AFLOW Prototype: A4B2C6D16E_cF232_203_e_d_f_eg_a
Prototype | : | C4Mg2Na6O16S |
AFLOW prototype label | : | A4B2C6D16E_cF232_203_e_d_f_eg_a |
Strukturbericht designation | : | None |
Pearson symbol | : | cF232 |
Space group number | : | 203 |
Space group symbol | : | $Fd\bar{3}$ |
AFLOW prototype command | : | aflow --proto=A4B2C6D16E_cF232_203_e_d_f_eg_a --params=$a$,$x_{3}$,$x_{4}$,$x_{5}$,$x_{6}$,$y_{6}$,$z_{6}$ |
Basis vectors:
\[ \begin{array}{ccccccc} & & \text{Lattice Coordinates} & & \text{Cartesian Coordinates} &\text{Wyckoff Position} & \text{Atom Type} \\ \mathbf{B}_{1} & = & \frac{1}{8} \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + \frac{1}{8} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(8a\right) & \text{S} \\ \mathbf{B}_{2} & = & \frac{7}{8} \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \frac{7}{8}a \, \mathbf{\hat{x}} + \frac{7}{8}a \, \mathbf{\hat{y}} + \frac{7}{8}a \, \mathbf{\hat{z}} & \left(8a\right) & \text{S} \\ \mathbf{B}_{3} & = & \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}a \, \mathbf{\hat{z}} & \left(16d\right) & \text{Mg} \\ \mathbf{B}_{4} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(16d\right) & \text{Mg} \\ \mathbf{B}_{5} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16d\right) & \text{Mg} \\ \mathbf{B}_{6} & = & \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16d\right) & \text{Mg} \\ \mathbf{B}_{7} & = & x_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{C} \\ \mathbf{B}_{8} & = & x_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} - 3x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{C} \\ \mathbf{B}_{9} & = & x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 3x_{3}\right) \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{C} \\ \mathbf{B}_{10} & = & \left(\frac{1}{2} - 3x_{3}\right) \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{C} \\ \mathbf{B}_{11} & = & -x_{3} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}}-x_{3}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{C} \\ \mathbf{B}_{12} & = & -x_{3} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +3x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}}-x_{3}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{C} \\ \mathbf{B}_{13} & = & -x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} +3x_{3}\right) \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{C} \\ \mathbf{B}_{14} & = & \left(\frac{1}{2} +3x_{3}\right) \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{C} \\ \mathbf{B}_{15} & = & x_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{O I} \\ \mathbf{B}_{16} & = & x_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} - 3x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{O I} \\ \mathbf{B}_{17} & = & x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 3x_{4}\right) \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{O I} \\ \mathbf{B}_{18} & = & \left(\frac{1}{2} - 3x_{4}\right) \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{O I} \\ \mathbf{B}_{19} & = & -x_{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}}-x_{4}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{O I} \\ \mathbf{B}_{20} & = & -x_{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +3x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}}-x_{4}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{O I} \\ \mathbf{B}_{21} & = & -x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} +3x_{4}\right) \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{O I} \\ \mathbf{B}_{22} & = & \left(\frac{1}{2} +3x_{4}\right) \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{O I} \\ \mathbf{B}_{23} & = & \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + x_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{24} & = & x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{5}\right)a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{25} & = & x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{2} + x_{5} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{26} & = & \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{27} & = & x_{5} \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + x_{5}a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{28} & = & \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{2} + x_{5} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{5}\right)a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{29} & = & \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2}-x_{5} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{30} & = & -x_{5} \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +x_{5}\right)a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{31} & = & -x_{5} \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{2}-x_{5} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{32} & = & \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{5}\right)a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{33} & = & -x_{5} \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}}-x_{5}a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{34} & = & \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{2}-x_{5} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +x_{5}\right)a \, \mathbf{\hat{z}} & \left(48f\right) & \text{Na} \\ \mathbf{B}_{35} & = & \left(-x_{6}+y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + y_{6}a \, \mathbf{\hat{y}} + z_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{36} & = & \left(x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}+y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{6}\right)a \, \mathbf{\hat{y}} + z_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{37} & = & \left(x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}+y_{6}+z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{6}\right)a \, \mathbf{\hat{x}} + y_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} - x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{39} & = & \left(x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}+y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{3} & = & z_{6}a \, \mathbf{\hat{x}} + x_{6}a \, \mathbf{\hat{y}} + y_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{40} & = & \left(\frac{1}{2} - x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}+y_{6}+z_{6}\right) \, \mathbf{a}_{3} & = & z_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{41} & = & \left(-x_{6}+y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{6}\right)a \, \mathbf{\hat{y}} + y_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{42} & = & \left(x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{6}\right)a \, \mathbf{\hat{x}} + x_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{43} & = & \left(x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}+y_{6}+z_{6}\right) \, \mathbf{a}_{3} & = & y_{6}a \, \mathbf{\hat{x}} + z_{6}a \, \mathbf{\hat{y}} + x_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{44} & = & \left(-x_{6}+y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{6}\right)a \, \mathbf{\hat{x}} + z_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{45} & = & \left(\frac{1}{2} - x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}+y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{3} & = & y_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{46} & = & \left(x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{6}\right)a \, \mathbf{\hat{y}} + x_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{47} & = & \left(x_{6}-y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}}-y_{6}a \, \mathbf{\hat{y}}-z_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{48} & = & \left(-x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}-y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{y}}-z_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{49} & = & \left(-x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}-y_{6}-z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{x}}-y_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{50} & = & \left(\frac{1}{2} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{51} & = & \left(-x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}-y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{3} & = & -z_{6}a \, \mathbf{\hat{x}}-x_{6}a \, \mathbf{\hat{y}}-y_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{52} & = & \left(\frac{1}{2} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}-y_{6}-z_{6}\right) \, \mathbf{a}_{3} & = & -z_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{53} & = & \left(x_{6}-y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{y}}-y_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{54} & = & \left(-x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{6}\right)a \, \mathbf{\hat{x}}-x_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{55} & = & \left(-x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}-y_{6}-z_{6}\right) \, \mathbf{a}_{3} & = & -y_{6}a \, \mathbf{\hat{x}}-z_{6}a \, \mathbf{\hat{y}}-x_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{56} & = & \left(x_{6}-y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{x}}-z_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{57} & = & \left(\frac{1}{2} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}-y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{3} & = & -y_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \mathbf{B}_{58} & = & \left(-x_{6}-y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}+y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{6}\right)a \, \mathbf{\hat{y}}-x_{6}a \, \mathbf{\hat{z}} & \left(96g\right) & \text{O II} \\ \end{array} \]