$\gamma$–Alum [AlNa(SO4)2·12H2O, $H4_{15}$] Structure : AB24CD20E2_cP192_205_a_4d_b_c3d_c

Picture of Structure; Click for Big Picture
Prototype : AlH24NaO20S2
AFLOW prototype label : AB24CD20E2_cP192_205_a_4d_b_c3d_c
Strukturbericht designation : $H4_{15}$
Pearson symbol : cP192
Space group number : 205
Space group symbol : $Pa\bar{3}$
AFLOW prototype command : aflow --proto=AB24CD20E2_cP192_205_a_4d_b_c3d_c
--params=
$a$,$x_{3}$,$x_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$,$x_{10}$,$y_{10}$,$z_{10}$,$x_{11}$,$y_{11}$,$z_{11}$


  • 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, $H4_{14}$, and
    • $\gamma$–alum, with small ions, prototype NaAl(SO4)2·12H2O, $H4_{15}$ (this structure).
  • 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.

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} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(4b\right) & \mbox{Na} \\ \mathbf{B}_{6} & = & \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{2}a \, \mathbf{\hat{y}} & \left(4b\right) & \mbox{Na} \\ \mathbf{B}_{7} & = & \frac{1}{2} \, \mathbf{a}_{1} & = & \frac{1}{2}a \, \mathbf{\hat{x}} & \left(4b\right) & \mbox{Na} \\ \mathbf{B}_{8} & = & \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{z}} & \left(4b\right) & \mbox{Na} \\ \mathbf{B}_{9} & = & 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{O I} \\ \mathbf{B}_{10} & = & \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{O I} \\ \mathbf{B}_{11} & = & -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{O I} \\ \mathbf{B}_{12} & = & \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{O I} \\ \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{O I} \\ \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{O I} \\ \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{O I} \\ \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{O I} \\ \mathbf{B}_{17} & = & 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{S} \\ \mathbf{B}_{18} & = & \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{S} \\ \mathbf{B}_{19} & = & -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{S} \\ \mathbf{B}_{20} & = & \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{S} \\ \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{S} \\ \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{S} \\ \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{S} \\ \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{S} \\ \mathbf{B}_{25} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + y_{5}a \, \mathbf{\hat{y}} + z_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{26} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{x}}-y_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{27} & = & -x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{28} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{5}\right)a \, \mathbf{\hat{y}}-z_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{29} & = & z_{5} \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + y_{5} \, \mathbf{a}_{3} & = & z_{5}a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} + y_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{30} & = & \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{2}-y_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{y}}-y_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{31} & = & \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-z_{5}\right)a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +y_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{32} & = & -z_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{3} & = & -z_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-y_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{33} & = & y_{5} \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + x_{5} \, \mathbf{a}_{3} & = & y_{5}a \, \mathbf{\hat{x}} + z_{5}a \, \mathbf{\hat{y}} + x_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{34} & = & -y_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{3} & = & -y_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{35} & = & \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{2}-x_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-z_{5}\right)a \, \mathbf{\hat{y}}-x_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{36} & = & \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{1}-z_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{5}\right)a \, \mathbf{\hat{x}}-z_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{37} & = & -x_{5} \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}}-y_{5}a \, \mathbf{\hat{y}}-z_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{x}} + y_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{39} & = & x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{40} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{5}\right)a \, \mathbf{\hat{y}} + z_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{41} & = & -z_{5} \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2}-y_{5} \, \mathbf{a}_{3} & = & -z_{5}a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}}-y_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{42} & = & \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{2} + y_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{y}} + y_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{43} & = & \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +z_{5}\right)a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-y_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{44} & = & z_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{3} & = & z_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +y_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{45} & = & -y_{5} \, \mathbf{a}_{1}-z_{5} \, \mathbf{a}_{2}-x_{5} \, \mathbf{a}_{3} & = & -y_{5}a \, \mathbf{\hat{x}}-z_{5}a \, \mathbf{\hat{y}}-x_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{46} & = & y_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{3} & = & y_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{47} & = & \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{2} + x_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +z_{5}\right)a \, \mathbf{\hat{y}} + x_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \mbox{H I} \\ \mathbf{B}_{48} & = & \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{5}\right)a \, \mathbf{\hat{x}} + z_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{5}\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 II} \\ \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 II} \\ \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 II} \\ \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 II} \\ \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 II} \\ \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 II} \\ \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 II} \\ \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 II} \\ \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 II} \\ \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 II} \\ \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 II} \\ \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 II} \\ \mathbf{B}_{61} & = & -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 II} \\ \mathbf{B}_{62} & = & \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 II} \\ \mathbf{B}_{63} & = & 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 II} \\ \mathbf{B}_{64} & = & \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 II} \\ \mathbf{B}_{65} & = & -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 II} \\ \mathbf{B}_{66} & = & \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 II} \\ \mathbf{B}_{67} & = & \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 II} \\ \mathbf{B}_{68} & = & 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 II} \\ \mathbf{B}_{69} & = & -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 II} \\ \mathbf{B}_{70} & = & 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 II} \\ \mathbf{B}_{71} & = & \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 II} \\ \mathbf{B}_{72} & = & \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 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 III} \\ \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 III} \\ \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 III} \\ \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 III} \\ \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 III} \\ \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 III} \\ \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 III} \\ \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 III} \\ \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 III} \\ \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 III} \\ \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 III} \\ \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 III} \\ \mathbf{B}_{85} & = & -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 III} \\ \mathbf{B}_{86} & = & \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 III} \\ \mathbf{B}_{87} & = & 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 III} \\ \mathbf{B}_{88} & = & \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 III} \\ \mathbf{B}_{89} & = & -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 III} \\ \mathbf{B}_{90} & = & \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 III} \\ \mathbf{B}_{91} & = & \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 III} \\ \mathbf{B}_{92} & = & 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 III} \\ \mathbf{B}_{93} & = & -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 III} \\ \mathbf{B}_{94} & = & 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 III} \\ \mathbf{B}_{95} & = & \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 III} \\ \mathbf{B}_{96} & = & \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 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 IV} \\ \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 IV} \\ \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 IV} \\ \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 IV} \\ \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 IV} \\ \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 IV} \\ \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 IV} \\ \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 IV} \\ \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 IV} \\ \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 IV} \\ \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 IV} \\ \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 IV} \\ \mathbf{B}_{109} & = & -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 IV} \\ \mathbf{B}_{110} & = & \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 IV} \\ \mathbf{B}_{111} & = & 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 IV} \\ \mathbf{B}_{112} & = & \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 IV} \\ \mathbf{B}_{113} & = & -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 IV} \\ \mathbf{B}_{114} & = & \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 IV} \\ \mathbf{B}_{115} & = & \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 IV} \\ \mathbf{B}_{116} & = & 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 IV} \\ \mathbf{B}_{117} & = & -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 IV} \\ \mathbf{B}_{118} & = & 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 IV} \\ \mathbf{B}_{119} & = & \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 IV} \\ \mathbf{B}_{120} & = & \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 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{O II} \\ \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{O II} \\ \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{O II} \\ \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{O II} \\ \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{O II} \\ \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{O II} \\ \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{O II} \\ \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{O II} \\ \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{O II} \\ \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{O II} \\ \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{O II} \\ \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{O II} \\ \mathbf{B}_{133} & = & -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{O II} \\ \mathbf{B}_{134} & = & \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{O II} \\ \mathbf{B}_{135} & = & 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{O II} \\ \mathbf{B}_{136} & = & \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{O II} \\ \mathbf{B}_{137} & = & -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{O II} \\ \mathbf{B}_{138} & = & \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{O II} \\ \mathbf{B}_{139} & = & \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{O II} \\ \mathbf{B}_{140} & = & 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{O II} \\ \mathbf{B}_{141} & = & -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{O II} \\ \mathbf{B}_{142} & = & 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{O II} \\ \mathbf{B}_{143} & = & \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{O II} \\ \mathbf{B}_{144} & = & \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{O II} \\ \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{O III} \\ \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{O III} \\ \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{O III} \\ \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{O III} \\ \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{O III} \\ \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{O III} \\ \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{O III} \\ \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{O III} \\ \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{O III} \\ \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{O III} \\ \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{O III} \\ \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{O III} \\ \mathbf{B}_{157} & = & -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{O III} \\ \mathbf{B}_{158} & = & \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{O III} \\ \mathbf{B}_{159} & = & 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{O III} \\ \mathbf{B}_{160} & = & \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{O III} \\ \mathbf{B}_{161} & = & -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{O III} \\ \mathbf{B}_{162} & = & \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{O III} \\ \mathbf{B}_{163} & = & \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{O III} \\ \mathbf{B}_{164} & = & 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{O III} \\ \mathbf{B}_{165} & = & -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{O III} \\ \mathbf{B}_{166} & = & 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{O III} \\ \mathbf{B}_{167} & = & \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{O III} \\ \mathbf{B}_{168} & = & \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{O III} \\ \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{O IV} \\ \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{O IV} \\ \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{O IV} \\ \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{O IV} \\ \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{O IV} \\ \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{O IV} \\ \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{O IV} \\ \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{O IV} \\ \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{O IV} \\ \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{O IV} \\ \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{O IV} \\ \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{O IV} \\ \mathbf{B}_{181} & = & -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{O IV} \\ \mathbf{B}_{182} & = & \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{O IV} \\ \mathbf{B}_{183} & = & 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{O IV} \\ \mathbf{B}_{184} & = & \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{O IV} \\ \mathbf{B}_{185} & = & -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{O IV} \\ \mathbf{B}_{186} & = & \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{O IV} \\ \mathbf{B}_{187} & = & \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{O IV} \\ \mathbf{B}_{188} & = & 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{O IV} \\ \mathbf{B}_{189} & = & -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{O IV} \\ \mathbf{B}_{190} & = & 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{O IV} \\ \mathbf{B}_{191} & = & \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{O IV} \\ \mathbf{B}_{192} & = & \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{O IV} \\ \end{array} \]

References

  • D. T. Cromer, M. I. Kay, and A. C. Larson, Refinement of the alum structures. II. X–ray and neutron diffraction of NaAl(SO4)2·12H2O, $\gamma$–alum, Acta Cryst. 22, 182–187 (1967), doi:10.1107/S0365110X67000313.
  • 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.

Found in

  • R. T. Downs and M. Hall–Wallace, The American Mineralogist Crystal Structure Database, Am. Mineral. 88, 247–250 (2003).

Geometry files


Prototype Generator

aflow --proto=AB24CD20E2_cP192_205_a_4d_b_c3d_c --params=

Species:

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