Library of Crystallographic Prototypes

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & x_{1} \, \mathbf{a}_{1} + y_{1} \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & x_{1}a \, \mathbf{\hat{x}} + y_{1}b \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C I} \\ \mathbf{B}_{2} & = & \left(\frac{1}{2} - x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{1}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C I} \\ \mathbf{B}_{3} & = & -x_{1} \, \mathbf{a}_{1} + y_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{1}\right) \, \mathbf{a}_{3} & = & -x_{1}a \, \mathbf{\hat{x}} + y_{1}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{1}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C I} \\ \mathbf{B}_{4} & = & \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{1}\right) \, \mathbf{a}_{2}-z_{1} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{1}\right)b \, \mathbf{\hat{y}}-z_{1}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C I} \\ \mathbf{B}_{5} & = & -x_{1} \, \mathbf{a}_{1}-y_{1} \, \mathbf{a}_{2}-z_{1} \, \mathbf{a}_{3} & = & -x_{1}a \, \mathbf{\hat{x}}-y_{1}b \, \mathbf{\hat{y}}-z_{1}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C I} \\ \mathbf{B}_{6} & = & \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{1}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{1}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C I} \\ \mathbf{B}_{7} & = & x_{1} \, \mathbf{a}_{1}-y_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & x_{1}a \, \mathbf{\hat{x}}-y_{1}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C I} \\ \mathbf{B}_{8} & = & \left(\frac{1}{2} - x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{1}\right) \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{1}\right)b \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C I} \\ \mathbf{B}_{9} & = & x_{2} \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}} + y_{2}b \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C II} \\ \mathbf{B}_{10} & = & \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{2}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C II} \\ \mathbf{B}_{11} & = & -x_{2} \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{2}\right) \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}} + y_{2}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{2}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C II} \\ \mathbf{B}_{12} & = & \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{2}\right) \, \mathbf{a}_{2}-z_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{2}\right)b \, \mathbf{\hat{y}}-z_{2}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C II} \\ \mathbf{B}_{13} & = & -x_{2} \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2}-z_{2} \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}}-y_{2}b \, \mathbf{\hat{y}}-z_{2}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C II} \\ \mathbf{B}_{14} & = & \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{2}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{2}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C II} \\ \mathbf{B}_{15} & = & x_{2} \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}}-y_{2}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C II} \\ \mathbf{B}_{16} & = & \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{2}\right) \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{2}\right)b \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C II} \\ \mathbf{B}_{17} & = & x_{3} \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + y_{3}b \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C III} \\ \mathbf{B}_{18} & = & \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{3}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C III} \\ \mathbf{B}_{19} & = & -x_{3} \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + y_{3}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{3}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C III} \\ \mathbf{B}_{20} & = & \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3}\right) \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{3}\right)b \, \mathbf{\hat{y}}-z_{3}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C III} \\ \mathbf{B}_{21} & = & -x_{3} \, \mathbf{a}_{1}-y_{3} \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}}-y_{3}b \, \mathbf{\hat{y}}-z_{3}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C III} \\ \mathbf{B}_{22} & = & \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{3}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{3}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C III} \\ \mathbf{B}_{23} & = & x_{3} \, \mathbf{a}_{1}-y_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}}-y_{3}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C III} \\ \mathbf{B}_{24} & = & \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3}\right) \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{3}\right)b \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C III} \\ \mathbf{B}_{25} & = & x_{4} \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + y_{4}b \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C IV} \\ \mathbf{B}_{26} & = & \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{4}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C IV} \\ \mathbf{B}_{27} & = & -x_{4} \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + y_{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{4}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C IV} \\ \mathbf{B}_{28} & = & \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{4}\right) \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{4}\right)b \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C IV} \\ \mathbf{B}_{29} & = & -x_{4} \, \mathbf{a}_{1}-y_{4} \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}}-y_{4}b \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C IV} \\ \mathbf{B}_{30} & = & \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{4}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{4}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C IV} \\ \mathbf{B}_{31} & = & x_{4} \, \mathbf{a}_{1}-y_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}}-y_{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C IV} \\ \mathbf{B}_{32} & = & \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{4}\right) \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{4}\right)b \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C IV} \\ \mathbf{B}_{33} & = & 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(8d\right) & \mbox{C V} \\ \mathbf{B}_{34} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{5}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C V} \\ \mathbf{B}_{35} & = & -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(8d\right) & \mbox{C V} \\ \mathbf{B}_{36} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{5}\right)b \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C V} \\ \mathbf{B}_{37} & = & -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(8d\right) & \mbox{C V} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{5}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{5}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C V} \\ \mathbf{B}_{39} & = & 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(8d\right) & \mbox{C V} \\ \mathbf{B}_{40} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{5}\right)b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C V} \\ \mathbf{B}_{41} & = & 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(8d\right) & \mbox{C VI} \\ \mathbf{B}_{42} & = & \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{6}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C VI} \\ \mathbf{B}_{43} & = & -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(8d\right) & \mbox{C VI} \\ \mathbf{B}_{44} & = & \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)b \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C VI} \\ \mathbf{B}_{45} & = & -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(8d\right) & \mbox{C VI} \\ \mathbf{B}_{46} & = & \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{6}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{6}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C VI} \\ \mathbf{B}_{47} & = & 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(8d\right) & \mbox{C VI} \\ \mathbf{B}_{48} & = & \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)b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C VI} \\ \mathbf{B}_{49} & = & 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(8d\right) & \mbox{C VII} \\ \mathbf{B}_{50} & = & \left(\frac{1}{2} - x_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{7}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C VII} \\ \mathbf{B}_{51} & = & -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(8d\right) & \mbox{C VII} \\ \mathbf{B}_{52} & = & \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)b \, \mathbf{\hat{y}}-z_{7}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C VII} \\ \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}b \, \mathbf{\hat{y}}-z_{7}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C VII} \\ \mathbf{B}_{54} & = & \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{7}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{7}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C VII} \\ \mathbf{B}_{55} & = & 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(8d\right) & \mbox{C VII} \\ \mathbf{B}_{56} & = & \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)b \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{C VII} \\ \mathbf{B}_{57} & = & 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(8d\right) & \mbox{H I} \\ \mathbf{B}_{58} & = & \left(\frac{1}{2} - x_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{8}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{59} & = & -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(8d\right) & \mbox{H I} \\ \mathbf{B}_{60} & = & \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)b \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \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}b \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{62} & = & \left(\frac{1}{2} +x_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{8}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{8}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{63} & = & 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(8d\right) & \mbox{H I} \\ \mathbf{B}_{64} & = & \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)b \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{65} & = & 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(8d\right) & \mbox{H II} \\ \mathbf{B}_{66} & = & \left(\frac{1}{2} - x_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{9}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{67} & = & -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(8d\right) & \mbox{H II} \\ \mathbf{B}_{68} & = & \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)b \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{69} & = & -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(8d\right) & \mbox{H II} \\ \mathbf{B}_{70} & = & \left(\frac{1}{2} +x_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{9}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{9}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{71} & = & 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(8d\right) & \mbox{H II} \\ \mathbf{B}_{72} & = & \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)b \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{73} & = & 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(8d\right) & \mbox{H III} \\ \mathbf{B}_{74} & = & \left(\frac{1}{2} - x_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{10}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{75} & = & -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(8d\right) & \mbox{H III} \\ \mathbf{B}_{76} & = & \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)b \, \mathbf{\hat{y}}-z_{10}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{77} & = & -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(8d\right) & \mbox{H III} \\ \mathbf{B}_{78} & = & \left(\frac{1}{2} +x_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{10}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{10}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{79} & = & 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(8d\right) & \mbox{H III} \\ \mathbf{B}_{80} & = & \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)b \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{81} & = & 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(8d\right) & \mbox{H IV} \\ \mathbf{B}_{82} & = & \left(\frac{1}{2} - x_{11}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{11}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{11}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{11}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{83} & = & -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(8d\right) & \mbox{H IV} \\ \mathbf{B}_{84} & = & \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)b \, \mathbf{\hat{y}}-z_{11}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{85} & = & -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(8d\right) & \mbox{H IV} \\ \mathbf{B}_{86} & = & \left(\frac{1}{2} +x_{11}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{11}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{11}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{11}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{11}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{87} & = & 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(8d\right) & \mbox{H IV} \\ \mathbf{B}_{88} & = & \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)b \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{89} & = & 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(8d\right) & \mbox{H V} \\ \mathbf{B}_{90} & = & \left(\frac{1}{2} - x_{12}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{12}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{12}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{12}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{91} & = & -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(8d\right) & \mbox{H V} \\ \mathbf{B}_{92} & = & \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)b \, \mathbf{\hat{y}}-z_{12}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{93} & = & -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(8d\right) & \mbox{H V} \\ \mathbf{B}_{94} & = & \left(\frac{1}{2} +x_{12}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{12}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{12}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{12}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{12}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{95} & = & 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(8d\right) & \mbox{H V} \\ \mathbf{B}_{96} & = & \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)b \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{97} & = & 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(8d\right) & \mbox{H VI} \\ \mathbf{B}_{98} & = & \left(\frac{1}{2} - x_{13}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{13}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{13}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{13}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{99} & = & -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(8d\right) & \mbox{H VI} \\ \mathbf{B}_{100} & = & \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)b \, \mathbf{\hat{y}}-z_{13}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{101} & = & -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(8d\right) & \mbox{H VI} \\ \mathbf{B}_{102} & = & \left(\frac{1}{2} +x_{13}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{13}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{13}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{13}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{13}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{103} & = & 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(8d\right) & \mbox{H VI} \\ \mathbf{B}_{104} & = & \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)b \, \mathbf{\hat{y}} + z_{13}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{105} & = & 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(8d\right) & \mbox{H VII} \\ \mathbf{B}_{106} & = & \left(\frac{1}{2} - x_{14}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{14}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{14}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{14}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{107} & = & -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(8d\right) & \mbox{H VII} \\ \mathbf{B}_{108} & = & \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)b \, \mathbf{\hat{y}}-z_{14}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{109} & = & -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(8d\right) & \mbox{H VII} \\ \mathbf{B}_{110} & = & \left(\frac{1}{2} +x_{14}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{14}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{14}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{14}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{14}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{111} & = & 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(8d\right) & \mbox{H VII} \\ \mathbf{B}_{112} & = & \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)b \, \mathbf{\hat{y}} + z_{14}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{113} & = & 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(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{114} & = & \left(\frac{1}{2} - x_{15}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{15}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{15}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{15}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{115} & = & -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(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{116} & = & \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)b \, \mathbf{\hat{y}}-z_{15}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{117} & = & -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(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{118} & = & \left(\frac{1}{2} +x_{15}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{15}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{15}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{15}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} - z_{15}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{119} & = & 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(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{120} & = & \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)b \, \mathbf{\hat{y}} + z_{15}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \end{array} \]