Te[OH]6 Structure: A12B6C_cF608_210_4h_2h_e
| Prototype | : | Te[OH]6 |
| AFLOW prototype label | : | A12B6C_cF608_210_4h_2h_e |
| Strukturbericht designation | : | None |
| Pearson symbol | : | cF608 |
| Space group number | : | 210 |
| Space group symbol | : | $F4_{1}32$ |
| AFLOW prototype command | : | aflow --proto=A12B6C_cF608_210_4h_2h_e --params=$a$,$x_{1}$,$x_{2}$,$y_{2}$,$z_{2}$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$ |
- All H sites are half occupation. Polytypes appear in space groups #14, #228, and #225.
Face-centered Cubic primitive vectors:
\[ \begin{array}{ccc} \mathbf{a}_1 & = & \frac12 \, a \, \mathbf{\hat{y}} + \frac12 \, a \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{y}} \\ \end{array} \]
Basis vectors:
\[ \begin{array}{ccccccc} & & \text{Lattice Coordinates} & & \text{Cartesian Coordinates} &\text{Wyckoff Position} & \text{Atom Type} \\ \mathbf{B}_{1} & = & x_{1} \, \mathbf{a}_{1} + x_{1} \, \mathbf{a}_{2} + x_{1} \, \mathbf{a}_{3} & = & x_{1}a \, \mathbf{\hat{x}} + x_{1}a \, \mathbf{\hat{y}} + x_{1}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{Te} \\ \mathbf{B}_{2} & = & x_{1} \, \mathbf{a}_{1} + x_{1} \, \mathbf{a}_{2}-3x_{1} \, \mathbf{a}_{3} & = & -x_{1}a \, \mathbf{\hat{x}}-x_{1}a \, \mathbf{\hat{y}} + x_{1}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{Te} \\ \mathbf{B}_{3} & = & x_{1} \, \mathbf{a}_{1}-3x_{1} \, \mathbf{a}_{2} + x_{1} \, \mathbf{a}_{3} & = & -x_{1}a \, \mathbf{\hat{x}} + x_{1}a \, \mathbf{\hat{y}}-x_{1}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{Te} \\ \mathbf{B}_{4} & = & -3x_{1} \, \mathbf{a}_{1} + x_{1} \, \mathbf{a}_{2} + x_{1} \, \mathbf{a}_{3} & = & x_{1}a \, \mathbf{\hat{x}}-x_{1}a \, \mathbf{\hat{y}}-x_{1}a \, \mathbf{\hat{z}} & \left(32e\right) & \text{Te} \\ \mathbf{B}_{5} & = & \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +3x_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{1}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{Te} \\ \mathbf{B}_{6} & = & \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{1}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{Te} \\ \mathbf{B}_{7} & = & \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +3x_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{1}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{Te} \\ \mathbf{B}_{8} & = & \left(\frac{1}{4} +3x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{1}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{1}\right)a \, \mathbf{\hat{z}} & \left(32e\right) & \text{Te} \\ \mathbf{B}_{9} & = & \left(-x_{2}+y_{2}+z_{2}\right) \, \mathbf{a}_{1} + \left(x_{2}-y_{2}+z_{2}\right) \, \mathbf{a}_{2} + \left(x_{2}+y_{2}-z_{2}\right) \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}} + y_{2}a \, \mathbf{\hat{y}} + z_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{10} & = & \left(x_{2}-y_{2}+z_{2}\right) \, \mathbf{a}_{1} + \left(-x_{2}+y_{2}+z_{2}\right) \, \mathbf{a}_{2} + \left(-x_{2}-y_{2}-z_{2}\right) \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}}-y_{2}a \, \mathbf{\hat{y}} + z_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{11} & = & \left(x_{2}+y_{2}-z_{2}\right) \, \mathbf{a}_{1} + \left(-x_{2}-y_{2}-z_{2}\right) \, \mathbf{a}_{2} + \left(-x_{2}+y_{2}+z_{2}\right) \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}} + y_{2}a \, \mathbf{\hat{y}}-z_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{12} & = & \left(-x_{2}-y_{2}-z_{2}\right) \, \mathbf{a}_{1} + \left(x_{2}+y_{2}-z_{2}\right) \, \mathbf{a}_{2} + \left(x_{2}-y_{2}+z_{2}\right) \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}}-y_{2}a \, \mathbf{\hat{y}}-z_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{13} & = & \left(x_{2}+y_{2}-z_{2}\right) \, \mathbf{a}_{1} + \left(-x_{2}+y_{2}+z_{2}\right) \, \mathbf{a}_{2} + \left(x_{2}-y_{2}+z_{2}\right) \, \mathbf{a}_{3} & = & z_{2}a \, \mathbf{\hat{x}} + x_{2}a \, \mathbf{\hat{y}} + y_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{14} & = & \left(-x_{2}-y_{2}-z_{2}\right) \, \mathbf{a}_{1} + \left(x_{2}-y_{2}+z_{2}\right) \, \mathbf{a}_{2} + \left(-x_{2}+y_{2}+z_{2}\right) \, \mathbf{a}_{3} & = & z_{2}a \, \mathbf{\hat{x}}-x_{2}a \, \mathbf{\hat{y}}-y_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{15} & = & \left(-x_{2}+y_{2}+z_{2}\right) \, \mathbf{a}_{1} + \left(x_{2}+y_{2}-z_{2}\right) \, \mathbf{a}_{2} + \left(-x_{2}-y_{2}-z_{2}\right) \, \mathbf{a}_{3} & = & -z_{2}a \, \mathbf{\hat{x}}-x_{2}a \, \mathbf{\hat{y}} + y_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{16} & = & \left(x_{2}-y_{2}+z_{2}\right) \, \mathbf{a}_{1} + \left(-x_{2}-y_{2}-z_{2}\right) \, \mathbf{a}_{2} + \left(x_{2}+y_{2}-z_{2}\right) \, \mathbf{a}_{3} & = & -z_{2}a \, \mathbf{\hat{x}} + x_{2}a \, \mathbf{\hat{y}}-y_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{17} & = & \left(x_{2}-y_{2}+z_{2}\right) \, \mathbf{a}_{1} + \left(x_{2}+y_{2}-z_{2}\right) \, \mathbf{a}_{2} + \left(-x_{2}+y_{2}+z_{2}\right) \, \mathbf{a}_{3} & = & y_{2}a \, \mathbf{\hat{x}} + z_{2}a \, \mathbf{\hat{y}} + x_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{18} & = & \left(-x_{2}+y_{2}+z_{2}\right) \, \mathbf{a}_{1} + \left(-x_{2}-y_{2}-z_{2}\right) \, \mathbf{a}_{2} + \left(x_{2}-y_{2}+z_{2}\right) \, \mathbf{a}_{3} & = & -y_{2}a \, \mathbf{\hat{x}} + z_{2}a \, \mathbf{\hat{y}}-x_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{19} & = & \left(-x_{2}-y_{2}-z_{2}\right) \, \mathbf{a}_{1} + \left(-x_{2}+y_{2}+z_{2}\right) \, \mathbf{a}_{2} + \left(x_{2}+y_{2}-z_{2}\right) \, \mathbf{a}_{3} & = & y_{2}a \, \mathbf{\hat{x}}-z_{2}a \, \mathbf{\hat{y}}-x_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{20} & = & \left(x_{2}+y_{2}-z_{2}\right) \, \mathbf{a}_{1} + \left(x_{2}-y_{2}+z_{2}\right) \, \mathbf{a}_{2} + \left(-x_{2}-y_{2}-z_{2}\right) \, \mathbf{a}_{3} & = & -y_{2}a \, \mathbf{\hat{x}}-z_{2}a \, \mathbf{\hat{y}} + x_{2}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{21} & = & \left(\frac{1}{4} +x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{22} & = & \left(\frac{1}{4} - x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{23} & = & \left(\frac{1}{4} - x_{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{24} & = & \left(\frac{1}{4} +x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{25} & = & \left(\frac{1}{4} - x_{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{26} & = & \left(\frac{1}{4} +x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{27} & = & \left(\frac{1}{4} +x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{28} & = & \left(\frac{1}{4} - x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{29} & = & \left(\frac{1}{4} - x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{30} & = & \left(\frac{1}{4} +x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{31} & = & \left(\frac{1}{4} +x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{32} & = & \left(\frac{1}{4} - x_{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{2}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H I} \\ \mathbf{B}_{33} & = & \left(-x_{3}+y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}-y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}+y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + y_{3}a \, \mathbf{\hat{y}} + z_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{34} & = & \left(x_{3}-y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}+y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}-y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}}-y_{3}a \, \mathbf{\hat{y}} + z_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{35} & = & \left(x_{3}+y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}-y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}+y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + y_{3}a \, \mathbf{\hat{y}}-z_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{36} & = & \left(-x_{3}-y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}+y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}-y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}}-y_{3}a \, \mathbf{\hat{y}}-z_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{37} & = & \left(x_{3}+y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}+y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}-y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & z_{3}a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}} + y_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{38} & = & \left(-x_{3}-y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}-y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}+y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & z_{3}a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}}-y_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{39} & = & \left(-x_{3}+y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}+y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}-y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & -z_{3}a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}} + y_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{40} & = & \left(x_{3}-y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}-y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}+y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & -z_{3}a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}}-y_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{41} & = & \left(x_{3}-y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}+y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}+y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & y_{3}a \, \mathbf{\hat{x}} + z_{3}a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{42} & = & \left(-x_{3}+y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}-y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}-y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & -y_{3}a \, \mathbf{\hat{x}} + z_{3}a \, \mathbf{\hat{y}}-x_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{43} & = & \left(-x_{3}-y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}+y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}+y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & y_{3}a \, \mathbf{\hat{x}}-z_{3}a \, \mathbf{\hat{y}}-x_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{44} & = & \left(x_{3}+y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}-y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}-y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & -y_{3}a \, \mathbf{\hat{x}}-z_{3}a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{45} & = & \left(\frac{1}{4} +x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{3} + y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{46} & = & \left(\frac{1}{4} - x_{3} + y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{3} - y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{47} & = & \left(\frac{1}{4} - x_{3} - y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{3} + y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{48} & = & \left(\frac{1}{4} +x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{3} - y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{49} & = & \left(\frac{1}{4} - x_{3} - y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{50} & = & \left(\frac{1}{4} +x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{3} + y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{3} - y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{51} & = & \left(\frac{1}{4} +x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{3} - y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{3} + y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{52} & = & \left(\frac{1}{4} - x_{3} + y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{53} & = & \left(\frac{1}{4} - x_{3} + y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{3} - y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{54} & = & \left(\frac{1}{4} +x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{3} - y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{55} & = & \left(\frac{1}{4} +x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{3} + y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{56} & = & \left(\frac{1}{4} - x_{3} - y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{3} + y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H II} \\ \mathbf{B}_{57} & = & \left(-x_{4}+y_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}-y_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}+y_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + y_{4}a \, \mathbf{\hat{y}} + z_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{58} & = & \left(x_{4}-y_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}+y_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}-y_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}}-y_{4}a \, \mathbf{\hat{y}} + z_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{59} & = & \left(x_{4}+y_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}-y_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}+y_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + y_{4}a \, \mathbf{\hat{y}}-z_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{60} & = & \left(-x_{4}-y_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}+y_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}-y_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}}-y_{4}a \, \mathbf{\hat{y}}-z_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{61} & = & \left(x_{4}+y_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}+y_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}-y_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & z_{4}a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}} + y_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{62} & = & \left(-x_{4}-y_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}-y_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}+y_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & z_{4}a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}}-y_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{63} & = & \left(-x_{4}+y_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}+y_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}-y_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & -z_{4}a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}} + y_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{64} & = & \left(x_{4}-y_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}-y_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}+y_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & -z_{4}a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}}-y_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{65} & = & \left(x_{4}-y_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}+y_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}+y_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & y_{4}a \, \mathbf{\hat{x}} + z_{4}a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{66} & = & \left(-x_{4}+y_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}-y_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}-y_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & -y_{4}a \, \mathbf{\hat{x}} + z_{4}a \, \mathbf{\hat{y}}-x_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{67} & = & \left(-x_{4}-y_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}+y_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}+y_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & y_{4}a \, \mathbf{\hat{x}}-z_{4}a \, \mathbf{\hat{y}}-x_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{68} & = & \left(x_{4}+y_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}-y_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}-y_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & -y_{4}a \, \mathbf{\hat{x}}-z_{4}a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{69} & = & \left(\frac{1}{4} +x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4} + y_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{70} & = & \left(\frac{1}{4} - x_{4} + y_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4} - y_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{71} & = & \left(\frac{1}{4} - x_{4} - y_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4} + y_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{72} & = & \left(\frac{1}{4} +x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4} - y_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{73} & = & \left(\frac{1}{4} - x_{4} - y_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{74} & = & \left(\frac{1}{4} +x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4} + y_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4} - y_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{75} & = & \left(\frac{1}{4} +x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4} - y_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4} + y_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{76} & = & \left(\frac{1}{4} - x_{4} + y_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{77} & = & \left(\frac{1}{4} - x_{4} + y_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4} - y_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{78} & = & \left(\frac{1}{4} +x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4} - y_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{79} & = & \left(\frac{1}{4} +x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4} + y_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{80} & = & \left(\frac{1}{4} - x_{4} - y_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4} + y_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H III} \\ \mathbf{B}_{81} & = & \left(-x_{5}+y_{5}+z_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}-y_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(x_{5}+y_{5}-z_{5}\right) \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + y_{5}a \, \mathbf{\hat{y}} + z_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{82} & = & \left(x_{5}-y_{5}+z_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}+y_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(-x_{5}-y_{5}-z_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}}-y_{5}a \, \mathbf{\hat{y}} + z_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{83} & = & \left(x_{5}+y_{5}-z_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}-y_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(-x_{5}+y_{5}+z_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + y_{5}a \, \mathbf{\hat{y}}-z_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{84} & = & \left(-x_{5}-y_{5}-z_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}+y_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(x_{5}-y_{5}+z_{5}\right) \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}}-y_{5}a \, \mathbf{\hat{y}}-z_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{85} & = & \left(x_{5}+y_{5}-z_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}+y_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(x_{5}-y_{5}+z_{5}\right) \, \mathbf{a}_{3} & = & z_{5}a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} + y_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{86} & = & \left(-x_{5}-y_{5}-z_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}-y_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(-x_{5}+y_{5}+z_{5}\right) \, \mathbf{a}_{3} & = & z_{5}a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}}-y_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{87} & = & \left(-x_{5}+y_{5}+z_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}+y_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(-x_{5}-y_{5}-z_{5}\right) \, \mathbf{a}_{3} & = & -z_{5}a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}} + y_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{88} & = & \left(x_{5}-y_{5}+z_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}-y_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(x_{5}+y_{5}-z_{5}\right) \, \mathbf{a}_{3} & = & -z_{5}a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}}-y_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{89} & = & \left(x_{5}-y_{5}+z_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}+y_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(-x_{5}+y_{5}+z_{5}\right) \, \mathbf{a}_{3} & = & y_{5}a \, \mathbf{\hat{x}} + z_{5}a \, \mathbf{\hat{y}} + x_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{90} & = & \left(-x_{5}+y_{5}+z_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}-y_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(x_{5}-y_{5}+z_{5}\right) \, \mathbf{a}_{3} & = & -y_{5}a \, \mathbf{\hat{x}} + z_{5}a \, \mathbf{\hat{y}}-x_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{91} & = & \left(-x_{5}-y_{5}-z_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}+y_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(x_{5}+y_{5}-z_{5}\right) \, \mathbf{a}_{3} & = & y_{5}a \, \mathbf{\hat{x}}-z_{5}a \, \mathbf{\hat{y}}-x_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{92} & = & \left(x_{5}+y_{5}-z_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}-y_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(-x_{5}-y_{5}-z_{5}\right) \, \mathbf{a}_{3} & = & -y_{5}a \, \mathbf{\hat{x}}-z_{5}a \, \mathbf{\hat{y}} + x_{5}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{93} & = & \left(\frac{1}{4} +x_{5} - y_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{5} + y_{5} - z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{5} + y_{5} + z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{94} & = & \left(\frac{1}{4} - x_{5} + y_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{5} - y_{5} - z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{5} - y_{5} + z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{95} & = & \left(\frac{1}{4} - x_{5} - y_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{5} + y_{5} + z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{5} + y_{5} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{96} & = & \left(\frac{1}{4} +x_{5} + y_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{5} - y_{5} + z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{5} - y_{5} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{97} & = & \left(\frac{1}{4} - x_{5} - y_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{5} - y_{5} - z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{5} + y_{5} + z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{98} & = & \left(\frac{1}{4} +x_{5} + y_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{5} + y_{5} - z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{5} - y_{5} + z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{99} & = & \left(\frac{1}{4} +x_{5} - y_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{5} - y_{5} + z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{5} + y_{5} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{100} & = & \left(\frac{1}{4} - x_{5} + y_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{5} + y_{5} + z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{5} - y_{5} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{101} & = & \left(\frac{1}{4} - x_{5} + y_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{5} - y_{5} + z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{5} + y_{5} + z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{102} & = & \left(\frac{1}{4} +x_{5} - y_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{5} + y_{5} + z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{5} - y_{5} + z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{103} & = & \left(\frac{1}{4} +x_{5} + y_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{5} - y_{5} - z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{5} + y_{5} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{104} & = & \left(\frac{1}{4} - x_{5} - y_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{5} + y_{5} - z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{5} - y_{5} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{5}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{H IV} \\ \mathbf{B}_{105} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{106} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{107} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{108} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{109} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{110} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{111} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{112} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{113} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{114} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{115} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{116} & = & \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(96h\right) & \text{O I} \\ \mathbf{B}_{117} & = & \left(\frac{1}{4} +x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{6} + y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +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} +x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{118} & = & \left(\frac{1}{4} - x_{6} + y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - 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} - x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{119} & = & \left(\frac{1}{4} - x_{6} - y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - 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} - x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{120} & = & \left(\frac{1}{4} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{6} - y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +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} +x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{121} & = & \left(\frac{1}{4} - x_{6} - y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +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} +z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{122} & = & \left(\frac{1}{4} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{6} + y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - 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} +z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{123} & = & \left(\frac{1}{4} +x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{6} - y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - 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} - z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{124} & = & \left(\frac{1}{4} - x_{6} + y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +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} - z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{125} & = & \left(\frac{1}{4} - x_{6} + y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{6} - y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +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} +y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{126} & = & \left(\frac{1}{4} +x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - 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} - y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{127} & = & \left(\frac{1}{4} +x_{6} + y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{6} - y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - 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} +y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{128} & = & \left(\frac{1}{4} - x_{6} - y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{6} + y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +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} - y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{6}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O I} \\ \mathbf{B}_{129} & = & \left(-x_{7}+y_{7}+z_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}-y_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}+y_{7}-z_{7}\right) \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}} + y_{7}a \, \mathbf{\hat{y}} + z_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{130} & = & \left(x_{7}-y_{7}+z_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}+y_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}-y_{7}-z_{7}\right) \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}}-y_{7}a \, \mathbf{\hat{y}} + z_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{131} & = & \left(x_{7}+y_{7}-z_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}-y_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}+y_{7}+z_{7}\right) \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}} + y_{7}a \, \mathbf{\hat{y}}-z_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{132} & = & \left(-x_{7}-y_{7}-z_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}+y_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}-y_{7}+z_{7}\right) \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}}-y_{7}a \, \mathbf{\hat{y}}-z_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{133} & = & \left(x_{7}+y_{7}-z_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}+y_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}-y_{7}+z_{7}\right) \, \mathbf{a}_{3} & = & z_{7}a \, \mathbf{\hat{x}} + x_{7}a \, \mathbf{\hat{y}} + y_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{134} & = & \left(-x_{7}-y_{7}-z_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}-y_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}+y_{7}+z_{7}\right) \, \mathbf{a}_{3} & = & z_{7}a \, \mathbf{\hat{x}}-x_{7}a \, \mathbf{\hat{y}}-y_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{135} & = & \left(-x_{7}+y_{7}+z_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}+y_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}-y_{7}-z_{7}\right) \, \mathbf{a}_{3} & = & -z_{7}a \, \mathbf{\hat{x}}-x_{7}a \, \mathbf{\hat{y}} + y_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{136} & = & \left(x_{7}-y_{7}+z_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}-y_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}+y_{7}-z_{7}\right) \, \mathbf{a}_{3} & = & -z_{7}a \, \mathbf{\hat{x}} + x_{7}a \, \mathbf{\hat{y}}-y_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{137} & = & \left(x_{7}-y_{7}+z_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}+y_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}+y_{7}+z_{7}\right) \, \mathbf{a}_{3} & = & y_{7}a \, \mathbf{\hat{x}} + z_{7}a \, \mathbf{\hat{y}} + x_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{138} & = & \left(-x_{7}+y_{7}+z_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}-y_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}-y_{7}+z_{7}\right) \, \mathbf{a}_{3} & = & -y_{7}a \, \mathbf{\hat{x}} + z_{7}a \, \mathbf{\hat{y}}-x_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{139} & = & \left(-x_{7}-y_{7}-z_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}+y_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}+y_{7}-z_{7}\right) \, \mathbf{a}_{3} & = & y_{7}a \, \mathbf{\hat{x}}-z_{7}a \, \mathbf{\hat{y}}-x_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{140} & = & \left(x_{7}+y_{7}-z_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}-y_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}-y_{7}-z_{7}\right) \, \mathbf{a}_{3} & = & -y_{7}a \, \mathbf{\hat{x}}-z_{7}a \, \mathbf{\hat{y}} + x_{7}a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{141} & = & \left(\frac{1}{4} +x_{7} - y_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{7} + y_{7} - z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{7} + y_{7} + z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{142} & = & \left(\frac{1}{4} - x_{7} + y_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{7} - y_{7} - z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{7} - y_{7} + z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - z_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{143} & = & \left(\frac{1}{4} - x_{7} - y_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{7} + y_{7} + z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{7} + y_{7} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{144} & = & \left(\frac{1}{4} +x_{7} + y_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{7} - y_{7} + z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{7} - y_{7} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{145} & = & \left(\frac{1}{4} - x_{7} - y_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{7} - y_{7} - z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{7} + y_{7} + z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{146} & = & \left(\frac{1}{4} +x_{7} + y_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{7} + y_{7} - z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{7} - y_{7} + z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{147} & = & \left(\frac{1}{4} +x_{7} - y_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{7} - y_{7} + z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{7} + y_{7} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{148} & = & \left(\frac{1}{4} - x_{7} + y_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{7} + y_{7} + z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{7} - y_{7} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{149} & = & \left(\frac{1}{4} - x_{7} + y_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{7} - y_{7} + z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{7} + y_{7} + z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{150} & = & \left(\frac{1}{4} +x_{7} - y_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{7} + y_{7} + z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{7} - y_{7} + z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{151} & = & \left(\frac{1}{4} +x_{7} + y_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{7} - y_{7} - z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{7} + y_{7} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \mathbf{B}_{152} & = & \left(\frac{1}{4} - x_{7} - y_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{7} + y_{7} - z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{7} - y_{7} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{7}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \text{O II} \\ \end{array} \]References
- D. F. Mullica, J. D. Korp, W. O. Milligan, G. W. Beall, and I. Bernal, Neutron structural refinement of cubic orthotelluric acid, Acta Crystallogr. Sect. B Struct. Sci. 36, 2565–2570 (1980), doi:10.1107/S0567740880009454.
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- P. Villars and K. Cenzual, Pearson's Crystal Data – Crystal Structure Database for Inorganic Compounds, ASM International (2013).