International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. D. ch. 3.1, pp. 352-357

Table 3.1.3.1 

V. Janovecb* and V. Kopskýe

Table 3.1.3.1 | top | pdf |
Point-group symmetry descents associated with irreducible representations

Property tensors that appear in this table: [\varepsilon] enantiomorphism, chirality; [P_i] dielectric polarization; [u_{\mu}] strain; [g_{\mu}] optical activity; [d_{i\mu}] piezoelectric tensor; [A_{i\mu}] electrogyration tensor; [\pi_{\mu\nu}] piezo-optic tensor ([i=1,2,3]; [\mu,\nu=1,2,\ldots,6]). Applications of this table to symmetry analysis of equitranslational phase transitions and to changes of property tensors at ferroic transitions are explained in Section 3.1.3.3[link].

(a) Triclinic parent groups

R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
[F_1] [n_F] [n_f] [n_a] [n_e]
Parent symmetry [\bi G]: [\quad {\bf 1}\quad {\bi C}_{\bf 1}]
No ferroic symmetry descent
Parent symmetry [\bi G]: [\quad{\bf\overline 1}\quad {\bi C}_{\bi i}]
[A_{u}] [{\sf x}_1^-] [1] [C_1] [1] All components of odd parity tensors 2 1 2

(b) Monoclinic parent groups

R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
[F_1] [n_F] [n_f] [n_a] [n_e]
Parent symmetry [\bi G]: [\quad{\bf 2}_{\bi z} \quad {\bi C}_{\bf 2{\bi z}}]
B [{\sf x}_3] 1 [C_1] 1 [P_{1}], [P_{2}]; [u_{4}, u_{5}] 2 2 2
Parent symmetry [\bi G]: [\quad{\bi m}_{\bi z}\quad {\bi C}_{\bi sz}]
[A''] [{\sf x}_3] 1 [C_1] 1 [\varepsilon]; [P_{3}]; [u_{4}, u_{5}] 2 2 2
Parent symmetry [\bi G]: [\quad {\bf 2}_{\bi z}{\bf /}{\bi m}_{\bi z}\quad {\bi C}_{\bf 2{\bi hz}}]
[B_g] [{\sf x}_3^+] [{ {\overline 1}}] [C_i] 1 [u_{4}, u_{5}] 2 2 0
[A_u] [{\sf x}_1^-] [2_z] [C_{2z}] 1 [\varepsilon]; [P_{3}] 2 1 2
[B_u] [{\sf x}_3^-] [m_z] [C_{sz}] 1 [P_{1}, P_2] 2 1 2

(c) Orthorhombic parent groups

R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
[F_1] [n_F] [n_f] [n_a] [n_e]
Parent symmetry [\bi G]: [\quad {\bf 2_{\bi x}2_{\bi y}2_{\bi z}\quad {\bi D}_{2}}]
[B_{1g}] [{\sf x}_{2}] [2_{z}] [C_{2z}] 1 [P_{3}]; [u_{6}] 2 2 2
[B_{3g}] [{\sf x}_{3}] [2_{x}] [C_{2x}] 1 [P_{1}]; [u_{4}] 2 2 2
[B_{2g}] [{\sf x}_{4}] [2_{y}] [C_{2y}] 1 [P_{2}]; [u_{5}] 2 2 2
Parent symmetry [\bi G]: [\quad{\bi m}_{\bi x}{\bi m}_{\bi y}{\bf 2}_{\bi z}\quad {\bi C}_{\bf 2{\bi vz}}]
[A_{2}] [{\sf x}_{2}] [2_{z}] [C_{2z}] 1 [u_{6}] 2 2 1
[B_{2}] [{\sf x}_{3}] [m_{x}] [C_{sx}] 1 [P_2]; [u_{4}] 2 2 2
[B_{1}] [{\sf x}_{4}] [m_{y}] [C_{sy}] 1 [P_{1}]; [u_5] 2 2 2
Parent symmetry [\bi G]: [\quad{\bi m_{x}m_{y}m_{z}\quad D_{\bf 2{\bi h}}}]
[B_{1g}] [{\sf x}_{2}^{+}] [2_{z}/m_{z}] [C_{2hz}] 1 [u_{6}] 2 2 0
[B_{3g}] [{\sf x}_{3}^{+}] [2_{x}/m_{x}] [C_{2hx}] 1 [u_{4}] 2 2 0
[B_{2g}] [{\sf x}_{4}^{+}] [2_{y}/m_{y}] [C_{2hy}] 1 [u_{5}] 2 2 0
[A_{1u}] [{\sf x}_{1}^{-}] [2_{x}2_{y}2_{z}] [D_{2}] 1 [\varepsilon]; [g_{1}], [g_{2}], [g_{3}]; [d_{14}], [d_{25}], [d_{36}] 2 1 0
[B_{1u}] [{\sf x}_{2}^{-}] [m_{x}m_{y}2_{z}] [C_{2vz}] 1 [P_{3}] 2 1 2
[B_{3u}] [{\sf x}_{3}^{-}] [2_{x}m_{y}m_{z}] [C_{2vx}] 1 [P_{1}] 2 1 2
[B_{2u}] [{\sf x}_{4}^{-}] [m_{x}2_{y}m_{z}] [C_{2vy}] 1 [P_{2}] 2 1 2

(d) Tetragonal parent groups

R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
[F_1] [n_F] [n_f] [n_a] [n_e]
Parent symmetry [\bi G]: [\quad{\bf 4}_{\bi z}\quad {\bi C}_{\bf 4{\bi z}}]
B [{\sf x}_{3}] [2_{z}] [C_{2z}] 1 [\delta u_{1}=-\delta u_{2}], [u_{6}] 2 2 1
[^{1}E\, \oplus\, ^{2}\kern-1pt E] [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{1},P_{2})]; [(u_{4},-u_{5})] 4 4 4
(Li)                
Parent symmetry [\bi G]: [\quad{\bf\overline 4}_{\bi z}\quad {\bi S}_{\bf 4{\bi z}}]
B [{\sf x}_{3}] [2_{z}] [C_{2z}] 1 [\varepsilon]; [P_{3}]; [\delta u_{1}=-\delta u_{2}], [u_{6}] 2 2 2
[^{1}E\, \oplus \, ^{2}\kern-1pt E] [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{1},-P_{2})]; [(u_{4},-u_{5})] 4 4 4
Parent symmetry [\bi G]: [\quad {\bf 4_{\bi z}/{\bi m}_{\bi z}\quad {\bi C}_{4{\bi hz}}}]
[B_{g}] [{\sf x}_{3}^{+}] [2_{z}/m_{z}] [C_{2hz}] 1 [\delta u_{1}=-\delta u_{2}], [u_{6}] 2 2 0
[A_{u}] [{\sf x}_{1}^{-}] [4_{z}] [C_{4z}] 1 [\varepsilon]; [P_{3}] 2 1 2
[B_{u}] [{\sf x}_{3}^{-}] [{\overline 4}_{z}] [S_{4z}] 1 [g_{1}=-g_{2}], [g_{6}]; [d_{31}=-d_{32}], [d_{36}], [d_{14}=d_{25}], [d_{15}=-d_{24}] 2 1 0
[^{1}E_{g}\, \oplus \, ^{2}\kern-1pt E_{g}] [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 [(u_{4},-u_{5})] 4 4 0
[^{1}E_{u} \, \oplus \, ^{2}\kern-1pt E_{u}] [(x_{1}^{-},y_{1}^{-})] [m_{z}] [C_{sz}] 1 [(P_{1},P_{2})] 4 2 4
Parent symmetry [\bi G]: [\quad{\bf 4_{\bi z}2_{\bi x}2_{\bi xy}\quad {\bi D}_{4{\bi z}}}]
[A_{2}] [{\sf x}_{2}] [4_{z}] [C_{4z}] 1 [P_{3}] 2 1 2
[B_{1}] [{\sf x}_{3}] [2_{x}2_{y}2_{z}] [D_{2}] 1 [\delta u_{1}=-\delta u_{2}] 2 2 0
[B_{2}] [{\sf x}_{4}] [2_{x{\overline y}}2_{xy}2_{z}] [{\hat D}_{2z}] 1 [u_{6}] 2 2 0
E [(x_{1},0)] [2_{x}] [C_{2x}] 2 [P_{1}]; [u_{4}] 4 4 4
  [(x_{1},x_{1})] [2_{xy}] [C_{2xy}] 2 [P_{1}=P_{2}]; [u_{4}=-u_{5}] 4 4 4
(Li) [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{1},P_{2})]; [(u_{4},-u_{5})] 8 8 8
Parent symmetry [\bi G]: [\quad {\bf 4_{\bi z}{\bi m}_{\bi x}{\bi m}_{\bi xy}\quad {\bi C}_{4{\bi vz}}}]
[A_{2}] [{\sf x}_{2}] [4_{z}] [C_{4z}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 1
[B_{1}] [{\sf x}_{3}] [m_{x}m_{y}2_{z}] [C_{2vz}] 1 [\delta u_{1}=-\delta u_{2}] 2 2 1
[B_{2}] [{\sf x}_{4}] [m_{x{\overline y}}m_{xy}2_{z}] [{\hat C}_{2vz}] 1 [u_{6}] 2 2 1
E [(x_{1},0)] [m_{x}] [C_{sx}] 2 [P_{2}]; [u_{4}] 4 4 4
  [(x_{1},x_{1})] [m_{xy}] [C_{sxy}] 2 [P_{2}=-P_{1}]; [u_{4}=-u_{5}] 4 4 4
  [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{2},-P_{1})]; [(u_{4},-u_{5})] 8 8 8
Parent symmetry [\bi G]: [\quad{\bf\overline 4}_{\bi z}{\bf 2}_{\bi x}{\bi m}_{\bi xy}\quad {\bi D}_{\bf 2{\bi dz}}]
[A_{2}] [{\sf x}_{2}] [{\overline 4}_{z}] [S_{4z}] 1 [g_{6}]; [d_{31}=-d_{32}], [d_{15}=-d_{24}] 2 1 0
[B_{1}] [{\sf x}_{3}] [2_{x}2_{y}2_{z}] [D_{2}] 1 [\varepsilon]; [\delta u_{1}=-\delta u_{2}] 2 2 0
[B_{2}] [{\sf x}_{4}] [m_{x{\overline y}}m_{xy}2_{z}] [{\hat C}_{2vz}] 1 [P_{3}]; [u_{6}] 2 2 2
E [(x_{1},0)] [2_{x}] [C_{2x}] 2 [P_{1}]; [u_{4}] 4 4 4
  [(x_{1},x_{1})] [m_{xy}] [C_{sxy}] 2 [P_{1}=-P_{2}]; [u_{4}=-u_{5}] 4 4 4
  [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{1},-P_{2})]; [(u_{4},-u_{5})] 8 8 8
Parent symmetry [\bi G]: [\quad{\bf\overline 4}_{\bi z}{\bi m}_{\bi x}{\bf 2}_{\bi xy}\quad {\hat {\bi D}}_{\bf 2{\bi dz}}]
[A_{2}] [{\sf x}_{2}] [{\overline 4}_{z}] [S_{4z}] 1 [g_{1}=-g_{2}]; [d_{36}], [d_{14}=d_{25}] 2 1 0
[B_{2}] [{\sf x}_{3}] [m_{x}m_{y}2_{z}] [C_{2vz}] 1 [P_{3}]; [\delta u_{1}=-\delta u_{2}] 2 2 2
[B_{1}] [{\sf x}_{4}] [2_{x{\overline y}}2_{xy}2_{z}] [{\hat D}_{2z}] 1 [\varepsilon]; [u_{6}] 2 2 0
E [(x_{1},0)] [m_{x}] [C_{sx}] 2 [P_{2}]; [u_{4}] 4 4 4
  [(x_{1},x_{1})] [2_{xy}] [C_{2xy}] 2 [P_{2}=P_{1}]; [u_{4}=-u_{5}] 4 4 4
  [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{2},P_{1})]; [(u_{4},-u_{5})] 8 8 8
Parent symmetry [\bi G]: [\quad{\bf 4_{\bi z}/{\bi m}_{\bi z}{\bi m}_{\bi x}{\bi m}_{\bi xy}\quad {\bi D}_{\bf 4{\bi hz}}}]
[A_{2g}] [{\sf x}_{2}^{+}] [4_{z}/m_{z}] [C_{4hz}] 1 [A_{31}=A_{32}], [A_{33}], [A_{15}=A_{24}] 2 1 0
[B_{1g}] [{\sf x}_{3}^{+}] [m_{x}m_{y}m_{z}] [D_{2h}] 1 [\delta u_{1}=-\delta u_{2}] 2 2 0
[B_{2g}] [{\sf x}_{4}^{+}] [m_{x{\overline y}}m_{xy}m_{z}] [{\hat D}_{2hz}] 1 [u_{6}] 2 2 0
[A_{1u}] [{\sf x}_{1}^{-}] [4_{z}2_{x}2_{xy}] [D_{4z}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 0
[A_{2u}] [{\sf x}_{2}^{-}] [4_{z}m_{x}m_{xy}] [C_{4vz}] 1 [P_{3}] 2 1 2
[B_{1u}] [{\sf x}_{3}^{-}] [{\overline 4}_{z}2_{x}m_{xy}] [D_{2dz}] 1 [g_{1}=-g_{2}]; [d_{14}=d_{25}], [d_{36}] 2 1 0
[B_{2u}] [{\sf x}_{4}^{-}] [{\overline 4}_{z}m_{x}2_{xy}] [{\hat D}_{2dz}] 1 [g_{6}]; [d_{31}=-d_{32}], [d_{15}=-d_{24}] 2 1 0
[E_{g}] [(x_{1}^{+},0)] [2_{x}/m_{x}] [C_{2hx}] 2 [u_{4}] 4 4 0
  [(x_{1}^{+},x_{1}^{+})] [2_{xy}/m_{xy}] [C_{2hxy}] 2 [u_{4}=-u_{5}] 4 4 0
  [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 [(u_{4},-u_{5})] 8 8 0
[E_{u}] [(x_{1}^{-},0)] [2_{x}m_{y}m_{z}] [C_{2vx}] 2 [P_{1}] 4 2 4
  [(x_{1}^{-},x_{1}^{-})] [m_{x{\overline y}}2_{xy}m_{z}] [C_{2vxy}] 2 [P_{1}=P_{2}] 4 2 4
  [(x_{1}^{-},y_{1}^{-})] [m_{z}] [C_{sz}] 1 [(P_{1},P_{2})] 8 8 8

(e) Trigonal parent groups

R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
[F_1] [n_F] [n_f] [n_a] [n_e]
Parent symmetry [\bi G]: [\quad\bf 3_{\bi z}\quad {\bi C}_{3}]
E [(x_{1},y_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]) 3 3 3
          ([u_{1}-u_{2}], [-2u_{6}]), ([u_{4}], [-u_{5}])      
(La, Li)         [\delta u_{1}=-\delta u_{2}]      
Parent symmetry [\bi G]: [\quad\bf{\overline 3}_{\bi z}\quad {\bi C}_{3{\bi i}}]
[A_{u}] [{\sf x}_{1}^{-}] [3_{z}] [C_{3}] 1 [\varepsilon]; [P_{3}] 2 1 2
[E_{g}] [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{1}-u_{2}], [-2u_{6}]), ([u_{4}], [-u_{5}]) 3 3 0
(La)         [\delta u_{1}=-\delta u_{2}]      
[E_{u}] [(x_{1}^{-},y_{1}^{-})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]) 6 3 6
Parent symmetry [\bi G]: [\quad\bf 3_{\bi z}2_{\bi x}\quad {\bi D}_{3{\bi x}}]
[A_{2}] [{\sf x}_{2}] [3_{z}] [C_{3}] 1 [P_{3}] 2 1 2
E [(x_{1}, 0)] [2_{x}] [C_{2x}] 3 [P_{1}]; [\delta u_{1}= -\delta u_{2}], [u_{4}] 3 3 3
(La, Li) [(x_{1}, y_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]); ([u_{1}-u_{2}], [-2u_{6}]), ([u_{4}], [-u_{5}]) 6 6 6
Parent symmetry [\bi G]: [\quad\bf 3_{\bi z}{\bi m_{x}\quad C}_{3{\bi vx}}]
[A_{2}] [{\sf x}_{2}] [3_{z}] [C_{3}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{11}=-d_{12}=-d_{26}], [d_{14}=-d_{25}] 2 1 1
E [(x_{1}, 0)] [m_{x}] [C_{sx}] 3 [P_{2}]; [\delta u_{1}=-\delta u_{2}], [u_{4}] 3 3 3
(La) [(x_{1}, y_{1})] [1] [C_{1}] 1 ([P_{2}], [-P_{1}]); ([u_{1}-u_{2}], [-2u_{6}]), ([u_{4}], [-u_{5}]) 6 6 6
Parent symmetry [\bi G]: [\quad\bf{\overline 3}_{\bi z}{\bi m_{x}\quad D}_{3{\bi dx}}]
[A_{2g}] [{\sf x}_{2}^{+}] [{\overline 3}_{z}] [C_{3i}] 1 [A_{22}=-A_{21}=-A_{16}], [A_{31}=A_{32}], [A_{33}], [A_{15}=A_{24}] 2 1 0
[A_{1u}] [{\sf x}_{1}^{-}] [3_{z}2_{x}] [D_{3x}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{11}=-d_{12}=-d_{26}], [d_{14}=-d_{25}] 2 1 0
[A_{2u}] [{\sf x}_{2}^{-}] [3_{z}m_{x}] [C_{3vx}] 1 [P_{3}] 2 1 2
[E_{g}] [(x_{1}^{+}, 0)] [2_{x}/m_{x}] [C_{2hx}] 3 [\delta u_{1}=-\delta u_{2}], [u_{4}] 3 3 0
(La) [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{1}-u_{2}], [-2u_{6}]), ([u_{4}], [-u_{5}]) 6 6 0
[E_{u}] [(0, y_{1}^{-})] [m_{x}] [C_{sx}] 3 [P_{2}] 6 3 6
  [(x_{1}^{-}, 0)] [2_{x}] [C_{2x}] 3 [P_{1}] 6 3 6
  [(x_{1}^{-},y_{1}^{-})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]) 12 6 12
Parent symmetry [\bi G]: [\quad\bf 3_{\bi z}2_{\bi y}\quad {\bi D}_{3{\bi y}}]
[A_{2}] [{\sf x}_{2}] [3_{z}] [C_{3}] 1 [P_{3}] 2 1 2
E [(0, y_{1})] [2_{y}] [C_{2y}] 3 [P_{2}]; [\delta u_{1}=-\delta u_{2}], [u_{5}] 3 3 3
(La, Li) [(x_{1}, y_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]); ([2u_{6}], [u_{1}-u_{2}]), ([u_{4}], [-u_{5}]) 6 6 6
Parent symmetry [\bi G]: [\quad\bf 3_{\bi z}{\bi m}_{\bi y}\quad {\bi C}_{3{\bi vy}}]
[A_{2}] [{\sf x}_{2}] [3_{z}] [C_{3}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{22}=-d_{21}=-d_{16}], [d_{14}=-d_{25}] 2 1 1
E [(0, y_{1})] [m_{y}] [C_{sy}] 3 [P_{1}]; [\delta u_{1}=-\delta u_{2}], [u_{5}] 3 3 3
(La) [(x_{1}, y_{1})] [1] [C_{1}] 1 ([P_{2}], [-P_{1}]); ([2u_{6}], [u_{1}-u_{2}]), ([u_{4}], [-u_{5}]) 6 6 6
Parent symmetry [\bi G]: [\quad\bf{\overline 3}_{\bi z}{\bi m}_{\bi y}\quad {\bi D}_{3{\bi dy}}]
[A_{2g}] [{\sf x}_{2}^{+}] [{\overline 3}_{z}] [C_{3i}] 1 [A_{11}=-A_{12}=-A_{26}], [A_{31}=A_{32}], [A_{33}], [A_{15}=A_{24}] 2 1 0
[A_{1u}] [{\sf x}_{1}^{-}] [3_{z}2_{y}] [D_{3y}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{22}=-d_{21}=-d_{16}], [d_{14}=-d_{25}] 2 1 0
[A_{2u}] [{\sf x}_{2}^{-}] [3_{z}m_{y}] [C_{3vy}] 1 [P_{3}] 2 1 2
[E_{g}] [(0, y_{1}^{+})] [2_{y}/m_{y}] [C_{2hy}] 3 [\delta u_{1}=-\delta u_{2}], [u_{5}] 3 3 0
(La) [(x_{1}^{+}, y_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([2u_{6}], [u_{1}-u_{2}]), ([u_{4}], [-u_{5}]) 6 6 0
[E_{u}] [(0, y_{1}^{-})] [2_{y}] [C_{2y}] 3 [P_{2}] 6 3 6
  [(x_{1}^{-}, 0)] [m_{y}] [C_{sy}] 3 [P_{1}] 6 3 6
  [(x_{1}^{-}, y_{1}^{-})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]) 12 6 12

(f) Hexagonal parent groups

Covariants with standardized labels and conversion equations:[\displaylines{{g}_{1}^{-}=g_{1}+g_{2}; \quad g_{2x}^{-}=g_{1}-g_{2}, \quad g_{2y}^{-}=2g_{6}\cr g_{1}=\textstyle{{1}\over{2}}({g}_{1}^{-}+g_{2x}^{-}), \quad g_{2}=\textstyle{{1}\over{2}}({g}_{1}^{-}-g_{2x}^{-}); \quad \delta g_{1}=-\delta g_{2}= \textstyle{{1}\over{2}}g_{2x}^{-}\cr {d}_{1}^{-}=d_{14}-d_{25}; \quad d_{2x,2}^{-}=d_{14}+d_{25}, \quad d_{2y,2}^{-}=d_{24}-d_{15}\cr {d}_{2,1}^{-}=d_{31}+d_{32}; \quad d_{2x,1}^{-}=2d_{36}, \quad d_{2y,1}^{-}=d_{32}-d_{31}\cr d_{14}=\textstyle{{1}\over{2}}({d}_{1}^{-}+d_{2x,2}^{-}), \quad d_{25}=\textstyle{{1}\over{2}}(-{d}_{1}^{-}+d_{2x,2}^{-}); \quad \delta d_{14}=\delta d_{25}=\textstyle{{1}\over{2}}d_{2x}^{-}\cr d_{36}=\textstyle{{1}\over{2}}d_{2x,1}^{-}, \quad d_{31}=\textstyle{{1}\over{2}}({d}_{2,1}^{-}-d_{2y,1}^{-}); \quad d_{32}=\textstyle{{1}\over{2}}({d}_{2,1}^{-}+d_{2y,1}^{-}).}]

R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
[F_1] [n_F] [n_f] [n_a] [n_e]
Parent symmetry [\bi G]: [\quad \bf 6_{\bi z}\quad {\bi C}_{6}]
B [{\sf x}_{3}] [3_{z}] [C_{3}] 1 [d_{11}=-d_{12}=-d_{26}], [d_{22}=-d_{21}=-d_{16}] 2 1 1
[E_{2}] [(x_{2},y_{2})] [2_{z}] [C_{2z}] 1 ([u_{1}-u_{2}], [2u_{6}]) [\delta u_{1}=-\delta u_{2}] 3 3 1
(La, Li)                
[E_{1}] [(x_{1},y_{1})] [1] [C_{1}] 1 ([P_{1}],[P_{2}]) 6 6 6
(Li)         ([u_{4}], [-u_{5}])      
Parent symmetry [\bi G]: [\quad\bf{\overline 6}_{\bi z}\quad {\bi C}_{3{\bi h}}]
[A''] [{\sf x}_{3}] [3_{z}] [C_{3}] 1 [\varepsilon]; [P_{3}] 2 1 2
[E'] [(x_{2},y_{2})] [m_{z}] [C_{sz}] 1 ([P_{2}], [P_{1}]) 3 3 3
(La)         ([u_{1}-u_{2}], [2u_{6}]) [\delta u_{1}=-\delta u_{2}]      
[E''] [(x_{1},y_{1})] [1] [C_{1}] 1 ([u_{4}], [-u_{5}]) 6 6 6
Parent symmetry [\bi G]: [\quad\bf 6_{\bi z}/{\bi m}_{\bi z}\quad {\bi C}_{6{\bi h}}]
[B_{g}] [{\sf x}_{3}^{+}] [{\overline 3}_{z}] [C_{3i}] 1 [A_{11}=-A_{12}=-A_{26}], [A_{22}=-A_{21}=-A_{16}] 2 1 0
[A_{u}] [{\sf x}_{1}^{-}] [6_{z}] [C_{6}] 1 [\varepsilon]; [P_{3}] 2 1 2
[B_{u}] [{\sf x}_{3}^{-}] [{\overline 6}_{z}] [C_{3h}] 1 [d_{11}=-d_{12}=-d_{26}], [d_{22}=-d_{21}=-d_{16}] 2 1 0
[E_{2g}] [(x_{2}^{+},y_{2}^{+})] [2_{z}/m_{z}] [C_{2hz}] 1 ([u_{1}-u_{2}], [2u_{6}]) [\delta u_{1}=- \delta u_{2}] 3 3 0
(La)                
[E_{1g}] [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{4}], [-u_{5}]) 6 6 0
[E_{2u}] [(x_{2}^{-},y_{2}^{-})] [2_{z}] [C_{2z}] 1 ([g_{1}-g_{2}], [2g_{6}]) [g_{1}=-g_{2}], [g_{6}] 6 3 2
          ([2d_{36}], [d_{32}-d_{31}]) [d_{32}=-d_{31}], [d_{36}]      
          ([d_{14}+d_{25}], [d_{24}-d_{15}]) [d_{14}=d_{25}], [d_{24}=-d_{15}]      
[E_{1u}] [(x_{1}^{-},y_{1}^{-})] [m_{z}] [C_{sz}] 1 ([P_{1}], [P_{2}]) 6 3 6
Parent symmetry [\bi G]: [\quad\bf 6_{\bi z}2_{\bi x}2_{\bi y}\quad {\bi D}_{6}]
[A_{2}] [{\sf x}_{2}] [6_{z}] [C_{6}] 1 [P_{3}] 2 1 2
[B_{1}] [{\sf x}_{3}] [3_{z}2_{x}] [D_{3x}] 1 [d_{11}=-d_{12}=-d_{26}] 2 1 0
[B_{2}] [{\sf x}_{4}] [3_{z}2_{y}] [D_{3y}] 1 [d_{22}=-d_{21}=-d_{16}] 2 1 0
[E_{2}] [(x_{2},0)] [2_{x}2_{y}2_{z}] [D_{2}] 3 [\delta u_{1}=-\delta u_{2}] 3 3 0
(La, Li) [(x_{2},y_{2})] [2_{z}] [C_{2z}] 1 ([u_{1}-u_{2}], [2u_{6}]) 6 6 2
[E_{1}] [(x_{1},0)] [2_{x}] [C_{2x}] 3 [P_{1}]; [u_{4}] 6 6 6
  [(0,y_{1})] [2_{y}] [C_{2y}] 3 [P_{2}]; [u_{5}] 6 6 6
(Li) [(x_{1},y_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]); ([u_{4}], [-u_{5})] 12 12 12
Parent symmetry [\bi G]: [\quad\bf 6_{\bi z}{\bi m_{x}m_{y}\quad C}_{6{\bi v}}]
[A_{2}] [{\sf x}_{2}] [6_{z}] [C_{6}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 1
[B_{2}] [{\sf x}_{3}] [3_{z}m_{x}] [C_{3vx}] 1 [d_{22}=-d_{21}=-d_{16}] 2 1 1
[B_{1}] [{\sf x}_{4}] [3_{z}m_{y}] [C_{3vy}] 1 [d_{11}=-d_{12}=-d_{26}] 2 1 1
[E_{2}] [(x_{2},0)] [m_{x}m_{y}2_{z}] [C_{2vz}] 3 [\delta u_{1}=-\delta u_{2}] 3 3 1
(La) [(x_{2},y_{2})] [2_{z}] [C_{2z}] 1 ([u_{1}-u_{2}], [2u_{6}]) 6 6 1
[E_{1}] [(x_{1},0)] [m_{x}] [C_{sx}] 3 [P_{2}]; [u_{4}] 6 6 6
  [(0,y_{1})] [m_{y}] [C_{sy}] 3 [P_{1}]; [u_{5}] 6 6 6
  [(x_{1},y_{1})] [1] [C_{1}] 1 ([P_{2}], [-P_{1}]); ([u_{4}], [-u_{5}]) 12 12 12
Parent symmetry [\bi G]: [\quad\bf{\overline 6}_{\bi z}2_{\bi x}{\bi m_{y}\quad D}_{3{\bi h}}]
[A_{2}'] [{\sf x}_{2}] [{\overline 6}_{z}] [C_{3h}] 1 [d_{22}=-d_{21}=-d_{16}] 2 1 0
[A_{1}''] [{\sf x}_{3}] [3_{z}2_{x}] [D_{3x}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 0
[A_{2}''] [{\sf x}_{4}] [3_{z}m_{y}] [C_{3vy}] 1 [P_{3}] 2 1 2
[E'] [(x_{2},0)] [2_{x}m_{y}m_{z}] [C_{2vx}] 3 [P_{1}]; [\delta u_{1}=-\delta u_{2}] 3 3 3
(La) [(x_{2},y_{2})] [m_{z}] [C_{sz}] 1 ([P_{1}],[-P_{2}]); ([u_{1}-u_{2}], [2u_{6}]) 6 6 6
[E''] [(x_{1},0)] [2_{x}] [C_{2x}] 3 [u_{4}] 6 6 3
  [(0,y_{1})] [m_{y}] [C_{sy}] 3 [u_{5}] 6 6 6
  [(x_{1},y_{1})] [1] [C_{1}] 1 ([u_{4}], [-u_{5}]) 12 12 12
Parent symmetry [\bi G]: [\quad{\bf\overline 6}_{\bi z}{\bi m}_{\bi x}{\bf 2}_{\bi y}\quad {\hat {\bi D}}_{{\bf3}{\bi h}}]
[A_{2}'] [{\sf x}_{2}] [{\overline 6}_{z}] [C_{3h}] 1 [d_{11}=-d_{12}=-d_{26}] 2 1 0
[A_{2}''] [{\sf x}_{3}] [3_{z}m_{x}] [C_{3vx}] 1 [P_{3}] 2 1 2
[A_{1}'] [{\sf x}_{4}] [3_{z}2_{y}] [D_{3y}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 0
[E'] [(x_{2},0)] [m_{x}2_{y}m_{z}] [C_{2vy}] 3 [P_{2}]; [\delta u_{1}=-\delta u_{2}] 3 3 3
(La) [(x_{2},y_{2})] [m_{z}] [C_{sz}] 1 ([P_{2}], [P_{1}]); ([u_{1}-u_{2}], [2u_{6}]) 6 6 6
[E''] [(x_{1},0)] [m_{x}] [C_{sx}] 3 [u_{4}] 6 6 6
  [(0,y_{1})] [2_{y}] [C_{2y}] 3 [u_{5}] 6 6 3
  [(x_{1},y_{1})] [1] [C_{1}] 1 ([u_{4}], [-u_{5}]) 12 12 12
Parent symmetry [\bi G]: [\quad\bf 6_{\bi z}/{\bi m_{z}m_{x}m_{y}\quad D}_{6{\bi h}}]
[A_{2g}] [{\sf x}_{2}^{+}] [6_{z}/m_{z}] [C_{6h}] 1 [A_{31}=A_{32}], [A_{33}], [A_{15}=A_{24}] 2 1 0
[B_{1g}] [{\sf x}_{3}^{+}] [{\overline 3}_{z}m_{x}] [D_{3dx}] 1 [A_{11}=-A_{12}=-A_{26}] 2 1 0
[B_{2g}] [{\sf x}_{4}^{+}] [{\overline 3}_{z}m_{y}] [D_{3dy}] 1 [A_{22}=-A_{21}=-A_{16}] 2 1 0
[A_{1u}] [{\sf x}_{1}^{-}] [6_{z}2_{x}2_{y}] [D_{6}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 0
[A_{2u}] [{\sf x}_{2}^{-}] [6_{z}m_{x}m_{y}] [C_{6v}] 1 [P_{3}] 2 1 2
[B_{1u}] [{\sf x}_{3}^{-}] [{\overline 6}_{z}2_{x}m_{y}] [D_{3h}] 1 [d_{11}=-d_{12}=-d_{26}] 2 1 0
[B_{2u}] [{\sf x}_{4}^{-}] [{\overline 6}_{z}m_{x}2_{y}] [{\hat D}_{3h}] 1 [d_{22}=-d_{21}=-d_{16}] 2 1 0
[E_{2g}] [(x_{2}^{+},0)] [m_{x}m_{y}m_{z}] [D_{2h}] 3 [\delta u_{1}=-\delta u_{2}] 3 3 0
(La) [(x_{2}^{+},y_{2}^{+})] [2_{z}/m_{z}] [C_{2hz}] 1 ([u_{1}-u_{2}], [2u_{6}]) 6 6 0
[E_{1g}] [(x_{1}^{+},0)] [2_{x}/m_{x}] [C_{2hx}] 3 [u_{4}] 6 6 0
  [(0,y_{1}^{+})] [2_{y}/m_{y}] [C_{2hy}] 3 [u_{5}] 6 6 0
  [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{4}], [-u_{5}]) 12 12 0
[E_{1u}] [(x_{1}^{-},0)] [2_{x}m_{y}m_{z}] [C_{2vx}] 3 [P_{1}] 6 3 6
  [(0,y_{1}^{-})] [m_{x}2_{y}m_{z}] [C_{2vy}] 3 [P_{2}] 6 3 6
  [(x_{1}^{-},y_{1}^{-})] [m_{z}] [C_{sz}] 1 ([P_{1}], [P_{2}]) 12 6 12
[E_{2u}] [(x_{2}^{-},0)] [2_{x}2_{y}2_{z}] [D_{2}] 3 [\delta g_{1}=-\delta g_{2}]; [d_{36}], [\delta d_{14}=\delta d_{25}] 6 3 0
  [(0,y_{2}^{-})] [m_{x}m_{y}2_{z}] [C_{2vz}] 3 [g_{6}]: [d_{32}=-d_{31}], [d_{24}=-d_{15}] 6 3 2
  [(x_{2}^{-},y_{2}^{-})] [2_{z}] [C_{2z}] 1 ([g_{1}-g_{2}], [2g_{6}]); [(2d_{36}, d_{32}-d_{31})], [(d_{14}+d_{25}, d_{24}-d_{15})] 12 6 2

(g) Cubic parent groups

Covariants with standardized labels and conversion equations:[\displaylines{ u_{3x}=u_{3x}^{+}=u_{3}-a(u_{1}+u_{2}); \quad u_{3y}=u_{3y}^{+}=b(u_{1}-u_{2})\cr \delta u_{1}=-\textstyle{{1}\over{3}}u_{3x}^{+}+ {\textstyle{1}\over{\sqrt 3}}u_{3y}^{+}; \quad \delta u_{2}=-\textstyle{{1}\over{3}}u_{3x}^{+}-\textstyle{{1}\over{\sqrt 3}}u_{3y}^{+}; \quad \delta u_{3}=\textstyle{{2}\over{3}}u_{3x}^{+}\cr {g}_{1}^{-}=g_{1}+g_{2}+g_{3}; \quad g_{3x}^{-}=g_{3}-a(g_{1}+g_{2}); \quad g_{3y}^{-}=b(g_{1}-g_{2}) \cr g_{1}=\textstyle{{1}\over{3}}{g}_{1}^{-}-\textstyle{{1}\over{3}}g_{3x}^{-}+ \textstyle{{1}\over{\sqrt 3}}g_{3y}^{-}; \quad g_{2}=\textstyle{{1}\over{3}}{g}_{1}^{-}-\textstyle{{1}\over{3}}g_{3x}^{-}- \textstyle{{1}\over{\sqrt 3}}g_{3y}^{-}; \quad g_{3}=\textstyle{{1}\over{3}}{g}_{1}^{-}+\textstyle{{2}\over{3}}g_{3x}^{-}\cr {d}_{1}^{-}=d_{14}+d_{25}+d_{36}; \quad d_{3x}^{-}=b(d_{14}-d_{25}), \quad d_{3y}^{-}=a(d_{14}+d_{25})-d_{36}\cr d_{14}=\textstyle{{1}\over{3}}{d}_{1}^{-}+\textstyle{{1}\over{\sqrt 3}}d_{3x}^{-}+ \textstyle{{1}\over{3}}d_{3y}^{-}; \quad d_{25}=\textstyle{{1}\over{3}}{d}_{1}^{-}-\textstyle{{1}\over{\sqrt 3}}d_{3x}^{-}+ \textstyle{{1}\over{3}}d_{3y}^{-}; \quad d_{36}=\textstyle{{1}\over{3}}{d}_{1}^{-}-\textstyle{{2}\over{3}}d_{3y}^{-}\cr d_{1x}=d_{13}-d_{12}; \quad d_{1y}=d_{21}-d_{23}; \quad d_{1z}=d_{32}-d_{31}\cr d_{2x}=d_{13}+d_{12}; \quad d_{2y}=d_{21}+d_{23}; \quad d_{2z}=d_{32}+d_{31}\cr d_{13}=\textstyle{{1}\over{2}}(d_{1x}+d_{2x}); \quad d_{21}=\textstyle{{1}\over{2}}(d_{1y}+d_{2y}); \quad d_{32}=\textstyle{{1}\over{2}}(d_{1z}+d_{2z})\cr d_{12}=\textstyle{{1}\over{2}}(d_{2x}-d_{1x}); \quad d_{23}=\textstyle{{1}\over{2}}(d_{2y}-d_{1y}); \quad d_{31}=\textstyle{{1}\over{2}}(d_{2z}-d_{1z})}]

[a=\textstyle{{1}\over{2}}], [b=\textstyle{{\sqrt{3}}\over {2}}], [\pi^a_{\mu\nu} = (\pi_{\mu\nu}-\pi_{\nu\mu})], [\mu=1,2,\ldots,6], [\nu=1,2,\ldots,6].

R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
[F_1] [n_F] [n_f] [n_a] [n_e]
Parent symmetry [\bi G]: [\quad\bf 23\quad {\bi T}]
E [(x_{3},y_{3})] [2_{x}2_{y}2_{z}] [D_{2}] 1 [[u_{3}-a(u_{1}+u_{2})], [b(u_{1}-u_{2})]] 3 3 0
(La)         [\delta u_{1}+\delta u_{2}+\delta u_{3}=0]      
T [(0,0,z_{1})] [2_{z}] [C_{2z}] 3 [P_{3}]; [u_{6}] 6 6 6
  [(x_{1},x_{1},x_{1})] [3_{p}] [C_{3p}] 4 [P_{1}=P_{2}=P_{3}]; [u_{4}=u_{5}=u_{6}] 4 4 4
(La, Li) [(x_{1},y_{1},z_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}], [P_{3}]); ([u_{4}], [u_{5}], [u_{6}]) 12 12 12
Parent symmetry [\bi G]: [\quad\bi m{\bf\overline 3}\quad T_{h}]
[A_{u}] [{\sf x}_{1}^{-}] [23] T 1 [\varepsilon]; [g_{1}=g_{2}=g_{3}]; [d_{14}=d_{25}=d_{36}] 2 1 0
[E_{g}] [(x_{3}^{+},y_{3}^{+})] [m_{x}m_{y}m_{z}] [D_{2h}] 1 [[u_{3}-a(u_{1}+u_{2})], [b(u_{1}-u_{2})]] 3 3 0
(La)         [\delta u_{1}+\delta u_{2}+\delta u_{3}=0]      
[E_{u}] [(x_{3}^{-},y_{3}^{-})] [2_{x}2_{y}2_{z}] [D_{2}] 1 [[g_{3}-a(g_{1}+g_{2})], [b(g_{1}-g_{2})]] 6 3 0
          [\delta g_{1}+\delta g_{2}+\delta g_{3}=0]      
          [[b(d_{14}-d_{25})], [a(d_{14}+d_{25})-d_{36}]]      
          [\delta d_{14}+\delta d_{25}+\delta d_{36}=0]      
[T_{g}] [(0,0,z_{1}^{+})] [2_{z}/m_{z}] [C_{2hz}] 3 [u_{6}] 6 6 0
  [(x_{1}^{+},x_{1}^{+},x_{1}^{+})] [{\overline 3}_{p}] [C_{3ip}] 4 [u_{4}=u_{5}=u_{6}] 4 4 0
(La) [(x_{1}^{+},y_{1}^{+},z_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{4}], [u_{5}], [u_{6}]) 12 12 0
[T_{u}] [(0,0,z_{1}^{-})] [m_{x}m_{y}2_{z}] [C_{2vz}] 3 [P_{3}] 6 3 6
  [(x_{1}^{-},x_{1}^{-},x_{1}^{-})] [3_{p}] [C_{3p}] 4 [P_{1}=P_{2}=P_{3}] 8 4 8
  [(x_{1}^{-},y_{1}^{-},z_{1}^{-})] [1] [C_{1}] 1 ([P_{1}], [P_{2}], [P_{3}]) 24 12 24
Parent symmetry [\bi G]: [\quad\bf 432\quad {\bi O}]
[A_{2}] [{\sf x}_{2}] [23] T 1 [d_{14}=d_{25}=d_{36}] 2 1 0
E [(x_{3},0)] [4_{z}2_{x}2_{xy}] [D_{4z}] 3 [\delta u_{1}=\delta u_{2}= -{{1}\over{2}}\delta u_{3}] 3 3 0
(La) [(x_{3},y_{3})] [2_{x}2_{y}2_{z}] [D_{2}] 1 [[u_{3}-a(u_{1}+u_{2})], [b(u_{1}-u_{2})]] 6 6 0
          [\delta u_{1}+\delta u_{2}+\delta u_{3}=0]      
[T_{1}] [(0,0,z_{1})] [4_{z}] [C_{4z}] 3 [P_{3}] 6 3 6
  [(x_{1},x_{1},0)] [2_{xy}] [C_{2xy}] 6 [P_{1}=P_{2}] 12 12 12
  [(x_{1},x_{1},x_{1})] [3_{p}] [C_{3p}] 4 [P_{1}=P_{2}=P_{3}] 8 4 8
(Li) [(x_{1},y_{1},z_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}], [P_{3}]) 24 24 24
[T_{2}] [(0,0,z_{2})] [2_{x{\overline y}}2_{xy}2_{z}] [{\hat D}_{2z}] 3 [u_{6}] 6 6 0
  [(x_{2}, -x_{2}, z_{2})] [2_{xy}] [C_{2xy}] 6 [u_{4}=-u_{5}], [u_{6}] 12 12 12
  [(x_{2}, x_{2}, x_{2})] [3_{p}2_{x{\overline y}}] [D_{3p}] 4 [u_{4}=u_{5}=u_{6}] 4 4 0
(La, Li) [(x_{2}, y_{2}, z_{2})] [1] [C_{1}] 1 ([u_{4}], [u_{5}], [u_{6}]) 24 24 24
Parent symmetry [\bi G]: [\quad\bf{\overline 4}3{\bi m\quad T_{d}}]
[A_{2}] [{\sf x}_{2}] [23] T 1 [\varepsilon]; [g_{1}=g_{2}=g_{3}] 2 1 0
          [A_{14}=A_{25}=A_{36}]; [\pi_{23}^{a}=\pi_{31}^{a}=\pi_{12}^{a}]      
E [(x_{3},0)] [{\overline 4}_{z}2_{x}m_{xy}] [D_{2dz}] 3 [\delta u_{1}=\delta u_{2}= -{{1}\over{2}}\delta u_{3}] 3 3 0
(La) [(x_{3},y_{3})] [2_{x}2_{y}2_{z}] [D_{2}] 1 [[u_{3}-a(u_{1}+u_{2})], [b(u_{1}-u_{2})]] 6 6 0
          [\delta u_{1}+\delta u_{2}+\delta u_{3}=0]      
[T_{1}] [(0,0,z_{1})] [{\overline 4}_{z}] [S_{4z}] 3 [g_{6}]; [d_{32}=-d_{31}], [d_{24}=-d_{15}] 6 3 0
  [(x_{1},x_{1},0)] [m_{xy}] [C_{sxy}] 6 [g_{4}=g_{5}] 12 12 12
          [d_{13}=-d_{23}], [d_{12}=-d_{21}]      
          [d_{35}=-d_{34}], [d_{26}=-d_{16}]      
  [(x_{1},x_{1},x_{1})] [3_{p}] [C_{3p}] 4 [g_{4}=g_{5}=g_{6}] 8 4 4
          [d_{13}=d_{21}=d_{32}, d_{12}=d_{23}=d_{31}]      
          [d_{35}=d_{16}=d_{24}, d_{26}=d_{34}=d_{15}]      
  [(x_{1},y_{1},z_{1})] [1] [C_{1}] 1 ([g_{4}], [g_{5}], [g_{6}]) 24 24 24
          ([d_{13}-d_{12}], [d_{21}-d_{23}], [d_{32}-d_{31}])      
          ([d_{35}-d_{26}], [d_{16}-d_{34}], [d_{24}-d_{15}])      
[T_{2}] [(0,0,z_{2})] [m_{x{\overline y}}m_{xy}2_{z}] [{\hat C}_{2vz}] 3 [P_{3}]; [u_{6}] 6 6 6
  [(x_{2}, -x_{2}, z_{2})] [m_{xy}] [C_{sxy}] 6 [P_{1}=-P_{2}], [P_{3}]; [u_{4}=-u_{5}], [u_{6}] 12 12 12
  [(x_{2}, x_{2}, x_{2})] [3_{p}m_{x{\overline y}}] [C_{3vp}] 4 [P_{1}=P_{2}=P_{3}]; [u_{4}=u_{5}=u_{6}] 4 4 4
(La) [(x_{2}, y_{2}, z_{2})] [1] [C_{1}] 1 ([P_{1}], [P_{2}], [P_{3}]); ([u_{4}], [u_{5}], [u_{6}]) 24 24 24
Parent symmetry [\bi G]: [\quad\bi m{\bf \overline 3}m\quad O_{h}]
[A_{2g}] [{\sf x}_{2}^{+}] [m{\overline 3}] [T_{h}] 1 [A_{14}=A_{25}=A_{36}]; [\pi_{23}^{a}=\pi_{31}^{a}=\pi_{12}^{a}] 2 1 0
[A_{1u}] [{\sf x}_{1}^{-}] [432] O 1 [\varepsilon]; [g_{1}=g_{2}=g_{3}]; 2 1 0
[A_{2u}] [{\sf x}_{2}^{-}] [{\overline 4}3m] [T_{d}] 1 [d_{14}=d_{25}=d_{36}] 2 1 0
[E_{g}] [(x_{3}^{+},0)] [4_{z}/m_{z}m_{x}m_{xy}] [D_{4hz}] 3 [\delta u_{3}] 3 3 0
(La) [(x_{3}^{+},y_{3}^{+})] [m_{x}m_{y}m_{z}] [D_{2h}] 1 [ [\delta u_{3}-a(\delta u_{1}+\delta u_{2})], [b(\delta u_{1}-\delta u_{2})]] 6 6 0
[E_{u}] [(x_{3}^{-}, 0)] [4_{z}2_{x}2_{xy}] [D_{4z}] 3 [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 12 3 0
  [(0, y_{3}^{-})] [{\overline 4}_{z}2_{x}m_{xy}] [D_{2dz}] 3 [g_{1}=-g_{2}]; [d_{14}=d_{25}=d_{36}] 6 3 0
  [(x_{3}^{-}, y_{3}^{-})] [2_{x}2_{y}2_{z}] [D_{2}] 1 [[g_{3}-a(g_{1}+g_{2})], [b(g_{1}-g_{2})]] 12 6 0
          [[b(d_{14}-d_{25}),a(d_{14}+d_{25})-d_{36}]]      
[T_{1g}] [(0,0,z_{1}^{+})] [4_{z}/m_{z}] [C_{4hz}] 3 [A_{33}], [A_{32}=A_{31}], [A_{24}=A_{15}, A_{14}=-A_{25}] 6 3 0
  [(x_{1}^{+},x_{1}^{+},0)] [2_{xy}/m_{xy}] [C_{2hxy}] 6 [A_{11}=A_{22}], 12 12 0
          [A_{13}=A_{23}], [A_{12}=A_{21}]      
          [A_{35}=A_{34}], [A_{26}=A_{16}]      
  [(x_{1}^{+},x_{1}^{+},x_{1}^{+})] [{\overline 3}_{p}] [C_{3ip}] 4 [A_{11}=A_{22}=A_{33}] 8 4 0
          [A_{13}=A_{21}=A_{32}], [A_{12}=A_{32}=A_{31}]      
          [A_{35}=A_{16}=A_{24}], [A_{26}=A_{34}=A_{15}]      
  [(x_{1}^{+},y_{1}^{+},z_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([A_{11}], [A_{22}], [A_{33}]) 24 24 0
          ([A_{13}+A_{12}], [A_{21}+A_{23}], [A_{32}+A_{31}])      
          ([A_{35}+A_{26}], [A_{16}+A_{34}], [A_{24}+A_{15}])      
[T_{2g}] [(0,0,z_{2}^{+})] [m_{x{\overline y}}m_{xy}m_{z}] [{\hat D}_{2hz}] 3 [u_{6}] 6 6 0
  [(x_{2}^{+},-x_{2}^{+},z_{2}^{+})] [2_{xy}/m_{xy}] [C_{2hxy}] 6 [u_{4}=-u_{5}], [u_{6}] 24 12 12
  [(x_{2}^{+},x_{2}^{+},x_{2}^{+})] [{\overline 3}_{p}m_{x{\overline y}}] [D_{3dp}] 4 [u_{4}=u_{5}=u_{6}] 4 4 0
(La) [(x_{2}^{+},y_{2}^{+},z_{2}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{4}], [u_{5}], [u_{6}]) 24 24 0
[T_{1u}] [(0,0,z_{1}^{-})] [4_{z}m_{x}m_{xy}] [C_{4vz}] 3 [P_{3}] 6 3 6
  [(x_{1}^{-},y_{1}^{-},0)] [m_{z}] [C_{sz}] 3 [P_{1}],[P_{2}] 24 12 24
  [(x_{1}^{-},x_{1}^{-},0)] [m_{x{\overline y}}2_{xy}m_{z}] [{\hat C}_{2vxy}] 6 [P_{1}=P_{2}] 12 6 12
  [(x_{1}^{-},-x_{1}^{-},z_{1}^{-})] [m_{xy}] [C_{sxy}] 6 [P_{1}=-P_{2}], [P_{3}] 24 12 24
  [(x_{1}^{-},x_{1}^{-},x_{1}^{-})] [3_{p}m_{x{\overline y}}] [C_{3vp}] 4 [P_{1}=P_{2}=P_{3}] 8 4 8
  [(x_{1}^{-},y_{1}^{-},z_{1}^{-})] [1] [C_{1}] 1 ([P_{1}], [P_{2}], [P_{3}]) 48 24 48
[T_{2u}] [(0,0,z_{2}^{-})] [{\overline 4}_{z}m_{x}2_{xy}] [{\hat D}_{2dz}] 3 [g_{6}]; [d_{32}=-d_{31}], [d_{24}=-d_{15}] 6 3 0
  [(x_{2}^{-},y_{2}^{-},0)] [m_{z}] [C_{sz}] 3 [g_{4}], [g_{5}]; [d_{13}], [d_{12}], [d_{21}], [d_{23}] 24 12 24
          [d_{35}], [d_{26}], [d_{16}], [d_{34}]      
  [(x_{2}^{-},-x_{2}^{-},0)] [m_{x{\overline y}}2_{xy}m_{z}] [{\hat C}_{2vxy}] 6 [g_{4}=-g_{5}]; [d_{13}=d_{23}], [d_{21}=d_{21}] 12 6 12
          [d_{35}=d_{34}], [d_{16}=d_{26}]      
  [(x_{2}^{-},-x_{2}^{-},z_{2}^{-})] [2_{xy}] [C_{2xy}] 6 [g_{4}=-g_{5}], [g_{6}]; [d_{13}=d_{23}], [d_{21}=d_{21}] 24 12 12
          [d_{35}=d_{34}], [d_{16}=d_{26}]      
          [d_{32}=-d_{31}], [d_{24}=-d_{15}]      
  [(x_{2}^{-},x_{2}^{-},x_{2}^{-})] [3_{p}2_{x{\overline y}}] [D_{3p}] 4 [g_{4}=g_{5}=g_{6}]; 8 4 0
          [d_{13}=-d_{12}=d_{21}=-d_{23}=d_{32}-d_{31}]      
          [d_{35}=-d_{26}=d_{16}=-d_{34}=d_{24}=-d_{15}]      
  [(x_{2}^{-},y_{2}^{-},z_{2}^{-})] [1] [C_{1}] 1 ([g_{4}], [g_{5}], [g_{6}]) 48 24 48
          ([d_{13}-d_{12}], [d_{21}-d_{23}], [d_{32}-d_{31}])      
          ([d_{35}-d_{26}], [d_{16}-d_{34}], [d_{24}-d_{15}])