# Encyclopedia of Crystallographic Prototypes

M. J. Mehl, D. Hicks, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 1, Comp. Mat. Sci. 136, S1-S828 (2017). (doi=10.1016/j.commatsci.2017.01.017)
D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 2, Comp. Mat. Sci. 161, S1-S1011 (2019). (doi=10.1016/j.commatsci.2018.10.043)

## $\beta$–Alum [Al(NH3CH3)2(SO4)2·12H2O, $H4_{14}$] Structure : AB2C36D2E20F2_cP252_205_a_c_6d_c_c3d_c

 Prototype : AlC2H36N2O20S2 AFLOW prototype label : AB2C36D2E20F2_cP252_205_a_c_6d_c_c3d_c Strukturbericht designation : $H4_{14}$ Pearson symbol : cP252 Space group number : 205 Space group symbol : $Pa\bar{3}$ AFLOW prototype command : aflow --proto=AB2C36D2E20F2_cP252_205_a_c_6d_c_c3d_c --params=$a$,$x_{2}$,$x_{3}$,$x_{4}$,$x_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$,$x_{10}$,$y_{10}$,$z_{10}$,$x_{11}$,$y_{11}$,$z_{11}$,$x_{12}$,$y_{12}$,$z_{12}$,$x_{13}$,$y_{13}$,$z_{13}$,$x_{14}$,$y_{14}$,$z_{14}$

### Other compounds with this structure

• AlCs(SO4)2·12H2O

• The alums have the general formula $AB$($X$O4)2·12H2O, where $A$ is a monovalent ion, $B$ is a trivalent ion, and $X$ is a chalcogen. In most cases atom $B$ is aluminum and atom $X$ is sulfur, leading to the name alum.
• All alums have their room–temperature form in space group $Pa\overline{3}$ #205, but the bonding between the $A$ and $B$ ions and the $X$O4 complex can be quite different.
• (Lipson, 1935ab) described three general forms of alum based on the sizes of the monovalent ions. Each of these forms was given a Strukturbericht designation by (Gottfried, 1937):
• $\alpha$–alum, with intermediate sized ions, prototype KAl(SO4)2·12H2O, $H4_{13}$,
• $\beta$–alum, with large ions, prototype (NH3CH3)Al(SO4)2·12H2O, $H4_{14}$ (this structure), and
• $\gamma$–alum, with small ions, prototype NaAl(SO4)2·12H2O, $H4_{15}$.
• This classification scheme is not complete, e.g., (Ledsham, 1968) points out that NaCr(SO4)2·12H2O does not fit into any of these categories, and that the actual structure depends on the combination of monovalent and trivalent ions.
• As noted above, the $Pa\overline{3}$ structures of alum are the room temperature form. As the temperature decreases the alum structure may transform. For example, in the temperature range 150–170 K the $\beta$–alum (NH3CH3)Al(SO4)2·12H2O transforms into an orthorhombic structure with fully ordered NH3CH3 ions.
• This structure was originally determined by (Lipsom, 1935c), who could only determine that the NH3CH3 ion occupied the ($4b$) Wyckoff position. (Abdeen, 1981) showed that the ion was statistically distributed at two possible sites. The C–N bond distance is 1.4 Å, slightly smaller than the 1.51 Å distance observed in the low temperature structure. At any site, one of the two nitrogen positions is occupied, along with the carbon position 1.4 Å away. Six hydrogen positions from the (H–I) and (H–II) sites are then occupied.

### Simple Cubic primitive vectors:

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

Basis vectors:

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

### References

• A. M. Abdeen, G. Will, W. Schäfer, A. Kirfel, M. O. Bargouth, and K. Recker, X–Ray and neutron diffraction study of alums, Zeitschrift für Kristallographie – Crystalline Materials 157, 147–166 (1981), doi:10.1524/zkri.1981.157.14.147.
• C. Gottfried and F. Schossberger, eds., Strukturbericht Band III 1933–1935 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1937).
• A. H. C. Ledsham and H. Steeple, The crystal structure of sodium chromium alum and caesium chromium alum, Acta Crystallogr. Sect. B Struct. Sci. 24, 1287–1289 (1968), doi:10.1107/S0567740868004188.
• R. O. W. Fletcher and H. Steeple, The crystal structure of the low–temperature phase of methylammonium alum, Acta Cryst. 17, 290–294 (1964), doi:10.1107/S0365110X64000706.

### Prototype Generator

aflow --proto=AB2C36D2E20F2_cP252_205_a_c_6d_c_c3d_c --params=