Low–Temperature (NH3CH3)Al(SO4)2·12H2O Structure : ABC30DE20F2_oP220_29_a_a_30a_a_20a_2a

Picture of Structure; Click for Big Picture
Prototype : AlCH30NO20S2
AFLOW prototype label : ABC30DE20F2_oP220_29_a_a_30a_a_20a_2a
Strukturbericht designation : None
Pearson symbol : oP220
Space group number : 29
Space group symbol : $Pca2_{1}$
AFLOW prototype command : aflow --proto=ABC30DE20F2_oP220_29_a_a_30a_a_20a_2a
--params=
$a$,$b/a$,$c/a$,$x_{1}$,$y_{1}$,$z_{1}$,$x_{2}$,$y_{2}$,$z_{2}$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$,$x_{10}$,$y_{10}$,$z_{10}$,$x_{11}$,$y_{11}$,$z_{11}$,$x_{12}$,$y_{12}$,$z_{12}$,$x_{13}$,$y_{13}$,$z_{13}$,$x_{14}$,$y_{14}$,$z_{14}$,$x_{15}$,$y_{15}$,$z_{15}$,$x_{16}$,$y_{16}$,$z_{16}$,$x_{17}$,$y_{17}$,$z_{17}$,$x_{18}$,$y_{18}$,$z_{18}$,$x_{19}$,$y_{19}$,$z_{19}$,$x_{20}$,$y_{20}$,$z_{20}$,$x_{21}$,$y_{21}$,$z_{21}$,$x_{22}$,$y_{22}$,$z_{22}$,$x_{23}$,$y_{23}$,$z_{23}$,$x_{24}$,$y_{24}$,$z_{24}$,$x_{25}$,$y_{25}$,$z_{25}$,$x_{26}$,$y_{26}$,$z_{26}$,$x_{27}$,$y_{27}$,$z_{27}$,$x_{28}$,$y_{28}$,$z_{28}$,$x_{29}$,$y_{29}$,$z_{29}$,$x_{30}$,$y_{30}$,$z_{30}$,$x_{31}$,$y_{31}$,$z_{31}$,$x_{32}$,$y_{32}$,$z_{32}$,$x_{33}$,$y_{33}$,$z_{33}$,$x_{34}$,$y_{34}$,$z_{34}$,$x_{35}$,$y_{35}$,$z_{35}$,$x_{36}$,$y_{36}$,$z_{36}$,$x_{37}$,$y_{37}$,$z_{37}$,$x_{38}$,$y_{38}$,$z_{38}$,$x_{39}$,$y_{39}$,$z_{39}$,$x_{40}$,$y_{40}$,$z_{40}$,$x_{41}$,$y_{41}$,$z_{41}$,$x_{42}$,$y_{42}$,$z_{42}$,$x_{43}$,$y_{43}$,$z_{43}$,$x_{44}$,$y_{44}$,$z_{44}$,$x_{45}$,$y_{45}$,$z_{45}$,$x_{46}$,$y_{46}$,$z_{46}$,$x_{47}$,$y_{47}$,$z_{47}$,$x_{48}$,$y_{48}$,$z_{48}$,$x_{49}$,$y_{49}$,$z_{49}$,$x_{50}$,$y_{50}$,$z_{50}$,$x_{51}$,$y_{51}$,$z_{51}$,$x_{52}$,$y_{52}$,$z_{52}$,$x_{53}$,$y_{53}$,$z_{53}$,$x_{54}$,$y_{54}$,$z_{54}$,$x_{55}$,$y_{55}$,$z_{55}$


  • 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 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 this orthorhombic structure with fully ordered NH3CH3 ions.
  • The data presented here was taken at 113 K.

Simple Orthorhombic primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & a \, \mathbf{\hat{x}} \\ \mathbf{a}_2 & = & b \, \mathbf{\hat{y}} \\ \mathbf{a}_3 & = & c \, \mathbf{\hat{z}} \\ \end{array} \]

Basis vectors:

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

References

  • 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.
  • 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.

Geometry files


Prototype Generator

aflow --proto=ABC30DE20F2_oP220_29_a_a_30a_a_20a_2a --params=

Species:

Running:

Output: