hat{x}}-y_{15}a \, \mathbf{\hat{y}}-z_{15}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{182} & = & \left(\frac{1}{2} +x_{15}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{15}\right) \, \mathbf{a}_{2}-z_{15} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{15}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{15}\right)a \, \mathbf{\hat{y}}-z_{15}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{183} & = & \left(\frac{1}{2} +y_{15}\right) \, \mathbf{a}_{1}-x_{15} \, \mathbf{a}_{2}-z_{15} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{15}\right)a \, \mathbf{\hat{x}}-x_{15}a \, \mathbf{\hat{y}}-z_{15}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{184} & = & -y_{15} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{15}\right) \, \mathbf{a}_{2}-z_{15} \, \mathbf{a}_{3} & = & -y_{15}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{15}\right)a \, \mathbf{\hat{y}}-z_{15}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{185} & = & \left(\frac{1}{2} +x_{15}\right) \, \mathbf{a}_{1}-y_{15} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{15}\right)a \, \mathbf{\hat{x}}-y_{15}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{186} & = & -x_{15} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{15}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{15}\right) \, \mathbf{a}_{3} & = & -x_{15}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{15}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{187} & = & -y_{15} \, \mathbf{a}_{1}-x_{15} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{15}\right) \, \mathbf{a}_{3} & = & -y_{15}a \, \mathbf{\hat{x}}-x_{15}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{188} & = & \left(\frac{1}{2} +y_{15}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{15}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{15}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{15}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O VIII} \\ \mathbf{B}_{189} & = & x_{16} \, \mathbf{a}_{1} + y_{16} \, \mathbf{a}_{2} + z_{16} \, \mathbf{a}_{3} & = & x_{16}a \, \mathbf{\hat{x}} + y_{16}a \, \mathbf{\hat{y}} + z_{16}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{190} & = & \left(\frac{1}{2} - x_{16}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{16}\right) \, \mathbf{a}_{2} + z_{16} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{16}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{16}\right)a \, \mathbf{\hat{y}} + z_{16}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{191} & = & \left(\frac{1}{2} - y_{16}\right) \, \mathbf{a}_{1} + x_{16} \, \mathbf{a}_{2} + z_{16} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{16}\right)a \, \mathbf{\hat{x}} + x_{16}a \, \mathbf{\hat{y}} + z_{16}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{192} & = & y_{16} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{16}\right) \, \mathbf{a}_{2} + z_{16} \, \mathbf{a}_{3} & = & y_{16}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{16}\right)a \, \mathbf{\hat{y}} + z_{16}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{193} & = & \left(\frac{1}{2} - x_{16}\right) \, \mathbf{a}_{1} + y_{16} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{16}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{16}\right)a \, \mathbf{\hat{x}} + y_{16}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{16}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{194} & = & x_{16} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{16}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{16}\right) \, \mathbf{a}_{3} & = & x_{16}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{16}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{16}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{195} & = & y_{16} \, \mathbf{a}_{1} + x_{16} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{16}\right) \, \mathbf{a}_{3} & = & y_{16}a \, \mathbf{\hat{x}} + x_{16}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{16}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{196} & = & \left(\frac{1}{2} - y_{16}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{16}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{16}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{16}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{16}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{16}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{197} & = & -x_{16} \, \mathbf{a}_{1}-y_{16} \, \mathbf{a}_{2}-z_{16} \, \mathbf{a}_{3} & = & -x_{16}a \, \mathbf{\hat{x}}-y_{16}a \, \mathbf{\hat{y}}-z_{16}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{198} & = & \left(\frac{1}{2} +x_{16}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{16}\right) \, \mathbf{a}_{2}-z_{16} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{16}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{16}\right)a \, \mathbf{\hat{y}}-z_{16}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{199} & = & \left(\frac{1}{2} +y_{16}\right) \, \mathbf{a}_{1}-x_{16} \, \mathbf{a}_{2}-z_{16} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{16}\right)a \, \mathbf{\hat{x}}-x_{16}a \, \mathbf{\hat{y}}-z_{16}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{200} & = & -y_{16} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{16}\right) \, \mathbf{a}_{2}-z_{16} \, \mathbf{a}_{3} & = & -y_{16}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{16}\right)a \, \mathbf{\hat{y}}-z_{16}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{201} & = & \left(\frac{1}{2} +x_{16}\right) \, \mathbf{a}_{1}-y_{16} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{16}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{16}\right)a \, \mathbf{\hat{x}}-y_{16}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{16}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{202} & = & -x_{16} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{16}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{16}\right) \, \mathbf{a}_{3} & = & -x_{16}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{16}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{16}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{203} & = & -y_{16} \, \mathbf{a}_{1}-x_{16} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{16}\right) \, \mathbf{a}_{3} & = & -y_{16}a \, \mathbf{\hat{x}}-x_{16}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{16}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{204} & = & \left(\frac{1}{2} +y_{16}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{16}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{16}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{16}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{16}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{16}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{O IX} \\ \mathbf{B}_{205} & = & x_{17} \, \mathbf{a}_{1} + y_{17} \, \mathbf{a}_{2} + z_{17} \, \mathbf{a}_{3} & = & x_{17}a \, \mathbf{\hat{x}} + y_{17}a \, \mathbf{\hat{y}} + z_{17}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{206} & = & \left(\frac{1}{2} - x_{17}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{17}\right) \, \mathbf{a}_{2} + z_{17} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{17}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{17}\right)a \, \mathbf{\hat{y}} + z_{17}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{207} & = & \left(\frac{1}{2} - y_{17}\right) \, \mathbf{a}_{1} + x_{17} \, \mathbf{a}_{2} + z_{17} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{17}\right)a \, \mathbf{\hat{x}} + x_{17}a \, \mathbf{\hat{y}} + z_{17}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{208} & = & y_{17} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{17}\right) \, \mathbf{a}_{2} + z_{17} \, \mathbf{a}_{3} & = & y_{17}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{17}\right)a \, \mathbf{\hat{y}} + z_{17}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{209} & = & \left(\frac{1}{2} - x_{17}\right) \, \mathbf{a}_{1} + y_{17} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{17}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{17}\right)a \, \mathbf{\hat{x}} + y_{17}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{17}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{210} & = & x_{17} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{17}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{17}\right) \, \mathbf{a}_{3} & = & x_{17}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{17}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{17}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{211} & = & y_{17} \, \mathbf{a}_{1} + x_{17} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{17}\right) \, \mathbf{a}_{3} & = & y_{17}a \, \mathbf{\hat{x}} + x_{17}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{17}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{212} & = & \left(\frac{1}{2} - y_{17}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{17}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{17}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{17}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{17}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{17}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{213} & = & -x_{17} \, \mathbf{a}_{1}-y_{17} \, \mathbf{a}_{2}-z_{17} \, \mathbf{a}_{3} & = & -x_{17}a \, \mathbf{\hat{x}}-y_{17}a \, \mathbf{\hat{y}}-z_{17}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{214} & = & \left(\frac{1}{2} +x_{17}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{17}\right) \, \mathbf{a}_{2}-z_{17} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{17}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{17}\right)a \, \mathbf{\hat{y}}-z_{17}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{215} & = & \left(\frac{1}{2} +y_{17}\right) \, \mathbf{a}_{1}-x_{17} \, \mathbf{a}_{2}-z_{17} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{17}\right)a \, \mathbf{\hat{x}}-x_{17}a \, \mathbf{\hat{y}}-z_{17}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{216} & = & -y_{17} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{17}\right) \, \mathbf{a}_{2}-z_{17} \, \mathbf{a}_{3} & = & -y_{17}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{17}\right)a \, \mathbf{\hat{y}}-z_{17}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{217} & = & \left(\frac{1}{2} +x_{17}\right) \, \mathbf{a}_{1}-y_{17} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{17}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{17}\right)a \, \mathbf{\hat{x}}-y_{17}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{17}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{218} & = & -x_{17} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{17}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{17}\right) \, \mathbf{a}_{3} & = & -x_{17}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{17}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{17}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{219} & = & -y_{17} \, \mathbf{a}_{1}-x_{17} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{17}\right) \, \mathbf{a}_{3} & = & -y_{17}a \, \mathbf{\hat{x}}-x_{17}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{17}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{220} & = & \left(\frac{1}{2} +y_{17}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{17}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{17}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{17}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{17}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{17}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{OH} \\ \mathbf{B}_{221} & = & x_{18} \, \mathbf{a}_{1} + y_{18} \, \mathbf{a}_{2} + z_{18} \, \mathbf{a}_{3} & = & x_{18}a \, \mathbf{\hat{x}} + y_{18}a \, \mathbf{\hat{y}} + z_{18}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{222} & = & \left(\frac{1}{2} - x_{18}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{18}\right) \, \mathbf{a}_{2} + z_{18} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{18}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{18}\right)a \, \mathbf{\hat{y}} + z_{18}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{223} & = & \left(\frac{1}{2} - y_{18}\right) \, \mathbf{a}_{1} + x_{18} \, \mathbf{a}_{2} + z_{18} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{18}\right)a \, \mathbf{\hat{x}} + x_{18}a \, \mathbf{\hat{y}} + z_{18}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{224} & = & y_{18} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{18}\right) \, \mathbf{a}_{2} + z_{18} \, \mathbf{a}_{3} & = & y_{18}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{18}\right)a \, \mathbf{\hat{y}} + z_{18}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{225} & = & \left(\frac{1}{2} - x_{18}\right) \, \mathbf{a}_{1} + y_{18} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{18}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{18}\right)a \, \mathbf{\hat{x}} + y_{18}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{18}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{226} & = & x_{18} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{18}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{18}\right) \, \mathbf{a}_{3} & = & x_{18}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{18}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{18}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{227} & = & y_{18} \, \mathbf{a}_{1} + x_{18} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{18}\right) \, \mathbf{a}_{3} & = & y_{18}a \, \mathbf{\hat{x}} + x_{18}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{18}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{228} & = & \left(\frac{1}{2} - y_{18}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{18}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{18}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{18}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{18}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{18}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{229} & = & -x_{18} \, \mathbf{a}_{1}-y_{18} \, \mathbf{a}_{2}-z_{18} \, \mathbf{a}_{3} & = & -x_{18}a \, \mathbf{\hat{x}}-y_{18}a \, \mathbf{\hat{y}}-z_{18}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{230} & = & \left(\frac{1}{2} +x_{18}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{18}\right) \, \mathbf{a}_{2}-z_{18} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{18}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{18}\right)a \, \mathbf{\hat{y}}-z_{18}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{231} & = & \left(\frac{1}{2} +y_{18}\right) \, \mathbf{a}_{1}-x_{18} \, \mathbf{a}_{2}-z_{18} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{18}\right)a \, \mathbf{\hat{x}}-x_{18}a \, \mathbf{\hat{y}}-z_{18}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{232} & = & -y_{18} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{18}\right) \, \mathbf{a}_{2}-z_{18} \, \mathbf{a}_{3} & = & -y_{18}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{18}\right)a \, \mathbf{\hat{y}}-z_{18}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{233} & = & \left(\frac{1}{2} +x_{18}\right) \, \mathbf{a}_{1}-y_{18} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{18}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{18}\right)a \, \mathbf{\hat{x}}-y_{18}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{18}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{234} & = & -x_{18} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{18}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{18}\right) \, \mathbf{a}_{3} & = & -x_{18}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{18}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{18}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{235} & = & -y_{18} \, \mathbf{a}_{1}-x_{18} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{18}\right) \, \mathbf{a}_{3} & = & -y_{18}a \, \mathbf{\hat{x}}-x_{18}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{18}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{236} & = & \left(\frac{1}{2} +y_{18}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{18}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{18}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{18}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{18}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{18}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si II} \\ \mathbf{B}_{237} & = & x_{19} \, \mathbf{a}_{1} + y_{19} \, \mathbf{a}_{2} + z_{19} \, \mathbf{a}_{3} & = & x_{19}a \, \mathbf{\hat{x}} + y_{19}a \, \mathbf{\hat{y}} + z_{19}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{238} & = & \left(\frac{1}{2} - x_{19}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{19}\right) \, \mathbf{a}_{2} + z_{19} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{19}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{19}\right)a \, \mathbf{\hat{y}} + z_{19}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{239} & = & \left(\frac{1}{2} - y_{19}\right) \, \mathbf{a}_{1} + x_{19} \, \mathbf{a}_{2} + z_{19} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{19}\right)a \, \mathbf{\hat{x}} + x_{19}a \, \mathbf{\hat{y}} + z_{19}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{240} & = & y_{19} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{19}\right) \, \mathbf{a}_{2} + z_{19} \, \mathbf{a}_{3} & = & y_{19}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{19}\right)a \, \mathbf{\hat{y}} + z_{19}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{241} & = & \left(\frac{1}{2} - x_{19}\right) \, \mathbf{a}_{1} + y_{19} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{19}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{19}\right)a \, \mathbf{\hat{x}} + y_{19}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{19}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{242} & = & x_{19} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{19}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{19}\right) \, \mathbf{a}_{3} & = & x_{19}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{19}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{19}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{243} & = & y_{19} \, \mathbf{a}_{1} + x_{19} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{19}\right) \, \mathbf{a}_{3} & = & y_{19}a \, \mathbf{\hat{x}} + x_{19}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{19}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{244} & = & \left(\frac{1}{2} - y_{19}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{19}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{19}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{19}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{19}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{19}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{245} & = & -x_{19} \, \mathbf{a}_{1}-y_{19} \, \mathbf{a}_{2}-z_{19} \, \mathbf{a}_{3} & = & -x_{19}a \, \mathbf{\hat{x}}-y_{19}a \, \mathbf{\hat{y}}-z_{19}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{246} & = & \left(\frac{1}{2} +x_{19}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{19}\right) \, \mathbf{a}_{2}-z_{19} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{19}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{19}\right)a \, \mathbf{\hat{y}}-z_{19}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{247} & = & \left(\frac{1}{2} +y_{19}\right) \, \mathbf{a}_{1}-x_{19} \, \mathbf{a}_{2}-z_{19} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{19}\right)a \, \mathbf{\hat{x}}-x_{19}a \, \mathbf{\hat{y}}-z_{19}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{248} & = & -y_{19} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{19}\right) \, \mathbf{a}_{2}-z_{19} \, \mathbf{a}_{3} & = & -y_{19}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{19}\right)a \, \mathbf{\hat{y}}-z_{19}c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{249} & = & \left(\frac{1}{2} +x_{19}\right) \, \mathbf{a}_{1}-y_{19} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{19}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{19}\right)a \, \mathbf{\hat{x}}-y_{19}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{19}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{250} & = & -x_{19} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{19}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{19}\right) \, \mathbf{a}_{3} & = & -x_{19}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{19}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{19}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{251} & = & -y_{19} \, \mathbf{a}_{1}-x_{19} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{19}\right) \, \mathbf{a}_{3} & = & -y_{19}a \, \mathbf{\hat{x}}-x_{19}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{19}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \mathbf{B}_{252} & = & \left(\frac{1}{2} +y_{19}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{19}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{19}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{19}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{19}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{19}\right)c \, \mathbf{\hat{z}} & \left(16k\right) & \mbox{Si III} \\ \end{array} \]

References

  • B. E. Warren and D. I. Modell, The Structure of Vesuvianite Ca10Al4(Mg,Fe)2Si9O34(OH)4, Zeitschrift für Kristallographie – Crystalline Materials 78, 422–432 (1931), doi:10.1524/zkri.1931.78.1.422.
  • F. M. Allen and C. W. Burnham, A comprehensive structure–model for vesuvianite; symmetry variations and crystal growth, Can. Mineral. 30, 1–18 (1992).
  • J. C. Rucklidge, V. Kocman, S. H. Whitlow, and E. J. Gabe, The Crystal Structures of Three Canadian Vesuvianites, Can. Mineral. 13, 15–21 (1975).

Found in

  • C. Hermann, O. Lohrmann, and H. Philipp, eds., Strukturbericht Band II 1928–1932 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1937).

Geometry files


Prototype Generator

aflow --proto=A4B10C2D34E4F9_tP252_126_k_ce2k_f_h8k_k_d2k --params=

Species:

Running:

Output: