AlPO$_{4}$-53(A) [24(AlPO$_{4}$)$\cdot$8.5CH$_{3}$NH$_{2}$$\cdot$14H$_{2}$O] Structure: A3BCD14E3_oP176_19_6a_2a_2a_28a_6a-001
| Prototype | Al$_{3}$CNO$_{14}$P$_{3}$ |
| AFLOW prototype label | A3BCD14E3_oP176_19_6a_2a_2a_28a_6a-001 |
| ICSD | 91680 |
| CCDC | 1651205 |
| Pearson symbol | oP176 |
| Space group number | 19 |
| Space group symbol | $P2_12_12_1$ |
| AFLOW prototype command |
aflow --proto=A3BCD14E3_oP176_19_6a_2a_2a_28a_6a-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak y_{1}, \allowbreak z_{1}, \allowbreak x_{2}, \allowbreak y_{2}, \allowbreak z_{2}, \allowbreak x_{3}, \allowbreak y_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak y_{4}, \allowbreak z_{4}, \allowbreak x_{5}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}, \allowbreak x_{9}, \allowbreak y_{9}, \allowbreak z_{9}, \allowbreak x_{10}, \allowbreak y_{10}, \allowbreak z_{10}, \allowbreak x_{11}, \allowbreak y_{11}, \allowbreak z_{11}, \allowbreak x_{12}, \allowbreak y_{12}, \allowbreak z_{12}, \allowbreak x_{13}, \allowbreak y_{13}, \allowbreak z_{13}, \allowbreak x_{14}, \allowbreak y_{14}, \allowbreak z_{14}, \allowbreak x_{15}, \allowbreak y_{15}, \allowbreak z_{15}, \allowbreak x_{16}, \allowbreak y_{16}, \allowbreak z_{16}, \allowbreak x_{17}, \allowbreak y_{17}, \allowbreak z_{17}, \allowbreak x_{18}, \allowbreak y_{18}, \allowbreak z_{18}, \allowbreak x_{19}, \allowbreak y_{19}, \allowbreak z_{19}, \allowbreak x_{20}, \allowbreak y_{20}, \allowbreak z_{20}, \allowbreak x_{21}, \allowbreak y_{21}, \allowbreak z_{21}, \allowbreak x_{22}, \allowbreak y_{22}, \allowbreak z_{22}, \allowbreak x_{23}, \allowbreak y_{23}, \allowbreak z_{23}, \allowbreak x_{24}, \allowbreak y_{24}, \allowbreak z_{24}, \allowbreak x_{25}, \allowbreak y_{25}, \allowbreak z_{25}, \allowbreak x_{26}, \allowbreak y_{26}, \allowbreak z_{26}, \allowbreak x_{27}, \allowbreak y_{27}, \allowbreak z_{27}, \allowbreak x_{28}, \allowbreak y_{28}, \allowbreak z_{28}, \allowbreak x_{29}, \allowbreak y_{29}, \allowbreak z_{29}, \allowbreak x_{30}, \allowbreak y_{30}, \allowbreak z_{30}, \allowbreak x_{31}, \allowbreak y_{31}, \allowbreak z_{31}, \allowbreak x_{32}, \allowbreak y_{32}, \allowbreak z_{32}, \allowbreak x_{33}, \allowbreak y_{33}, \allowbreak z_{33}, \allowbreak x_{34}, \allowbreak y_{34}, \allowbreak z_{34}, \allowbreak x_{35}, \allowbreak y_{35}, \allowbreak z_{35}, \allowbreak x_{36}, \allowbreak y_{36}, \allowbreak z_{36}, \allowbreak x_{37}, \allowbreak y_{37}, \allowbreak z_{37}, \allowbreak x_{38}, \allowbreak y_{38}, \allowbreak z_{38}, \allowbreak x_{39}, \allowbreak y_{39}, \allowbreak z_{39}, \allowbreak x_{40}, \allowbreak y_{40}, \allowbreak z_{40}, \allowbreak x_{41}, \allowbreak y_{41}, \allowbreak z_{41}, \allowbreak x_{42}, \allowbreak y_{42}, \allowbreak z_{42}, \allowbreak x_{43}, \allowbreak y_{43}, \allowbreak z_{43}, \allowbreak x_{44}, \allowbreak y_{44}, \allowbreak z_{44}$ |
- (Kirchner, 2000) found the aluminum phospate zeolite AlPO$_{4}$-53 in three states:
- As-synthesized AlPO$_{4}$-53(A) [24(AlPO$_{4}$)·8.5CH$_{3}$NH$_{2}$·14H$_{2}$O] (this structure) includes methyl and ammonium ions as well as water.
- Orthorhombic AlPO$_{4}$-53(B) is calcined (heated below the melting temperature) to remove those ions and molecules, leaving the framework structure.
- Monoclinic AlPO$_{4}$-53(C) is produced by heating AlPO$_{4}$-53(B) above 700$^\circ$C.
- (Kirchner, 2000) did not give the positions of any hydrogen atoms. The final two oxygen atoms, O XXVII and O XXVIII, represent the positions of the water molecules.
- Space group $P2_{1}2_{1}2_{1}$ #19 is chiral, Sohncke Class III. Any mirror operation on the coordinates, e.g., $(x y z) \rightarrow (-x -y -z)$, will produce a mirror image of the current structure, still in space group $P2_{1}2_{1}2_{1}$, with the same interatomic distances and bond angles.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&a \,\mathbf{\hat{x}}\\\mathbf{a_{2}}&=&b \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+y_{1} \, \mathbf{a}_{2}+z_{1} \, \mathbf{a}_{3}$ | = | $a x_{1} \,\mathbf{\hat{x}}+b y_{1} \,\mathbf{\hat{y}}+c z_{1} \,\mathbf{\hat{z}}$ | (4a) | Al I |
| $\mathbf{B_{2}}$ | = | $- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{1} \, \mathbf{a}_{2}+\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{1} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{1} \,\mathbf{\hat{y}}+c \left(z_{1} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al I |
| $\mathbf{B_{3}}$ | = | $- x_{1} \, \mathbf{a}_{1}+\left(y_{1} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{1} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{1} \,\mathbf{\hat{x}}+b \left(y_{1} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{1} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al I |
| $\mathbf{B_{4}}$ | = | $\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{1} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{1} \, \mathbf{a}_{3}$ | = | $a \left(x_{1} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{1} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{1} \,\mathbf{\hat{z}}$ | (4a) | Al I |
| $\mathbf{B_{5}}$ | = | $x_{2} \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $a x_{2} \,\mathbf{\hat{x}}+b y_{2} \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (4a) | Al II |
| $\mathbf{B_{6}}$ | = | $- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{2} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{2} \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al II |
| $\mathbf{B_{7}}$ | = | $- x_{2} \, \mathbf{a}_{1}+\left(y_{2} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{2} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{2} \,\mathbf{\hat{x}}+b \left(y_{2} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{2} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al II |
| $\mathbf{B_{8}}$ | = | $\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{2} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}$ | = | $a \left(x_{2} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{2} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{2} \,\mathbf{\hat{z}}$ | (4a) | Al II |
| $\mathbf{B_{9}}$ | = | $x_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}+b y_{3} \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (4a) | Al III |
| $\mathbf{B_{10}}$ | = | $- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}+\left(z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{3} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{3} \,\mathbf{\hat{y}}+c \left(z_{3} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al III |
| $\mathbf{B_{11}}$ | = | $- x_{3} \, \mathbf{a}_{1}+\left(y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}+b \left(y_{3} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{3} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al III |
| $\mathbf{B_{12}}$ | = | $\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{3} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}$ | = | $a \left(x_{3} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{3} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{3} \,\mathbf{\hat{z}}$ | (4a) | Al III |
| $\mathbf{B_{13}}$ | = | $x_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}+b y_{4} \,\mathbf{\hat{y}}+c z_{4} \,\mathbf{\hat{z}}$ | (4a) | Al IV |
| $\mathbf{B_{14}}$ | = | $- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}+\left(z_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{4} \,\mathbf{\hat{y}}+c \left(z_{4} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al IV |
| $\mathbf{B_{15}}$ | = | $- x_{4} \, \mathbf{a}_{1}+\left(y_{4} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{4} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}+b \left(y_{4} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{4} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al IV |
| $\mathbf{B_{16}}$ | = | $\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{4} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}$ | = | $a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{4} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{4} \,\mathbf{\hat{z}}$ | (4a) | Al IV |
| $\mathbf{B_{17}}$ | = | $x_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}+b y_{5} \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (4a) | Al V |
| $\mathbf{B_{18}}$ | = | $- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{5} \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al V |
| $\mathbf{B_{19}}$ | = | $- x_{5} \, \mathbf{a}_{1}+\left(y_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}+b \left(y_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{5} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al V |
| $\mathbf{B_{20}}$ | = | $\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{5} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{5} \,\mathbf{\hat{z}}$ | (4a) | Al V |
| $\mathbf{B_{21}}$ | = | $x_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $a x_{6} \,\mathbf{\hat{x}}+b y_{6} \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (4a) | Al VI |
| $\mathbf{B_{22}}$ | = | $- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{6} \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al VI |
| $\mathbf{B_{23}}$ | = | $- x_{6} \, \mathbf{a}_{1}+\left(y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{6} \,\mathbf{\hat{x}}+b \left(y_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{6} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al VI |
| $\mathbf{B_{24}}$ | = | $\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{6} \,\mathbf{\hat{z}}$ | (4a) | Al VI |
| $\mathbf{B_{25}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+b y_{7} \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (4a) | C I |
| $\mathbf{B_{26}}$ | = | $- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{7} \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | C I |
| $\mathbf{B_{27}}$ | = | $- x_{7} \, \mathbf{a}_{1}+\left(y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}+b \left(y_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{7} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | C I |
| $\mathbf{B_{28}}$ | = | $\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{7} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{7} \,\mathbf{\hat{z}}$ | (4a) | C I |
| $\mathbf{B_{29}}$ | = | $x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+b y_{8} \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (4a) | C II |
| $\mathbf{B_{30}}$ | = | $- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{8} \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | C II |
| $\mathbf{B_{31}}$ | = | $- x_{8} \, \mathbf{a}_{1}+\left(y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}+b \left(y_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{8} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | C II |
| $\mathbf{B_{32}}$ | = | $\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{8} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{8} \,\mathbf{\hat{z}}$ | (4a) | C II |
| $\mathbf{B_{33}}$ | = | $x_{9} \, \mathbf{a}_{1}+y_{9} \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $a x_{9} \,\mathbf{\hat{x}}+b y_{9} \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{34}}$ | = | $- \left(x_{9} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{9} \, \mathbf{a}_{2}+\left(z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{9} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{9} \,\mathbf{\hat{y}}+c \left(z_{9} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{35}}$ | = | $- x_{9} \, \mathbf{a}_{1}+\left(y_{9} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{9} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{9} \,\mathbf{\hat{x}}+b \left(y_{9} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{9} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{36}}$ | = | $\left(x_{9} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{9} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{9} \, \mathbf{a}_{3}$ | = | $a \left(x_{9} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{9} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{9} \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{37}}$ | = | $x_{10} \, \mathbf{a}_{1}+y_{10} \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $a x_{10} \,\mathbf{\hat{x}}+b y_{10} \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (4a) | N II |
| $\mathbf{B_{38}}$ | = | $- \left(x_{10} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{10} \, \mathbf{a}_{2}+\left(z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{10} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{10} \,\mathbf{\hat{y}}+c \left(z_{10} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | N II |
| $\mathbf{B_{39}}$ | = | $- x_{10} \, \mathbf{a}_{1}+\left(y_{10} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{10} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{10} \,\mathbf{\hat{x}}+b \left(y_{10} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{10} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | N II |
| $\mathbf{B_{40}}$ | = | $\left(x_{10} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{10} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{10} \, \mathbf{a}_{3}$ | = | $a \left(x_{10} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{10} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{10} \,\mathbf{\hat{z}}$ | (4a) | N II |
| $\mathbf{B_{41}}$ | = | $x_{11} \, \mathbf{a}_{1}+y_{11} \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $a x_{11} \,\mathbf{\hat{x}}+b y_{11} \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{42}}$ | = | $- \left(x_{11} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{11} \, \mathbf{a}_{2}+\left(z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{11} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{11} \,\mathbf{\hat{y}}+c \left(z_{11} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{43}}$ | = | $- x_{11} \, \mathbf{a}_{1}+\left(y_{11} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{11} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{11} \,\mathbf{\hat{x}}+b \left(y_{11} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{11} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{44}}$ | = | $\left(x_{11} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{11} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{11} \, \mathbf{a}_{3}$ | = | $a \left(x_{11} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{11} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{11} \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{45}}$ | = | $x_{12} \, \mathbf{a}_{1}+y_{12} \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $a x_{12} \,\mathbf{\hat{x}}+b y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (4a) | O II |
| $\mathbf{B_{46}}$ | = | $- \left(x_{12} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{12} \, \mathbf{a}_{2}+\left(z_{12} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{12} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{12} \,\mathbf{\hat{y}}+c \left(z_{12} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O II |
| $\mathbf{B_{47}}$ | = | $- x_{12} \, \mathbf{a}_{1}+\left(y_{12} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{12} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{12} \,\mathbf{\hat{x}}+b \left(y_{12} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{12} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O II |
| $\mathbf{B_{48}}$ | = | $\left(x_{12} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{12} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{12} \, \mathbf{a}_{3}$ | = | $a \left(x_{12} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{12} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{12} \,\mathbf{\hat{z}}$ | (4a) | O II |
| $\mathbf{B_{49}}$ | = | $x_{13} \, \mathbf{a}_{1}+y_{13} \, \mathbf{a}_{2}+z_{13} \, \mathbf{a}_{3}$ | = | $a x_{13} \,\mathbf{\hat{x}}+b y_{13} \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (4a) | O III |
| $\mathbf{B_{50}}$ | = | $- \left(x_{13} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{13} \, \mathbf{a}_{2}+\left(z_{13} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{13} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{13} \,\mathbf{\hat{y}}+c \left(z_{13} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O III |
| $\mathbf{B_{51}}$ | = | $- x_{13} \, \mathbf{a}_{1}+\left(y_{13} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{13} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{13} \,\mathbf{\hat{x}}+b \left(y_{13} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{13} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O III |
| $\mathbf{B_{52}}$ | = | $\left(x_{13} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{13} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{13} \, \mathbf{a}_{3}$ | = | $a \left(x_{13} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{13} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{13} \,\mathbf{\hat{z}}$ | (4a) | O III |
| $\mathbf{B_{53}}$ | = | $x_{14} \, \mathbf{a}_{1}+y_{14} \, \mathbf{a}_{2}+z_{14} \, \mathbf{a}_{3}$ | = | $a x_{14} \,\mathbf{\hat{x}}+b y_{14} \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (4a) | O IV |
| $\mathbf{B_{54}}$ | = | $- \left(x_{14} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{14} \, \mathbf{a}_{2}+\left(z_{14} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{14} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{14} \,\mathbf{\hat{y}}+c \left(z_{14} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O IV |
| $\mathbf{B_{55}}$ | = | $- x_{14} \, \mathbf{a}_{1}+\left(y_{14} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{14} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{14} \,\mathbf{\hat{x}}+b \left(y_{14} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{14} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O IV |
| $\mathbf{B_{56}}$ | = | $\left(x_{14} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{14} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{14} \, \mathbf{a}_{3}$ | = | $a \left(x_{14} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{14} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{14} \,\mathbf{\hat{z}}$ | (4a) | O IV |
| $\mathbf{B_{57}}$ | = | $x_{15} \, \mathbf{a}_{1}+y_{15} \, \mathbf{a}_{2}+z_{15} \, \mathbf{a}_{3}$ | = | $a x_{15} \,\mathbf{\hat{x}}+b y_{15} \,\mathbf{\hat{y}}+c z_{15} \,\mathbf{\hat{z}}$ | (4a) | O V |
| $\mathbf{B_{58}}$ | = | $- \left(x_{15} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{15} \, \mathbf{a}_{2}+\left(z_{15} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{15} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{15} \,\mathbf{\hat{y}}+c \left(z_{15} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O V |
| $\mathbf{B_{59}}$ | = | $- x_{15} \, \mathbf{a}_{1}+\left(y_{15} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{15} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{15} \,\mathbf{\hat{x}}+b \left(y_{15} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{15} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O V |
| $\mathbf{B_{60}}$ | = | $\left(x_{15} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{15} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{15} \, \mathbf{a}_{3}$ | = | $a \left(x_{15} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{15} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{15} \,\mathbf{\hat{z}}$ | (4a) | O V |
| $\mathbf{B_{61}}$ | = | $x_{16} \, \mathbf{a}_{1}+y_{16} \, \mathbf{a}_{2}+z_{16} \, \mathbf{a}_{3}$ | = | $a x_{16} \,\mathbf{\hat{x}}+b y_{16} \,\mathbf{\hat{y}}+c z_{16} \,\mathbf{\hat{z}}$ | (4a) | O VI |
| $\mathbf{B_{62}}$ | = | $- \left(x_{16} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{16} \, \mathbf{a}_{2}+\left(z_{16} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{16} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{16} \,\mathbf{\hat{y}}+c \left(z_{16} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VI |
| $\mathbf{B_{63}}$ | = | $- x_{16} \, \mathbf{a}_{1}+\left(y_{16} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{16} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{16} \,\mathbf{\hat{x}}+b \left(y_{16} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{16} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VI |
| $\mathbf{B_{64}}$ | = | $\left(x_{16} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{16} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{16} \, \mathbf{a}_{3}$ | = | $a \left(x_{16} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{16} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{16} \,\mathbf{\hat{z}}$ | (4a) | O VI |
| $\mathbf{B_{65}}$ | = | $x_{17} \, \mathbf{a}_{1}+y_{17} \, \mathbf{a}_{2}+z_{17} \, \mathbf{a}_{3}$ | = | $a x_{17} \,\mathbf{\hat{x}}+b y_{17} \,\mathbf{\hat{y}}+c z_{17} \,\mathbf{\hat{z}}$ | (4a) | O VII |
| $\mathbf{B_{66}}$ | = | $- \left(x_{17} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{17} \, \mathbf{a}_{2}+\left(z_{17} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{17} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{17} \,\mathbf{\hat{y}}+c \left(z_{17} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VII |
| $\mathbf{B_{67}}$ | = | $- x_{17} \, \mathbf{a}_{1}+\left(y_{17} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{17} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{17} \,\mathbf{\hat{x}}+b \left(y_{17} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{17} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VII |
| $\mathbf{B_{68}}$ | = | $\left(x_{17} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{17} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{17} \, \mathbf{a}_{3}$ | = | $a \left(x_{17} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{17} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{17} \,\mathbf{\hat{z}}$ | (4a) | O VII |
| $\mathbf{B_{69}}$ | = | $x_{18} \, \mathbf{a}_{1}+y_{18} \, \mathbf{a}_{2}+z_{18} \, \mathbf{a}_{3}$ | = | $a x_{18} \,\mathbf{\hat{x}}+b y_{18} \,\mathbf{\hat{y}}+c z_{18} \,\mathbf{\hat{z}}$ | (4a) | O VIII |
| $\mathbf{B_{70}}$ | = | $- \left(x_{18} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{18} \, \mathbf{a}_{2}+\left(z_{18} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{18} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{18} \,\mathbf{\hat{y}}+c \left(z_{18} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VIII |
| $\mathbf{B_{71}}$ | = | $- x_{18} \, \mathbf{a}_{1}+\left(y_{18} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{18} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{18} \,\mathbf{\hat{x}}+b \left(y_{18} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{18} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VIII |
| $\mathbf{B_{72}}$ | = | $\left(x_{18} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{18} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{18} \, \mathbf{a}_{3}$ | = | $a \left(x_{18} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{18} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{18} \,\mathbf{\hat{z}}$ | (4a) | O VIII |
| $\mathbf{B_{73}}$ | = | $x_{19} \, \mathbf{a}_{1}+y_{19} \, \mathbf{a}_{2}+z_{19} \, \mathbf{a}_{3}$ | = | $a x_{19} \,\mathbf{\hat{x}}+b y_{19} \,\mathbf{\hat{y}}+c z_{19} \,\mathbf{\hat{z}}$ | (4a) | O IX |
| $\mathbf{B_{74}}$ | = | $- \left(x_{19} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{19} \, \mathbf{a}_{2}+\left(z_{19} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{19} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{19} \,\mathbf{\hat{y}}+c \left(z_{19} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O IX |
| $\mathbf{B_{75}}$ | = | $- x_{19} \, \mathbf{a}_{1}+\left(y_{19} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{19} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{19} \,\mathbf{\hat{x}}+b \left(y_{19} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{19} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O IX |
| $\mathbf{B_{76}}$ | = | $\left(x_{19} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{19} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{19} \, \mathbf{a}_{3}$ | = | $a \left(x_{19} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{19} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{19} \,\mathbf{\hat{z}}$ | (4a) | O IX |
| $\mathbf{B_{77}}$ | = | $x_{20} \, \mathbf{a}_{1}+y_{20} \, \mathbf{a}_{2}+z_{20} \, \mathbf{a}_{3}$ | = | $a x_{20} \,\mathbf{\hat{x}}+b y_{20} \,\mathbf{\hat{y}}+c z_{20} \,\mathbf{\hat{z}}$ | (4a) | O X |
| $\mathbf{B_{78}}$ | = | $- \left(x_{20} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{20} \, \mathbf{a}_{2}+\left(z_{20} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{20} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{20} \,\mathbf{\hat{y}}+c \left(z_{20} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O X |
| $\mathbf{B_{79}}$ | = | $- x_{20} \, \mathbf{a}_{1}+\left(y_{20} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{20} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{20} \,\mathbf{\hat{x}}+b \left(y_{20} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{20} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O X |
| $\mathbf{B_{80}}$ | = | $\left(x_{20} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{20} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{20} \, \mathbf{a}_{3}$ | = | $a \left(x_{20} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{20} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{20} \,\mathbf{\hat{z}}$ | (4a) | O X |
| $\mathbf{B_{81}}$ | = | $x_{21} \, \mathbf{a}_{1}+y_{21} \, \mathbf{a}_{2}+z_{21} \, \mathbf{a}_{3}$ | = | $a x_{21} \,\mathbf{\hat{x}}+b y_{21} \,\mathbf{\hat{y}}+c z_{21} \,\mathbf{\hat{z}}$ | (4a) | O XI |
| $\mathbf{B_{82}}$ | = | $- \left(x_{21} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{21} \, \mathbf{a}_{2}+\left(z_{21} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{21} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{21} \,\mathbf{\hat{y}}+c \left(z_{21} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XI |
| $\mathbf{B_{83}}$ | = | $- x_{21} \, \mathbf{a}_{1}+\left(y_{21} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{21} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{21} \,\mathbf{\hat{x}}+b \left(y_{21} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{21} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XI |
| $\mathbf{B_{84}}$ | = | $\left(x_{21} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{21} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{21} \, \mathbf{a}_{3}$ | = | $a \left(x_{21} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{21} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{21} \,\mathbf{\hat{z}}$ | (4a) | O XI |
| $\mathbf{B_{85}}$ | = | $x_{22} \, \mathbf{a}_{1}+y_{22} \, \mathbf{a}_{2}+z_{22} \, \mathbf{a}_{3}$ | = | $a x_{22} \,\mathbf{\hat{x}}+b y_{22} \,\mathbf{\hat{y}}+c z_{22} \,\mathbf{\hat{z}}$ | (4a) | O XII |
| $\mathbf{B_{86}}$ | = | $- \left(x_{22} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{22} \, \mathbf{a}_{2}+\left(z_{22} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{22} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{22} \,\mathbf{\hat{y}}+c \left(z_{22} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XII |
| $\mathbf{B_{87}}$ | = | $- x_{22} \, \mathbf{a}_{1}+\left(y_{22} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{22} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{22} \,\mathbf{\hat{x}}+b \left(y_{22} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{22} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XII |
| $\mathbf{B_{88}}$ | = | $\left(x_{22} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{22} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{22} \, \mathbf{a}_{3}$ | = | $a \left(x_{22} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{22} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{22} \,\mathbf{\hat{z}}$ | (4a) | O XII |
| $\mathbf{B_{89}}$ | = | $x_{23} \, \mathbf{a}_{1}+y_{23} \, \mathbf{a}_{2}+z_{23} \, \mathbf{a}_{3}$ | = | $a x_{23} \,\mathbf{\hat{x}}+b y_{23} \,\mathbf{\hat{y}}+c z_{23} \,\mathbf{\hat{z}}$ | (4a) | O XIII |
| $\mathbf{B_{90}}$ | = | $- \left(x_{23} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{23} \, \mathbf{a}_{2}+\left(z_{23} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{23} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{23} \,\mathbf{\hat{y}}+c \left(z_{23} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIII |
| $\mathbf{B_{91}}$ | = | $- x_{23} \, \mathbf{a}_{1}+\left(y_{23} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{23} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{23} \,\mathbf{\hat{x}}+b \left(y_{23} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{23} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIII |
| $\mathbf{B_{92}}$ | = | $\left(x_{23} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{23} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{23} \, \mathbf{a}_{3}$ | = | $a \left(x_{23} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{23} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{23} \,\mathbf{\hat{z}}$ | (4a) | O XIII |
| $\mathbf{B_{93}}$ | = | $x_{24} \, \mathbf{a}_{1}+y_{24} \, \mathbf{a}_{2}+z_{24} \, \mathbf{a}_{3}$ | = | $a x_{24} \,\mathbf{\hat{x}}+b y_{24} \,\mathbf{\hat{y}}+c z_{24} \,\mathbf{\hat{z}}$ | (4a) | O XIV |
| $\mathbf{B_{94}}$ | = | $- \left(x_{24} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{24} \, \mathbf{a}_{2}+\left(z_{24} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{24} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{24} \,\mathbf{\hat{y}}+c \left(z_{24} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIV |
| $\mathbf{B_{95}}$ | = | $- x_{24} \, \mathbf{a}_{1}+\left(y_{24} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{24} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{24} \,\mathbf{\hat{x}}+b \left(y_{24} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{24} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIV |
| $\mathbf{B_{96}}$ | = | $\left(x_{24} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{24} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{24} \, \mathbf{a}_{3}$ | = | $a \left(x_{24} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{24} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{24} \,\mathbf{\hat{z}}$ | (4a) | O XIV |
| $\mathbf{B_{97}}$ | = | $x_{25} \, \mathbf{a}_{1}+y_{25} \, \mathbf{a}_{2}+z_{25} \, \mathbf{a}_{3}$ | = | $a x_{25} \,\mathbf{\hat{x}}+b y_{25} \,\mathbf{\hat{y}}+c z_{25} \,\mathbf{\hat{z}}$ | (4a) | O XV |
| $\mathbf{B_{98}}$ | = | $- \left(x_{25} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{25} \, \mathbf{a}_{2}+\left(z_{25} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{25} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{25} \,\mathbf{\hat{y}}+c \left(z_{25} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XV |
| $\mathbf{B_{99}}$ | = | $- x_{25} \, \mathbf{a}_{1}+\left(y_{25} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{25} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{25} \,\mathbf{\hat{x}}+b \left(y_{25} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{25} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XV |
| $\mathbf{B_{100}}$ | = | $\left(x_{25} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{25} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{25} \, \mathbf{a}_{3}$ | = | $a \left(x_{25} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{25} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{25} \,\mathbf{\hat{z}}$ | (4a) | O XV |
| $\mathbf{B_{101}}$ | = | $x_{26} \, \mathbf{a}_{1}+y_{26} \, \mathbf{a}_{2}+z_{26} \, \mathbf{a}_{3}$ | = | $a x_{26} \,\mathbf{\hat{x}}+b y_{26} \,\mathbf{\hat{y}}+c z_{26} \,\mathbf{\hat{z}}$ | (4a) | O XVI |
| $\mathbf{B_{102}}$ | = | $- \left(x_{26} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{26} \, \mathbf{a}_{2}+\left(z_{26} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{26} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{26} \,\mathbf{\hat{y}}+c \left(z_{26} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVI |
| $\mathbf{B_{103}}$ | = | $- x_{26} \, \mathbf{a}_{1}+\left(y_{26} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{26} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{26} \,\mathbf{\hat{x}}+b \left(y_{26} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{26} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVI |
| $\mathbf{B_{104}}$ | = | $\left(x_{26} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{26} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{26} \, \mathbf{a}_{3}$ | = | $a \left(x_{26} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{26} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{26} \,\mathbf{\hat{z}}$ | (4a) | O XVI |
| $\mathbf{B_{105}}$ | = | $x_{27} \, \mathbf{a}_{1}+y_{27} \, \mathbf{a}_{2}+z_{27} \, \mathbf{a}_{3}$ | = | $a x_{27} \,\mathbf{\hat{x}}+b y_{27} \,\mathbf{\hat{y}}+c z_{27} \,\mathbf{\hat{z}}$ | (4a) | O XVII |
| $\mathbf{B_{106}}$ | = | $- \left(x_{27} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{27} \, \mathbf{a}_{2}+\left(z_{27} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{27} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{27} \,\mathbf{\hat{y}}+c \left(z_{27} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVII |
| $\mathbf{B_{107}}$ | = | $- x_{27} \, \mathbf{a}_{1}+\left(y_{27} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{27} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{27} \,\mathbf{\hat{x}}+b \left(y_{27} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{27} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVII |
| $\mathbf{B_{108}}$ | = | $\left(x_{27} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{27} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{27} \, \mathbf{a}_{3}$ | = | $a \left(x_{27} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{27} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{27} \,\mathbf{\hat{z}}$ | (4a) | O XVII |
| $\mathbf{B_{109}}$ | = | $x_{28} \, \mathbf{a}_{1}+y_{28} \, \mathbf{a}_{2}+z_{28} \, \mathbf{a}_{3}$ | = | $a x_{28} \,\mathbf{\hat{x}}+b y_{28} \,\mathbf{\hat{y}}+c z_{28} \,\mathbf{\hat{z}}$ | (4a) | O XVIII |
| $\mathbf{B_{110}}$ | = | $- \left(x_{28} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{28} \, \mathbf{a}_{2}+\left(z_{28} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{28} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{28} \,\mathbf{\hat{y}}+c \left(z_{28} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVIII |
| $\mathbf{B_{111}}$ | = | $- x_{28} \, \mathbf{a}_{1}+\left(y_{28} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{28} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{28} \,\mathbf{\hat{x}}+b \left(y_{28} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{28} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVIII |
| $\mathbf{B_{112}}$ | = | $\left(x_{28} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{28} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{28} \, \mathbf{a}_{3}$ | = | $a \left(x_{28} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{28} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{28} \,\mathbf{\hat{z}}$ | (4a) | O XVIII |
| $\mathbf{B_{113}}$ | = | $x_{29} \, \mathbf{a}_{1}+y_{29} \, \mathbf{a}_{2}+z_{29} \, \mathbf{a}_{3}$ | = | $a x_{29} \,\mathbf{\hat{x}}+b y_{29} \,\mathbf{\hat{y}}+c z_{29} \,\mathbf{\hat{z}}$ | (4a) | O XIX |
| $\mathbf{B_{114}}$ | = | $- \left(x_{29} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{29} \, \mathbf{a}_{2}+\left(z_{29} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{29} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{29} \,\mathbf{\hat{y}}+c \left(z_{29} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIX |
| $\mathbf{B_{115}}$ | = | $- x_{29} \, \mathbf{a}_{1}+\left(y_{29} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{29} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{29} \,\mathbf{\hat{x}}+b \left(y_{29} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{29} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIX |
| $\mathbf{B_{116}}$ | = | $\left(x_{29} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{29} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{29} \, \mathbf{a}_{3}$ | = | $a \left(x_{29} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{29} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{29} \,\mathbf{\hat{z}}$ | (4a) | O XIX |
| $\mathbf{B_{117}}$ | = | $x_{30} \, \mathbf{a}_{1}+y_{30} \, \mathbf{a}_{2}+z_{30} \, \mathbf{a}_{3}$ | = | $a x_{30} \,\mathbf{\hat{x}}+b y_{30} \,\mathbf{\hat{y}}+c z_{30} \,\mathbf{\hat{z}}$ | (4a) | O XX |
| $\mathbf{B_{118}}$ | = | $- \left(x_{30} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{30} \, \mathbf{a}_{2}+\left(z_{30} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{30} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{30} \,\mathbf{\hat{y}}+c \left(z_{30} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XX |
| $\mathbf{B_{119}}$ | = | $- x_{30} \, \mathbf{a}_{1}+\left(y_{30} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{30} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{30} \,\mathbf{\hat{x}}+b \left(y_{30} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{30} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XX |
| $\mathbf{B_{120}}$ | = | $\left(x_{30} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{30} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{30} \, \mathbf{a}_{3}$ | = | $a \left(x_{30} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{30} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{30} \,\mathbf{\hat{z}}$ | (4a) | O XX |
| $\mathbf{B_{121}}$ | = | $x_{31} \, \mathbf{a}_{1}+y_{31} \, \mathbf{a}_{2}+z_{31} \, \mathbf{a}_{3}$ | = | $a x_{31} \,\mathbf{\hat{x}}+b y_{31} \,\mathbf{\hat{y}}+c z_{31} \,\mathbf{\hat{z}}$ | (4a) | O XXI |
| $\mathbf{B_{122}}$ | = | $- \left(x_{31} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{31} \, \mathbf{a}_{2}+\left(z_{31} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{31} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{31} \,\mathbf{\hat{y}}+c \left(z_{31} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXI |
| $\mathbf{B_{123}}$ | = | $- x_{31} \, \mathbf{a}_{1}+\left(y_{31} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{31} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{31} \,\mathbf{\hat{x}}+b \left(y_{31} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{31} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXI |
| $\mathbf{B_{124}}$ | = | $\left(x_{31} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{31} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{31} \, \mathbf{a}_{3}$ | = | $a \left(x_{31} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{31} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{31} \,\mathbf{\hat{z}}$ | (4a) | O XXI |
| $\mathbf{B_{125}}$ | = | $x_{32} \, \mathbf{a}_{1}+y_{32} \, \mathbf{a}_{2}+z_{32} \, \mathbf{a}_{3}$ | = | $a x_{32} \,\mathbf{\hat{x}}+b y_{32} \,\mathbf{\hat{y}}+c z_{32} \,\mathbf{\hat{z}}$ | (4a) | O XXII |
| $\mathbf{B_{126}}$ | = | $- \left(x_{32} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{32} \, \mathbf{a}_{2}+\left(z_{32} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{32} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{32} \,\mathbf{\hat{y}}+c \left(z_{32} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXII |
| $\mathbf{B_{127}}$ | = | $- x_{32} \, \mathbf{a}_{1}+\left(y_{32} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{32} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{32} \,\mathbf{\hat{x}}+b \left(y_{32} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{32} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXII |
| $\mathbf{B_{128}}$ | = | $\left(x_{32} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{32} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{32} \, \mathbf{a}_{3}$ | = | $a \left(x_{32} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{32} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{32} \,\mathbf{\hat{z}}$ | (4a) | O XXII |
| $\mathbf{B_{129}}$ | = | $x_{33} \, \mathbf{a}_{1}+y_{33} \, \mathbf{a}_{2}+z_{33} \, \mathbf{a}_{3}$ | = | $a x_{33} \,\mathbf{\hat{x}}+b y_{33} \,\mathbf{\hat{y}}+c z_{33} \,\mathbf{\hat{z}}$ | (4a) | O XXIII |
| $\mathbf{B_{130}}$ | = | $- \left(x_{33} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{33} \, \mathbf{a}_{2}+\left(z_{33} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{33} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{33} \,\mathbf{\hat{y}}+c \left(z_{33} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXIII |
| $\mathbf{B_{131}}$ | = | $- x_{33} \, \mathbf{a}_{1}+\left(y_{33} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{33} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{33} \,\mathbf{\hat{x}}+b \left(y_{33} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{33} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXIII |
| $\mathbf{B_{132}}$ | = | $\left(x_{33} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{33} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{33} \, \mathbf{a}_{3}$ | = | $a \left(x_{33} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{33} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{33} \,\mathbf{\hat{z}}$ | (4a) | O XXIII |
| $\mathbf{B_{133}}$ | = | $x_{34} \, \mathbf{a}_{1}+y_{34} \, \mathbf{a}_{2}+z_{34} \, \mathbf{a}_{3}$ | = | $a x_{34} \,\mathbf{\hat{x}}+b y_{34} \,\mathbf{\hat{y}}+c z_{34} \,\mathbf{\hat{z}}$ | (4a) | O XXIV |
| $\mathbf{B_{134}}$ | = | $- \left(x_{34} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{34} \, \mathbf{a}_{2}+\left(z_{34} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{34} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{34} \,\mathbf{\hat{y}}+c \left(z_{34} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXIV |
| $\mathbf{B_{135}}$ | = | $- x_{34} \, \mathbf{a}_{1}+\left(y_{34} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{34} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{34} \,\mathbf{\hat{x}}+b \left(y_{34} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{34} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXIV |
| $\mathbf{B_{136}}$ | = | $\left(x_{34} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{34} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{34} \, \mathbf{a}_{3}$ | = | $a \left(x_{34} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{34} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{34} \,\mathbf{\hat{z}}$ | (4a) | O XXIV |
| $\mathbf{B_{137}}$ | = | $x_{35} \, \mathbf{a}_{1}+y_{35} \, \mathbf{a}_{2}+z_{35} \, \mathbf{a}_{3}$ | = | $a x_{35} \,\mathbf{\hat{x}}+b y_{35} \,\mathbf{\hat{y}}+c z_{35} \,\mathbf{\hat{z}}$ | (4a) | O XXV |
| $\mathbf{B_{138}}$ | = | $- \left(x_{35} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{35} \, \mathbf{a}_{2}+\left(z_{35} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{35} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{35} \,\mathbf{\hat{y}}+c \left(z_{35} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXV |
| $\mathbf{B_{139}}$ | = | $- x_{35} \, \mathbf{a}_{1}+\left(y_{35} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{35} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{35} \,\mathbf{\hat{x}}+b \left(y_{35} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{35} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXV |
| $\mathbf{B_{140}}$ | = | $\left(x_{35} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{35} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{35} \, \mathbf{a}_{3}$ | = | $a \left(x_{35} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{35} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{35} \,\mathbf{\hat{z}}$ | (4a) | O XXV |
| $\mathbf{B_{141}}$ | = | $x_{36} \, \mathbf{a}_{1}+y_{36} \, \mathbf{a}_{2}+z_{36} \, \mathbf{a}_{3}$ | = | $a x_{36} \,\mathbf{\hat{x}}+b y_{36} \,\mathbf{\hat{y}}+c z_{36} \,\mathbf{\hat{z}}$ | (4a) | O XXVI |
| $\mathbf{B_{142}}$ | = | $- \left(x_{36} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{36} \, \mathbf{a}_{2}+\left(z_{36} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{36} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{36} \,\mathbf{\hat{y}}+c \left(z_{36} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXVI |
| $\mathbf{B_{143}}$ | = | $- x_{36} \, \mathbf{a}_{1}+\left(y_{36} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{36} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{36} \,\mathbf{\hat{x}}+b \left(y_{36} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{36} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXVI |
| $\mathbf{B_{144}}$ | = | $\left(x_{36} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{36} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{36} \, \mathbf{a}_{3}$ | = | $a \left(x_{36} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{36} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{36} \,\mathbf{\hat{z}}$ | (4a) | O XXVI |
| $\mathbf{B_{145}}$ | = | $x_{37} \, \mathbf{a}_{1}+y_{37} \, \mathbf{a}_{2}+z_{37} \, \mathbf{a}_{3}$ | = | $a x_{37} \,\mathbf{\hat{x}}+b y_{37} \,\mathbf{\hat{y}}+c z_{37} \,\mathbf{\hat{z}}$ | (4a) | O XXVII |
| $\mathbf{B_{146}}$ | = | $- \left(x_{37} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{37} \, \mathbf{a}_{2}+\left(z_{37} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{37} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{37} \,\mathbf{\hat{y}}+c \left(z_{37} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXVII |
| $\mathbf{B_{147}}$ | = | $- x_{37} \, \mathbf{a}_{1}+\left(y_{37} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{37} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{37} \,\mathbf{\hat{x}}+b \left(y_{37} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{37} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXVII |
| $\mathbf{B_{148}}$ | = | $\left(x_{37} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{37} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{37} \, \mathbf{a}_{3}$ | = | $a \left(x_{37} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{37} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{37} \,\mathbf{\hat{z}}$ | (4a) | O XXVII |
| $\mathbf{B_{149}}$ | = | $x_{38} \, \mathbf{a}_{1}+y_{38} \, \mathbf{a}_{2}+z_{38} \, \mathbf{a}_{3}$ | = | $a x_{38} \,\mathbf{\hat{x}}+b y_{38} \,\mathbf{\hat{y}}+c z_{38} \,\mathbf{\hat{z}}$ | (4a) | O XXVIII |
| $\mathbf{B_{150}}$ | = | $- \left(x_{38} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{38} \, \mathbf{a}_{2}+\left(z_{38} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{38} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{38} \,\mathbf{\hat{y}}+c \left(z_{38} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXVIII |
| $\mathbf{B_{151}}$ | = | $- x_{38} \, \mathbf{a}_{1}+\left(y_{38} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{38} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{38} \,\mathbf{\hat{x}}+b \left(y_{38} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{38} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XXVIII |
| $\mathbf{B_{152}}$ | = | $\left(x_{38} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{38} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{38} \, \mathbf{a}_{3}$ | = | $a \left(x_{38} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{38} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{38} \,\mathbf{\hat{z}}$ | (4a) | O XXVIII |
| $\mathbf{B_{153}}$ | = | $x_{39} \, \mathbf{a}_{1}+y_{39} \, \mathbf{a}_{2}+z_{39} \, \mathbf{a}_{3}$ | = | $a x_{39} \,\mathbf{\hat{x}}+b y_{39} \,\mathbf{\hat{y}}+c z_{39} \,\mathbf{\hat{z}}$ | (4a) | P I |
| $\mathbf{B_{154}}$ | = | $- \left(x_{39} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{39} \, \mathbf{a}_{2}+\left(z_{39} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{39} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{39} \,\mathbf{\hat{y}}+c \left(z_{39} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P I |
| $\mathbf{B_{155}}$ | = | $- x_{39} \, \mathbf{a}_{1}+\left(y_{39} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{39} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{39} \,\mathbf{\hat{x}}+b \left(y_{39} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{39} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P I |
| $\mathbf{B_{156}}$ | = | $\left(x_{39} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{39} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{39} \, \mathbf{a}_{3}$ | = | $a \left(x_{39} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{39} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{39} \,\mathbf{\hat{z}}$ | (4a) | P I |
| $\mathbf{B_{157}}$ | = | $x_{40} \, \mathbf{a}_{1}+y_{40} \, \mathbf{a}_{2}+z_{40} \, \mathbf{a}_{3}$ | = | $a x_{40} \,\mathbf{\hat{x}}+b y_{40} \,\mathbf{\hat{y}}+c z_{40} \,\mathbf{\hat{z}}$ | (4a) | P II |
| $\mathbf{B_{158}}$ | = | $- \left(x_{40} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{40} \, \mathbf{a}_{2}+\left(z_{40} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{40} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{40} \,\mathbf{\hat{y}}+c \left(z_{40} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P II |
| $\mathbf{B_{159}}$ | = | $- x_{40} \, \mathbf{a}_{1}+\left(y_{40} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{40} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{40} \,\mathbf{\hat{x}}+b \left(y_{40} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{40} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P II |
| $\mathbf{B_{160}}$ | = | $\left(x_{40} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{40} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{40} \, \mathbf{a}_{3}$ | = | $a \left(x_{40} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{40} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{40} \,\mathbf{\hat{z}}$ | (4a) | P II |
| $\mathbf{B_{161}}$ | = | $x_{41} \, \mathbf{a}_{1}+y_{41} \, \mathbf{a}_{2}+z_{41} \, \mathbf{a}_{3}$ | = | $a x_{41} \,\mathbf{\hat{x}}+b y_{41} \,\mathbf{\hat{y}}+c z_{41} \,\mathbf{\hat{z}}$ | (4a) | P III |
| $\mathbf{B_{162}}$ | = | $- \left(x_{41} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{41} \, \mathbf{a}_{2}+\left(z_{41} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{41} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{41} \,\mathbf{\hat{y}}+c \left(z_{41} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P III |
| $\mathbf{B_{163}}$ | = | $- x_{41} \, \mathbf{a}_{1}+\left(y_{41} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{41} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{41} \,\mathbf{\hat{x}}+b \left(y_{41} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{41} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P III |
| $\mathbf{B_{164}}$ | = | $\left(x_{41} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{41} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{41} \, \mathbf{a}_{3}$ | = | $a \left(x_{41} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{41} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{41} \,\mathbf{\hat{z}}$ | (4a) | P III |
| $\mathbf{B_{165}}$ | = | $x_{42} \, \mathbf{a}_{1}+y_{42} \, \mathbf{a}_{2}+z_{42} \, \mathbf{a}_{3}$ | = | $a x_{42} \,\mathbf{\hat{x}}+b y_{42} \,\mathbf{\hat{y}}+c z_{42} \,\mathbf{\hat{z}}$ | (4a) | P IV |
| $\mathbf{B_{166}}$ | = | $- \left(x_{42} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{42} \, \mathbf{a}_{2}+\left(z_{42} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{42} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{42} \,\mathbf{\hat{y}}+c \left(z_{42} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P IV |
| $\mathbf{B_{167}}$ | = | $- x_{42} \, \mathbf{a}_{1}+\left(y_{42} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{42} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{42} \,\mathbf{\hat{x}}+b \left(y_{42} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{42} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P IV |
| $\mathbf{B_{168}}$ | = | $\left(x_{42} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{42} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{42} \, \mathbf{a}_{3}$ | = | $a \left(x_{42} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{42} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{42} \,\mathbf{\hat{z}}$ | (4a) | P IV |
| $\mathbf{B_{169}}$ | = | $x_{43} \, \mathbf{a}_{1}+y_{43} \, \mathbf{a}_{2}+z_{43} \, \mathbf{a}_{3}$ | = | $a x_{43} \,\mathbf{\hat{x}}+b y_{43} \,\mathbf{\hat{y}}+c z_{43} \,\mathbf{\hat{z}}$ | (4a) | P V |
| $\mathbf{B_{170}}$ | = | $- \left(x_{43} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{43} \, \mathbf{a}_{2}+\left(z_{43} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{43} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{43} \,\mathbf{\hat{y}}+c \left(z_{43} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P V |
| $\mathbf{B_{171}}$ | = | $- x_{43} \, \mathbf{a}_{1}+\left(y_{43} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{43} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{43} \,\mathbf{\hat{x}}+b \left(y_{43} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{43} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P V |
| $\mathbf{B_{172}}$ | = | $\left(x_{43} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{43} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{43} \, \mathbf{a}_{3}$ | = | $a \left(x_{43} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{43} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{43} \,\mathbf{\hat{z}}$ | (4a) | P V |
| $\mathbf{B_{173}}$ | = | $x_{44} \, \mathbf{a}_{1}+y_{44} \, \mathbf{a}_{2}+z_{44} \, \mathbf{a}_{3}$ | = | $a x_{44} \,\mathbf{\hat{x}}+b y_{44} \,\mathbf{\hat{y}}+c z_{44} \,\mathbf{\hat{z}}$ | (4a) | P VI |
| $\mathbf{B_{174}}$ | = | $- \left(x_{44} - \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{44} \, \mathbf{a}_{2}+\left(z_{44} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{44} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{44} \,\mathbf{\hat{y}}+c \left(z_{44} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P VI |
| $\mathbf{B_{175}}$ | = | $- x_{44} \, \mathbf{a}_{1}+\left(y_{44} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{44} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{44} \,\mathbf{\hat{x}}+b \left(y_{44} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c \left(z_{44} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | P VI |
| $\mathbf{B_{176}}$ | = | $\left(x_{44} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{44} - \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{44} \, \mathbf{a}_{3}$ | = | $a \left(x_{44} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{44} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{44} \,\mathbf{\hat{z}}$ | (4a) | P VI |
References
- R. M. Kirchner, R. W. Grosse-Kunstleve, J. J. Pluth, S. T. Wilson, R. W. Broach, and J. V. Smith, The structures of as-synthesized AlPO$_{4}$-53(A), calcined dehydrated AlPO$_{4}$-53(B), and AlPO$_{4}$-53(C), a new phase determined by the FOCUS method, Micro. Meso. Mater. 39, 319–332 (2000), doi:10.1016/S1387-1811(00)00205-5.
First cited in
- N. Anderson, M. J. Mehl, H. Eckert, S. Divilov, X. Campilongo, S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 5. Submitted to Computational Materials Science (2026).
Geometry files
Prototype Generator
aflow --proto=A3BCD14E3_oP176_19_6a_2a_2a_28a_6a --params=$a,b/a,c/a,x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},x_{3},y_{3},z_{3},x_{4},y_{4},z_{4},x_{5},y_{5},z_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7},x_{8},y_{8},z_{8},x_{9},y_{9},z_{9},x_{10},y_{10},z_{10},x_{11},y_{11},z_{11},x_{12},y_{12},z_{12},x_{13},y_{13},z_{13},x_{14},y_{14},z_{14},x_{15},y_{15},z_{15},x_{16},y_{16},z_{16},x_{17},y_{17},z_{17},x_{18},y_{18},z_{18},x_{19},y_{19},z_{19},x_{20},y_{20},z_{20},x_{21},y_{21},z_{21},x_{22},y_{22},z_{22},x_{23},y_{23},z_{23},x_{24},y_{24},z_{24},x_{25},y_{25},z_{25},x_{26},y_{26},z_{26},x_{27},y_{27},z_{27},x_{28},y_{28},z_{28},x_{29},y_{29},z_{29},x_{30},y_{30},z_{30},x_{31},y_{31},z_{31},x_{32},y_{32},z_{32},x_{33},y_{33},z_{33},x_{34},y_{34},z_{34},x_{35},y_{35},z_{35},x_{36},y_{36},z_{36},x_{37},y_{37},z_{37},x_{38},y_{38},z_{38},x_{39},y_{39},z_{39},x_{40},y_{40},z_{40},x_{41},y_{41},z_{41},x_{42},y_{42},z_{42},x_{43},y_{43},z_{43},x_{44},y_{44},z_{44}$