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

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 495-497

Section 3.4.2.4. Synoptic table of ferroic transitions and domain states

V. Janoveca* and J. Přívratskáb

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic, and bDepartment of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic
Correspondence e-mail:  janovec@fzu.cz

3.4.2.4. Synoptic table of ferroic transitions and domain states

| top | pdf |

The considerations of this and all following sections can be applied to any phase transition with point-group symmetry descent [G\supset F]. All such non-magnetic crystallographically non-equivalent symmetry descents are listed in Table 3.4.2.7[link] together with some other data associated with symmetry reduction at a ferroic phase transition. These symmetry descents can also be traced in lattices of subgroups of crystallographic point groups, which are displayed in Figs. 3.1.3.1[link] and 3.1.3.2[link] .

The symmetry descents [G\supset F_1] listed in Table 3.4.2.7[link] are analogous to Aizu's `species' (Aizu, 1970a[link]), in which the symbol F stands for the symbol [\supset ] in our symmetry descent, and the orientation of symmetry elements of the group [F_1] with respect to G is specified by letters p, s, ps, pp etc. A list of 212 non-ferromagnetic species together with their property tensors is available online (Janovec, 2012[link]).

As we have already stated, any systematic analysis of domain structures requires an unambiguous specification of the orientation and location of symmetry elements in space. Moreover, in a continuum approach, the description of crystal properties is performed in a rectangular (Cartesian) coordinate system, which differs in hexagonal and trigonal crystals from the crystallographic coordinate system common in crystallography. Last but not least, a ready-to-use and user-friendly presentation calls for symbols that are explicit and concise.

To meet these requirements, we use in this chapter, in Section 3.1.3[link] and in the software GI[\star]KoBo-1 a symbolism in which the orientations of crystallographic elements and operations are expressed by means of suffixes related to a reference Cartesian coordinate system. The relation of this reference Cartesian coordinate system – called a crystallophysical coordinate system – to the usual crystallographic coordinate system is a matter of convention. We adhere to the generally accepted rules [see Nye (1985[link]) Appendix B, Sirotin & Shaskolskaya (1982[link]), Shuvalov (1988[link]), and IEEE Standards on Piezoelectricity, 1987[link]].

We list all symbols of crystallographic symmetry operations and a comparison of these symbols with other notations in Tables 3.4.2.5[link] and 3.4.2.6[link] and in Figs. 3.4.2.3[link] and 3.4.2.4[link].

Table 3.4.2.5| top | pdf |
Symbols of symmetry operations of the point group [m\bar 3m ]

Standard: symbols used in Section 3.1.3[link] , in the present chapter and in the software; all symbols refer to the cubic crystallographic (Cartesian) basis, [p\equiv[111]] (all [{\underline{p}}]ositive), [q\equiv[\bar1\bar11], \ r\equiv [1\bar1\bar1], \ s\equiv [\bar11\bar1] ]. BC: Bradley & Cracknell (1972[link]). AH: Altmann & Herzig (1994[link]). IT A: IT A (2005[link]). Jones: Jones' faithful representation symbols express the action of a symmetry operation on a vector [(xyz)] (see e.g. Bradley & Cracknell, 1972[link]).

StandardBCAHIT AJonesStandardBCAHIT AJones
1 or e E E 1 [x,y,z] [\bar{1}] or i I i [{\bar 1}]   [0,0,0] [\bar{x},\bar{y},\bar{z}]
[2_{z}] [C_{2z}] [C_{2z}] 2   [0,0,z] [{\bar x},{\bar y},z] [m_{z}] [\sigma_{z}] [\sigma_{z}] m   [x,y,0 ] [x,y,{\bar z}]
[2_{x}] [C_{2x}] [C_{2x}] 2   [x,0,0] [x,{\bar {y},{\bar z}}] [m_{x}] [\sigma_{x}] [\sigma_{x}] m   [0,y,z ] [{\bar x},y,z]
[2_{y}] [C_{2y}] [C_{2y}] 2   [0,y,0] [{\bar x},y,{\bar z}] [m_{y}] [\sigma_{y}] [\sigma_{y}] m   [x,0,z ] [x,{\bar y},z]
[2_{xy}] [C_{2a}] [C_{2a}^{\prime}] 2   [x,x,0] [y,x,{\bar z}] [m_{xy}] [\sigma_{da}] [\sigma_{d1}] m   [x,{\bar x},z ] [{\bar y},{\bar x},z]
[2_{x{\bar y}}] [C_{2b}] [C_{2b}^{\prime}] 2   [x,{\bar x},0 ] [{\bar y},{\bar x},{\bar z}] [m_{x{\bar y}}] [\sigma_{db}] [\sigma_{d2}] m   [x,x,z ] [y,x,z]
[2_{zx}] [C_{2c}] [C_{2c}^{\prime}] 2   [x,0,x,] [z,{\bar y},x] [m_{zx}] [\sigma_{dc}] [\sigma_{d3}] m   [{\bar x},y,x, ] [{\bar z},y,{\bar x}]
[2_{z{\bar x}}] [C_{2e}] [C_{2e}^{\prime}] 2   [{\bar x},0,x ] [{\bar z},{\bar y},{\bar x}] [m_{z{\bar x}}] [\sigma_{de}] [\sigma_{d5}] m   [x,y,x ] [z,y,x]
[2_{yz}] [C_{2d}] [C_{2d}^{\prime}] 2   [0,y,y] [{\bar x},z,y] [m_{yz}] [\sigma_{dd}] [\sigma_{d4}] m   [x,y,{\bar y} ] [x,{\bar z},{\bar y}]
[2_{y{\bar z}}] [C_{2f}] [C_{2f}^{\prime}] [2]   [0,y,{\bar y} ] [{\bar x},{\bar z},{\bar y}] [m_{y{\bar z}}] [\sigma_{df}] [\sigma_{d6}] m   [x,y,y ] [x,z,y]
[3_{p}] [C_{31}^{+}] [C_{31}^{+}] [3^{+}]   [x,x,x] [z,x,y] [{\bar 3}_{p}] [S_{61}^{-}] [S_{61}^{-}] [{\bar 3}^{+}]   [x,x,x] [{\bar z},{\bar x},{\bar y}]
[3_{q}] [C_{32}^{+}] [C_{32}^{+}] [3^{+}]   [{\bar x},{\bar x},x] [{\bar z},x,{\bar y}] [{\bar 3}_{q}] [S_{62}^{-}] [S_{62}^{-}] [{\bar 3}^{+}]   [{\bar x},{\bar x},x] [z,{\bar x},y]
[3_{r}] [C_{33}^{+}] [C_{33}^{+}] [3^{+}]   [x,{\bar x},{\bar x}] [{\bar z},{\bar x},y] [{\bar 3}_{r}] [S_{63}^{-}] [S_{63}^{-}] [{\bar 3}^{+}]   [x,{\bar x},{\bar x}] [z,x,{\bar y}]
[3_{s}] [C_{34}^{+}] [C_{34}^{+}] [3^{+}]   [{\bar x},x,{\bar x}] [z,{\bar x},{\bar y}] [{\bar 3}_{s}] [S_{64}^{-}] [S_{64}^{-}] [{\bar 3}^{+}]   [{\bar x},x,{\bar x}] [{\bar z},x,y]
[3_{p}^{2}] [C_{31}^{-}] [C_{31}^{-}] [3^{-}]   [x,x,x] [y,z,x] [{\bar 3}_{p}^{5}] [S_{61}^{+}] [S_{61}^{+}] [{\bar 3}^{-}]   [x,x,x] [{\bar y},{\bar z},{\bar x}]
[3_{q}^{2}] [C_{32}^{-}] [C_{32}^{-}] [3^{-}]   [{\bar x},{\bar x},x] [y,{\bar z},{\bar x}] [{\bar 3}_{q}^{5}] [S_{62}^{+}] [S_{62}^{+}] [{\bar 3}^{-}]   [{\bar x},{\bar x},x] [{\bar y},z,x]
[3_{r}^{2}] [C_{33}^{-}] [C_{33}^{-}] [3^{-}]   [x,{\bar x},{\bar x}] [{\bar y},z,{\bar x}] [{\bar 3}_{r}^{5}] [S_{63}^{+}] [S_{63}^{+}] [{\bar 3}^{-}]   [x,{\bar x},{\bar x}] [y,{\bar z},x]
[3_{s}^{2}] [C_{34}^{-}] [C_{34}^{-}] [3^{-}]   [{\bar x},x,{\bar x}] [{\bar y},{\bar z},x] [{\bar 3}_{s}^{5}] [S_{64}^{+}] [S_{64}^{+}] [{\bar 3}^{-}]   [{\bar x},x,{\bar x}] [y,z,{\bar x}]
[4_{z}] [C_{4z}^{+}] [C_{4z}^{+}] [4^{+}]   [0,0,z] [{\bar y},x,z] [{\bar 4}_{z}] [S_{4z}^{-}] [S_{4z}^{-}] [{\bar 4}^{+}]   [0,0,z] [y,{\bar x},{\bar z}]
[4_{x}] [C_{4x}^{+}] [C_{4x}^{+}] [4^{+}]   [x,0,0] [x,{\bar z},y] [{\bar 4}_{x}] [S_{4x}^{-}] [S_{4x}^{-}] [{\bar 4}^{+}]   [x,0,0] [{\bar x},z,{\bar y}]
[4_{y}] [C_{4y}^{+}] [C_{4y}^{+}] [4^{+}]   [0,y,0] [z,y,{\bar x}] [{\bar 4}_{y}] [S_{4y}^{-}] [S_{4y}^{-}] [{\bar 4}^{+}]   [0,y,0] [{\bar z},{\bar y},x]
[4_{z}^{3}] [C_{4z}^{-}] [C_{4z}^{-}] [4^{-}]   [0,0,z] [y,{\bar x},z] [{\bar 4}_{z}^{3}] [S_{4z}^{+}] [S_{4z}^{+}] [{\bar 4}^{-}]   [0,0,z] [{\bar y},x,{\bar z}]
[4_{x}^{3}] [C_{4x}^{-}] [C_{4x}^{-}] [4^{-}]   [x,0,0] [x,z,{\bar y}] [{\bar 4}_{x}^{3}] [S_{4x}^{+}] [S_{4x}^{+}] [{\bar 4}^{-}]   [x,0,0] [{\bar x},{\bar z},y]
[4_{y}^{3}] [C_{4y}^{-}] [C_{4y}^{-}] [4^{-}]   [0,y,0] [{\bar z},y,x] [{\bar 4}_{y}^{3}] [S_{4y}^{+}] [S_{4y}^{+}] [{\bar 4}^{-}]   [0,y,0] [z,{\bar y},{\bar x}]

Table 3.4.2.6| top | pdf |
Symbols of symmetry operations of the point group [6/mmm]

Standard: symbols used in Section 3.1.3[link] , in the present chapter and in the software; suffixes (in italic) refer to the Cartesian crystallophysical coordinate system. BC: Bradley & Cracknell (1972[link]). AH: Altmann & Herzig (1994[link]). IT A: IT A (2005[link]), coordinates (in Sans Serif) are expressed in a crystallographic hexagonal basis. Jones: Jones' faithful representation symbols express the action of a symmetry operation of a vector [({\sf x}{\sf y}{\sf z})] in a crystallographic basis (see e.g. Bradley & Cracknell, 1972[link]).

StandardBCAHIT AJonesStandardBCAHIT AJones
1 or e E E [{\sf 1}] [{\sf x},{\sf y},{\sf z}] [\bar 1] or i I I [{\bar{\sf 1}}]   [{\sf 0},{\sf 0},{\sf 0}] [\bar{\sf x},\bar{\sf y},\bar{\sf z}]
[6_{ z}] [C_{6}^{+}] [C_{6}^{+}] [{\sf 6^{+}}]   [{\sf 0},{\sf 0},{\sf z}] [{\sf x}-{\sf y},{\sf x},{\sf z}] [{\bar 6}_{ z}] [S_{3}^{-}] [S_{3}^{-}] [{\bar{\sf 6}^{+}}]   [{\sf 0},{\sf 0},{\sf z}] [{\sf y}-{\sf x},\bar{\sf x},\bar{\sf z}]
[3_{ z}] [C_{3}^{+}] [C_{3}^{+}] [{\sf 3^{+}}]   [{\sf 0},{\sf 0},{\sf z}] [{\bar {\sf y}},{\sf x}-{\sf y},{\sf z}] [{\bar 3}_{ z}] [S_{6}^{-}] [S_{6}^{-}] [{\bar {\sf 3}^{+}}]   [{\sf 0},{\sf 0},{\sf z}] [{\sf y},{\sf y}-{\sf x},\bar{\sf z}]
[2_{ z}] [C_{2}] [C_{2}] [{\sf 2}]   [{\sf 0},{\sf 0},{\sf z}] [\bar{\sf x},\bar{\sf y},{\sf z}] [m_{ z}] [\sigma_{h}] [\sigma_{h}] [{\sf m}]   [{\sf x},{\sf y},{\sf 0}] [{\sf x},{\sf y},\bar{\sf z}]
[3_{ z}^{2}] [C_{3}^{-}] [C_{3}^{-}] [{\sf 3^{-}}]   [{\sf 0},{\sf 0},{\sf z}] [{\sf y}-{\sf x},\bar{\sf x},{\sf z}] [{\bar 3}_{ z}^{5}] [S_{6}^{+}] [S_{6}^{+}] [{\bar {\sf 3}^{-}}]   [{\sf 0},{\sf 0},{\sf z}] [{\sf x}-{\sf y},{\sf x},\bar{\sf z}]
[6_{ z}^{5}] [C_{6}^{-}] [C_{6}^{-}] [{\sf 6^{-}}]   [{\sf 0},{\sf 0},{\sf z}] [{\sf y},{\sf y}-{\sf x},{\sf z}] [{\bar 6}_{ z}^{5}] [S_{3}^{+}] [S_{3}^{+}] [\bar{\sf 6}^{-}]   [{\sf 0},{\sf 0},{\sf z}] [\bar{\sf y},{\sf x}-{\sf y},\bar{\sf z}]
[2_{ x}] [C_{21}{^\prime}{^\prime}] [C_{21}{^\prime}{^\prime}] [{\sf 2}]   [{\sf x},{\sf 0},{\sf 0}] [{\sf x}-{\sf y},\bar{\sf y},\bar{\sf z}] [m_{ x}] [\sigma_{v1}] [\sigma_{v1}] [{\sf m}]   [{\sf x},{\sf 2}{\sf x},{\sf z}] [{\sf y}-{\sf x},{\sf y},{\sf z}]
[2_{x^\prime}] [C_{22}{^\prime}{^\prime}] [C_{22}{^\prime}{^\prime}] [{\sf 2}]   [{\sf 0},{\sf y},{\sf 0}] [\bar{\sf x},{\sf y}-{\sf x},\bar{\sf z}] [m_{x^\prime}] [\sigma_{v2}] [\sigma_{v2}] [{\sf m}]   [{\sf 2}{\sf x},{\sf x},{\sf z}] [{\sf x},{\sf x}-{\sf y},{\sf z}]
[2_{x{^\prime}{^\prime}}] [C_{23}{^\prime}{^\prime}] [C_{23}{^\prime}{^\prime}] [{\sf 2}]   [{\sf x},{\sf x},{\sf 0}] [{\sf y},{\sf x},\bar{\sf z}] [m_{x{^\prime}{^\prime}}] [\sigma_{v3}] [\sigma_{v3}] [{\sf m}]   [{\sf x},\bar{\sf x},{\sf z}] [\bar{\sf y},\bar{\sf x},{\sf z}]
[2_{y}] [C_{21}{^\prime}] [C_{21}{^\prime}] [{\sf 2}]   [{\sf x},{\sf 2}{\sf x},{\sf 0}] [{\sf y}-{\sf x},{\sf y},\bar{\sf z}] [m_{y}] [\sigma_{d1}] [\sigma_{d1}] [{\sf m}]   [{\sf x},{\sf 0},{\sf z}] [{\sf x}-{\sf y},\bar{\sf y},{\sf z}]
[2_{y{^\prime}}] [C_{22}{^\prime}] [C_{22}{^\prime}] [{\sf 2}]   [{\sf 2}{\sf x},{\sf x},{\sf 0}] [{\sf x},{\sf x}-{\sf y},\bar{\sf z}] [m_{y{^\prime}}] [\sigma_{d2}] [\sigma_{d2}] [{\sf m}]   [{\sf 0},{\sf y},{\sf z}] [\bar{\sf x},{\sf y}-{\sf x},{\sf z}]
[2_{y{^\prime}{^\prime}}] [C_{23}{^\prime}] [C_{23}{^\prime}] [{\sf 2}]   [{\sf x},\bar{\sf x},{\sf 0}] [\bar{\sf y},\bar{\sf x},\bar{\sf z}] [m_{y{^\prime}{^\prime}}] [\sigma_{d3}] [\sigma_{d3}] [{\sf m}]   [{\sf x},{\sf x},{\sf z}] [{\sf y},{\sf x},{\sf z}]
[Figure 3.4.2.3]

Figure 3.4.2.3 | top | pdf |

Oriented symmetry operations of the cubic group [m\bar3m] and of its subgroups. The Cartesian (rectangular) coordinate system [x, y, z ] is identical with the crystallographic and crystallophysical coordinate systems. Correlation with other notations is given in Table 3.4.2.5[link].

[Figure 3.4.2.4]

Figure 3.4.2.4 | top | pdf |

Oriented symmetry operations of the hexagonal group [6/mmm ] and of its hexagonal and trigonal subgroups. The coordinate system [x, y, z] corresponds to the Cartesian crystallophysical coordinate system, the axes [{\sf x}, {\sf y}, {\sf z}] of the crystallographic coordinate system are parallel to the twofold rotation axes [2_x, 2_{x^\prime}] and to the sixfold rotation axis [6_z]. Correlation with other notations is given in Table 3.4.2.6[link].

Now we can present the synoptic Table 3.4.2.7[link].

3.4.2.4.1. Explanation of Table 3.4.2.7[link]

| top | pdf |

  • G: point group expressing the symmetry of the parent (prototypic) phase. Subscripts of generators in the group symbol specify their orientation in the Cartesian (rectangular) crystallophysical coordinate system of the group G (see Tables 3.4.2.5[link] and 3.4.2.6[link], and Figs. 3.4.2.3[link] and 3.4.2.4[link]).

    Table 3.4.2.7| top | pdf |
    Group–subgroup symmetry descents [G \supset F_1]

    G: point-group symmetry of parent phase; [F_1]: point-group symmetry of single-domain state [{\bf S}_1]; [\Gamma_{\eta} ]: representation of G; [N_{G}(F_1)]: normalizer of [F_1] in G; [K_G(F_1, g_{1j})]: twinning groups, [g_{1j}\in G]; n: number of principal single-domain states; [d_F]: number of principal domain states with the same symmetry; [n_e]: number of ferroelectric single-domain states; [n_a]: number of ferroelastic single-domain states.

    G[F_1][\Gamma_{\eta}][N_{G}(F_1)][K_G(F_1, g_{1j})]n[d_F ][n_e ][n_a ]
    [ {\bar {\bf 1}}] 1 [A_u] [\bar1] [\bar1^{\star}] 2 2 2 1
    [{\bf 2}_{\bi u}] 1 B [2_u] [2_u^\star] 2 2 2 2
    [{\bi m_u}] 1 [A{^\prime}{^\prime}] [m_u] [m_u^\star] 2 2 2 2
    [{\bf 2}_{\bi u}/{\bi m_u} ] [m_u] [B_u] [2_u/m_u] [2_u^\star/m_u] 2 2 2 1
    [2_u] [A_u] [2_u/m_u] [2_u/m_u^\star] 2 2 2 1
    [\bar 1 ] [B_g ] [2_u/m_u] [2_u^\star/m_u^\star] 2 2 0 2
    1 Reducible [2_u/m_u] [m_u^{\star}], [ 2_u^{\star}], [\bar 1^{\star}] 4 4 4 2
    [{\bf 2}_{\bi x}{\bf 2}_{\bi y}{\bf 2}_{\bi z} ] [2_z] [B_{1g} ] [2_x2_y2_z ] [2_x^\star2_y^\star2_z ] 2 2 2 2
    [2_x] [B_{3g} ] [2_x2_y2_z] [2_x2_y^\star2_z^\star ] 2 2 2 2
    [2_y] [B_{2g} ] [2_x2_y2_z] [2_x^\star2_y2_z^\star] 2 2 2 2
    1 Reducible [2_x2_y2_z] [2_z^{\star}], [2_x^{\star}], [2_y^{\star}] 4 4 4 4
    [{\bi m_xm_y}{\bf 2}_{\bi z} ] [m_x] [B_2] [m_xm_y2_z] [m_xm_y^{\star}2_z^{\star}] 2 2 2 2
    [m_y ] [B_1] [m_xm_y2_z] [m_x^{\star}m_y2_z^{\star}] 2 2 2 2
    [2_z] [A_2] [m_xm_y2_z] [m_x^{\star}m_y^{\star}2_z] 2 2 1 2
    1 Reducible [m_xm_y2_z] [m_x^{\star}], [m_y^{\star}], [2_z^{\star} ] 4 2 4 4
    [{\bi m_xm_ym_z}] [m_xm_y2_z ] [B_{1u} ] [m_xm_ym_z] [m_xm_ym_z^\star] 2 2 2 1
    [2_xm_ym_z ] [B_{3u} ] [m_xm_ym_z] [m_x^{\star}m_ym_z] 2 2 2 1
    [m_x2_ym_z ] [B_{2u} ] [m_xm_ym_z] [m_xm_y^{\star}m_z] 2 2 2 1
    [2_x2_y2_z ] [A_{1u} ] [m_xm_ym_z] [m_x^{\star}m_y^{\star}m_z^{\star}] 2 2 0 1
    [2_z/m_z] [B_{1g} ] [m_xm_ym_z] [m_x^{\star}m_y^{\star}m_z ] 2 2 0 2
    [2_x/m_x] [B_{3g} ] [m_xm_ym_z] [m_xm_y^{\star}m_z^{\star}] 2 2 0 2
    [2_y/m_y] [B_{2g} ] [m_xm_ym_z] [m_x^{\star}m_ym_z^{\star}] 2 2 0 2
    [m_z ] Reducible [m_xm_ym_z] [2_x^{\star}m_y^{\star}m_z], [m_x^{\star}2_y^{\star}m_z], [2_z^{\star}/m_z] 4 4 4 2
    [m_x ] Reducible [m_xm_ym_z] [m_xm_y^{\star}2_z^{\star}], [m_x2_y^{\star}m_z^{\star}], [2_x^{\star}/m_x] 4 4 4 2
    [m_y ] Reducible [m_xm_ym_z] [m_x^{\star}m_y2_z^{\star}], [2_x^{\star}m_ym_z^{\star}], [2_y^{\star}/m_y] 4 4 4 2
    [2_z ] Reducible [m_xm_ym_z] [m_x^{\star}m_y^{\star}2_z], [2_x^{\star}2_y^{\star}2_z], [2_z/m_z^{\star}] 4 4 2 2
    [2_x ] Reducible [m_xm_ym_z] [2_xm_y^{\star}m_z^{\star}], [2_x2_y^{\star}2_z^{\star}], [2_x/m_x^{\star}] 4 4 2 2
    [2_y ] Reducible [m_xm_ym_z] [m_x^{\star}2_ym_z^{\star}], [2_x^{\star}2_y2_z^{\star}], [2_y/m_y^{\star}] 4 4 2 2
    [\bar 1] Reducible [m_xm_ym_z] [2_z^{\star}/m_z^{\star}], [2_x^{\star}/m_x^{\star} ], [2_y^{\star}/m_y^{\star}] 4 4 0 4
    1 Reducible [m_xm_ym_z] [m_z^{\star}], [m_x^{\star}], [m_y^{\star}], [2_z^{\star}], [2_x^{\star}], [2_y^{\star} ], [\bar 1^{\star}] 8 8 8 4
    [{\bf 4}_{\bi z} ] [2_z ] B [4_z] [4_z^{\star} ] 2 2 1 2
    1 [^1E\oplus ^2E] [4_z] [4_z, 2_z^{\star}] 4 4 4 4
    [{\bar {\bf 4}}_{\bi z} ] [2_z ] B [\bar4_z ] [\bar 4_z^{\star}] 2 2 2 2
    1 [^1E\oplus ^2E ] [\bar4_z ] [\bar 4_z,] [2_z^{\star}] 4 2 4 4
    [{\bf 4}_{\bi z}/{\bi m_z} ] [\bar4_z] [B_u] [4_z/m_z ] [4_z^{\star}/m_z^{\star} ] 2 2 0 1
    [4_z ] [A_u] [4_z/m_z ] [4_z/m_z^{\star} ] 2 2 2 1
    [2_z/m_z ] [B_g] [4_z/m_z ] [4_z^{\star}/m_z ] 2 2 0 2
    [m_z] [^1E_u \oplus ^2E_u] [4_z/m_z] [4_z/m_z], [2_z^{\star}/m_z] 4 4 4 2
    [2_z] Reducible [4_z/m_z] [\bar 4_z^{\star}], [4_z^{\star} ], [2_z/m_z^{\star}] 4 4 2 2
    [\bar 1] [^1E_g \oplus ^2E_g ] [4_z/m_z] [4_z/m_z], [2_z^{\star}/m_z^{\star} ] 4 4 0 4
    1 Reducible [4_z/m_z] [\bar4_z], [4_z], [m_z^{\star} ], [2_z^{\star}], [\bar1^{\star}] 8 8 8 4
    [{\bf 4}_{\bi z}{\bf 2}_{\bi x}{\bf 2}_{\bi xy} ] [4_z] [A_2] [4_z2_x2_{xy}] [4_z2_x^{\star}2_{xy}^{\star}] 2 2 2 1
    [2_{x\bar{y}}2_{xy}2_z] [B_2] [4_z2_x2_{xy}] [4_z^{\star}2_x^{\star}2_{xy}] 2 2 0 2
    [2_x2_y2_z ] [B_1] [4_z2_x2_{xy} ] [4_z^{\star}2_x2_{xy}^{\star}] 2 2 0 2
    [2_{xy}] [(2_{x\bar{y}})] E [2_{x\bar{y}}2_{xy}2_z ] [4_z2_x2_{xy}], [2_{x\bar{y}}^{\star}2_{xy}2_z^{\star} ] 4 2 2 2
    [2_z ] Reducible [4_z2_x2_{xy}] [4_z^{\star}], [2_x^{\star}2_y^{\star}2_z ], [2_{x\bar{y}}^{\star}2_{xy}^{\star}2_z] 4 4 2 2
    [2_x] [(2_y)] E [2_{x\bar{y}}2_{xy}2_z] [4_z2_x2_{xy}], [2_x2_y^{\star}2_z^{\star} ] 4 2 2 2
    1 E [4_z2_x2_{xy}] [4_z], [2_z^{\star}], [2_x^{\star}(2)], [2_{xy}^{\star}(2)] 8 8 8 8
    [{\bf 4}_{\bi z}{\bi m}_{\bi x}{\bi m}_{\bi xy} ] [4_z ] [A_2] [4_zm_xm_{xy}] [4_zm_x^{\star}m_{xy}^{\star} ] 2 2 1 1
    [m_{x\bar{y}}m_{xy}2_z] [B_2] [4_zm_xm_{xy} ] [4_z^{\star}m_x^{\star}m_{xy} ] 2 2 1 2
    [m_xm_y2_z ] [B_1] [4_zm_xm_{xy} ] [4_z^{\star}m_xm_{xy}^{\star} ] 2 2 1 2
    [m_{xy}] [(m_{x\bar{y}})] E [m_{x\bar{y}}m_{xy}2_z] [4_zm_xm_{xy}], [m_{x\bar{y}}^{\star}m_{xy}2_z^{\star} ] 4 2 4 4
    [m_x] [(m_y)] E [m_xm_y2_z ] [4_zm_xm_{xy}], [m_xm_y^{\star}2_z^{\star} ] 4 2 4 4
    [2_z] Reducible [4_zm_xm_{xy} ] [4_z^{\star}], [m_x^{\star}m_y^{\star}2_z ], [m_{x\bar{y}}^{\star}m_{xy}^{\star}2_z] 4 4 2 2
    1 E [4_zm_xm_{xy} ] [4_z], [m_x^{\star}(2)], [m_{xy}^{\star}(2)], [2_z^{\star}] 8 8 8 8
    [{\bar {\bf 4}}_{\bi z}{\bf 2}_{\bi x}{\bi m_{xy}} ] [\bar4_z] [A_2] [\bar4_z2_xm_{xy}] [\bar4_z2_x^{\star}m_{xy}^{\star}] 2 2 0 1
    [m_{x\bar{y}}m_{xy}2_z] [B_2] [\bar4_z2_xm_{xy}] [\bar4_z^{\star}2_x^{\star}m_{xy}] 2 2 2 2
    [2_x2_y2_z ] [B_1] [\bar4_z2_xm_{xy}] [\bar4_z^{\star}2_xm_{xy}^{\star}] 2 2 0 2
    [m_{xy}] [(m_{x\bar{y}})] E [m_{x\bar{y}}m_{xy}2_z] [\bar4_z2_xm_{xy}], [m_{x\bar{y}}^{\star}m_{xy}2_z^{\star} ] 4 2 4 4
    [2_z ] Reducible [\bar4_z2_xm_{xy}] [\bar4_z^{\star}], [m_{x\bar{y}}^{\star}m_{xy}^{\star}2_z ], [2_x^{\star}2_y^{\star}2_z] 4 4 2 2
    [2_x ] [(2_y) ] E [2_x2_y2_z ] [\bar4_z2_xm_{xy}], [2_x2_y^{\star}2_z^{\star} ] 4 2 4 4
    1 E [\bar4_z2_xm_{xy}] [\bar4_z], [m_{xy}^{\star}(2)], [2_z^{\star}], [2_x^{\star}(2)] 8 8 8 8
    [\bar4_zm_x2_{xy}] [\bar4_z] [A_2] [\bar4_zm_x2_{xy}] [\bar4_zm_x^{\star}2_{xy}^{\star}] 2 2 0 1
    [m_xm_y2_z ] [B_2] [\bar4_zm_x2_{xy}] [\bar4_z^{\star}m_x2_{xy}^{\star}] 2 2 2 2
    [2_{x\bar{y}}2_{xy}2_z] [B_1] [\bar4_zm_x2_{xy}] [\bar4_z^{\star}m_x^{\star}2_{xy}] 2 2 0 2
    [m_x] [(m_y)] E [m_xm_y2_z ] [\bar4_zm_x2_{xy}], [m_xm_y^{\star}2_z^{\star} ] 4 2 4 4
    [2_{xy}] [(2_{x\bar{y}})] E [2_{x\bar{y}}2_{xy}2_z] [\bar4_zm_x2_{xy}], [2_{x\bar{y}}^{\star}2_{xy}2_z^{\star} ] 4 2 4 4
    [2_z ] Reducible [\bar4_zm_x2_{xy}] [\bar4_z^{\star} ], [m_x^{\star}m_y^{\star}2_z ], [2_{x\bar{y}}^{\star}2_{xy}^{\star}2_z] 4 4 2 2
    1 E [\bar4_zm_x2_{xy}] [\bar4_z], [m_x^{\star}(2)], [2_{xy}^{\star}(2)], [2_z^{\star}] 8 8 8 8
    [{\bf 4}_{\bi z}/{\bi m_zm_xm_{xy}} ] [\bar4_zm_x2_{xy}] [B_{2u}] [4_z/m_zm_xm_{xy} ] [4_z^{\star}/m_z^{\star}m_xm_{xy}^{\star} ] 2 2 0 1
    [\bar4_z2_xm_{xy} ] [B_{1u} ] [4_z/m_zm_xm_{xy} ] [4_z^{\star}/m_z^{\star}m_x^{\star}m_{xy} ] 2 2 0 1
    [4_zm_xm_{xy} ] [A_{2u} ] [4_z/m_zm_xm_{xy} ] [4_z^{\star}/m_z^{\star}m_xm_{xy}] 2 2 2 1
    [4_z2_x2_{xy} ] [A_{1u} ] [4_z/m_zm_xm_{xy} ] [4_z^{\star}/m_z^{\star}m_x^{\star}m_{xy} ] 2 2 0 1
    [4_z/m_z ] [A_{2g} ] [4_z/m_zm_xm_{xy} ] [4_z^{\star}/m_zm_x^{\star}m_{xy}^{\star} ] 2 2 0 1
    [\bar4_z ] Reducible [4_z/m_zm_xm_{xy} ] [\bar4_z2_x^{\star}m_{xy}^{\star}], [\bar4_zm_x^{\star}2_{xy}^{\star}], [4_z^{\star}/m_z^{\star}] 4 4 0 1
    [4_z] Reducible [4_z/m_zm_xm_{xy} ] [4_zm_x^{\star}m_{xy}^{\star} ], [4_z2_x^{\star}2_{xy}^{\star}], [4_z/m_z^{\star}] 4 4 2 1
    [m_{x\bar{y}}m_{xy}m_z ] [B_{2g} ] [4_z/m_zm_xm_{xy}] [4_z^{\star}/m_zm_x^{\star}m_{xy}] 2 2 0 2
    [m_xm_ym_z] [B_{1g} ] [m_xm_ym_z ] [4_z^{\star}/m_zm_xm_{xy}^{\star}] 2 2 0 2
    [2_{x\bar{y}}m_{xy}m_z] [(m_{x\bar{y}}2_{xy}m_z)] [E_u] [m_{x\bar{y}}m_{xy}m_z ] [4_z/m_zm_xm_{xy}], [m_{x\bar{y}}^{\star}m_{xy}m_z ] 4 2 4 2
    [2_xm_ym_z] [(m_x2_ym_z)] [E_u] [m_xm_ym_z] [4_z/m_zm_xm_{xy}], [m_x^{\star}m_ym_z ] 4 2 4 2
    [m_{x\bar{y}}m_{xy}2_z ] Reducible [4_z/m_zm_xm_{xy}] [\bar4_z^{\star}2_x^{\star}m_{xy}], [4_z^{\star}m_x^{\star}m_{xy}], [m_{x\bar{y}}m_{xy}m_z^{\star} ] 4 4 2 2
    [m_xm_y2_z] Reducible [4_z/m_zm_xm_{xy}] [\bar4_z^{\star}m_x2_{xy}^{\star}], [4_z^{\star}m_xm_{xy}^{\star}], [m_xm_ym_z^{\star} ] 4 4 2 2
    [2_{x\bar{y}}2_{xy}2_z ] Reducible [4_z/m_zm_xm_{xy}] [\bar4_z^{\star}m_x^{\star}2_{xy}], [4_z^{\star}2_x^{\star}2_{xy}], [m_{x\bar{y}}^{\star}m_{xy}^{\star}m_z^{\star} ] 4 4 0 2
    [2_x2_y2_z] Reducible [4_z/m_zm_xm_{xy} ] [\bar4_z^{\star}2_xm_{xy}^{\star}], [4_z^{\star}2_x2_{xy}^{\star}], [m_x^{\star}m_y^{\star}m_z^{\star}] 4 4 0 2
    [2_{xy}/m_{xy}] [(2_{x\bar{y}}/m_{x\bar{y}})] [E_g] [m_{x\bar{y}}m_{xy}m_z ] [4_z/m_zm_xm_{xy}], [m_{x\bar{y}}^{\star}m_{xy}m_z^{\star} ] 4 2 0 4
    [2_z/m_z ] Reducible [4_z/m_zm_xm_{xy}] [4_z^{\star}/m_z], [m_{x\bar{y}}^{\star}m_{xy}^{\star}m_z ], [m_x^{\star}m_y^{\star}m_z ] 4 4 0 4
    [2_x/m_x] [(2_y/m_y)] [E_g] [m_xm_ym_z] [4_z/m_zm_xm_{xy}], [m_xm_y^{\star}m_z^{\star} ] 4 2 0 4
    [m_{xy} (m_{x \bar y})] Reducible [m_{x\bar{y}}m_{xy}m_z ] [\bar4_z2_xm_{xy}], [4_zm_xm_{xy} ], [2_{x\bar{y}}^{\star}m_{xy}m_z^{\star}], [m_{x\bar{y}}^{\star}m_{xy}2_z^{\star} ], [2_{xy}^{\star}/m_{xy}] 8 4 8 4
    [m_z ] [E_u] [4_z/m_zm_xm_{xy} ] [4_z/m_z], [2_{x\bar{y}}^{\star}m_{xy}^{\star}m_z(2) ], [2_x^{\star}m_y^{\star}m_z(2)], [2_z^{\star}/m_z ] 8 8 8 4
    [m_x] [(m_y) ] Reducible [m_xm_ym_z] [\bar4_zm_x2_{xy}], [4_zm_xm_{xy} ], [m_xm_y^{\star}2_z^{\star}], [m_x2_y^{\star}m_z^{\star}], [2_x^{\star}/m_x] 8 4 8 4
    [2_{xy}] [(2_{x\bar{y}})] Reducible [m_{x\bar{y}}m_{xy}m_z ] [\bar4_zm_x2_{xy}], [4_z2_x2_{xy} ], [m_{x\bar{y}}^{\star}2_{xy}m_z^{\star}], [2_{x\bar{y}}^{\star}2_{xy}2_z^{\star} ], [2_{xy}/m_{xy}^{\star}] 8 4 8 4
    [2_z] Reducible [4_z/m_zm_xm_{xy}] [\bar4_z^{\star}], [4_z^{\star} ], [m_x^{\star}m_y^{\star}2_z], [m_{x\bar{y}}^{\star}m_{xy}^{\star}2_z ], [2_x^{\star}2_y^{\star}2_z], [2_{x\bar{y}}^{\star}2_{xy}^{\star}2_z ], [2_z/m_z^{\star}] 8 8 2 4
    [2_x] [(2_y)] Reducible [m_xm_ym_z] [\bar4_z2_xm_{xy}], [4_z2_x2_{xy} ], [2_xm_y^{\star}m_z^{\star}], [2_x2_y^{\star}2_z^{\star}], [2_x/m_x^{\star}] 8 4 4 4
    [\bar1] [E_g] [4_z/m_zm_xm_{xy} ] [4_z/m_z], [2_{xy}^{\star}/m_{xy}^{\star}(2) ], [2_z^{\star}/m_z^{\star}], [2_x^{\star}/m_x^{\star}(2)] 8 8 0 8
    1 Reducible [4_z/m_zm_xm_{xy}] [\bar4_z], [4_z], [m_{xy}^{\star}(2) ], [m_z^{\star}], [m_x^{\star}(2)], [2_{xy}^{\star}(2) ], [2_z^{\star}], [2_x^{\star}(2)], [\bar1^{\star}] 16 16 16 8
    [{\bf 3}_{\bi z} ] 1 E [3_z ] [3_z ] 3 3 3 3
    [{\bar {\bf 3}}_{\bi z} ] [3_z ] [A_u] [\bar3_z ] [\bar 3_z^{\star} ] 2 2 2 1
    [\bar 1 ] [E_g] [\bar3_z ] [\bar 3_z] 3 3 0 3
    1 [E_u] [\bar3_z ] [\bar 3_z], [3_z], [\bar 1^{\star} ] 6 6 6 3
    [{\bf 3}_{\bi z}{\bf 2}_{\bi x} ] [3_z] [A_2] [3_z2_x] [3_z2_x^{\star}] 2 2 2 1
    [2_x] [(2_{x^{\prime}}], [2_{x^{{\prime}{\prime}}})] E [2_x ] [3_z2_x ] 3 1 3 3
    1 E [3_z2_x] [3_z, 2_x^{\star}(3) ] 6 6 6 6
    [3_z2_y ] [3_z] [A_2] [3_z2_y] [3_z2_y^{\star}] 2 2 2 1
    [2_y] [(2_{y^{\prime}}], [2_{y^{{\prime}{\prime}}})] E [2_y] [3_z2_y ] 3 1 3 3
    1 E [3_z2_y] [3_z], [2_y^{\star}(3)] 6 6 6 6
    [{\bf 3}_{\bi z}{\bi m_x} ] [3_z] [A_2] [3_zm_x] [3_zm_x^{\star}] 2 2 1 1
    [m_x] [(m_{x^\prime}], [m_{x{^\prime}{^\prime}})] E [m_x ] [3_zm_x ] 3 1 3 3
    1 E [3_zm_x] [3_z], [m_x^{\star}(3)] 6 6 6 6
    [3_zm_y] [3_z] [A_2] [3_zm_y] [3_zm_y^{\star}] 2 2 1 1
    [m_y ] [(m_{y{^\prime}}], [m_{y{^\prime}{^\prime}})] E [m_y] [3_zm_y ] 3 1 3 3
    1 E [3_zm_y] [3_z], [m_y^{\star}(3)] 6 6 6 6
    [{\bar {\bf 3}}_{\bi z}{\bi m_x} ] [3_zm_x ] [A_{2u} ] [\bar3_zm_x ] [\bar3_z^{\star}m_x ] 2 2 2 1
    [3_z2_x ] [A_{1u} ] [\bar3_zm_x ] [\bar3_z^{\star}m_x^{\star}] 2 2 0 1
    [\bar 3_z ] [A_{2g} ] [\bar3_zm_x ] [\bar3_zm_x^{\star} ] 2 2 0 1
    [3_z ] Reducible [\bar3_zm_x] [3_zm_x^{\star}], [3_z2_x^{\star} ], [\bar 3_z^{\star} ] 4 4 2 1
    [2_x/m_x] [(2_{x^\prime}/m_{x^\prime} ], [2_{x{^\prime}{^\prime}}/m_{x{^\prime}{^\prime}})] [E_g] [2_x/m_x] [\bar3_zm_x ] 3 1 0 3
    [m_x] [(m_{x{^\prime}}], [m_{x{^\prime}{^\prime}})] [E_u] [2_x/m_x] [\bar3_zm_x], [3_zm_x], [2_x^{\star}/m_x(3)] 6 2 6 3
    [2_x ] [(2_{x{^\prime}}], [2_{x{^\prime}{^\prime}})] [E_u] [2_x/m_x] [\bar3_zm_x], [3_z2_x], [2_x/m_x^{\star}(3) ] 6 2 6 3
    [\bar 1] [E_g] [\bar3_zm_x] [\bar 3_z], [2_x^{\star}/m_x^{\star}(3) ] 6 6 0 6
    1 [E_u] [\bar3_zm_x] [\bar 3_z], [3_z], [m_x^{\star}(3) ], [2_x^{\star}(3)], [\bar 1^{\star} ] 12 12 12 6
    [\bar3_zm_y] [3_zm_y] [A_{2u}] [\bar3_zm_y] [\bar3_z^{\star}m_y] 2 2 2 1
    [3_z2_y] [A_{1u}] [\bar3_zm_y] [\bar3_z^{\star}m_y^{\star}] 2 2 0 1
    [\bar 3_z] [A_{2g}] [\bar3_zm_y] [\bar3_zm_y^{\star}] 2 2 0 1
    [3_z] Reducible [\bar3_zm_y] [3_zm_y^{\star}], [3_z2_y^{\star} ], [\bar 3_z^{\star}] 4 4 0 1
    [2_y/m_y] [(2_{y{^\prime}}/m_{y{^\prime}} ], [2_{y{^\prime}{^\prime}}/m_{y{^\prime}{^\prime}})] [E_g] [2_y/m_y] [\bar3_zm_y ] 3 1 2 1
    [m_y] [(m_{y{^\prime}}], [m_{y{^\prime}{^\prime}})] [E_u] [2_y/m_y] [\bar3_zm_y], [3_zm_y], [2_y^{\star}/m_y(3)] 6 2 0 3
    [2_y ] [(2_{y{^\prime}}], [2_{y{^\prime}{^\prime}}) ] [E_u] [2_y/m_y] [\bar3_zm_y], [3_z2_y], [2_y/m_y^{\star}(3) ] 6 2 6 3
    [\bar 1] [E_g] [\bar3_zm_y] [\bar 3_z], [2_y^{\star}/m_y^{\star}(3) ] 6 6 0 3
    1 [E_u] [\bar3_zm_y] [\bar 3_z], [3_z], [m_y^{\star}(3) ], [2_y^{\star}(3)], [\bar 1^{\star}] 12 12 12 6
    [{\bf 6}_{\bi z} ] [3_z] B [6_z ] [6_z^{\star}] 2 2 1 1
    [2_z] [E_2] [6_z] [6_z ] 3 3 1 3
    1 [E_1] [6_z] [6_z], [3_z], [2_z^{\star} ] 6 6 6 6
    [{\bar{\bf 6}}_{\bi z} ] [3_z] [A{^\prime}{^\prime}] [\bar6_z] [\bar 6_z^{\star}] 2 2 2 1
    [m_z] [E{^\prime}] [\bar6_z] [\bar 6_z] 3 2 3 3
    1 [E{^\prime}{^\prime}] [\bar6_z] [\bar 6_z ], [3_z], [m_z^{\star} ] 6 6 6 6
    [{\bf 6}_{\bi z}/{\bi m_z} ] [\bar 6_z] [B_u ] [6_z/m_z] [6_z^{\star}/m_z] 2 2 0 1
    [6_z ] [A_u ] [6_z/m_z] [6_z/m_z^{\star}] 2 2 2 1
    [\bar 3_z ] [B_g ] [6_z/m_z] [6_z^{\star}/m_z^{\star} ] 2 2 0 1
    [3_z ] Reducible [6_z/m_z] [\bar 6_z^{\star}], [6_z^{\star} ], [\bar 3_z^{\star}] 4 4 2 1
    [2_z/m_z] [E_{2g} ] [6_z/m_z] [6_z/m_z] 3 3 0 3
    [m_z ] [E_{1u} ] [6_z/m_z] [6_z/m_z], [\bar 6_z], [2_z^{\star}/m_z ] 6 6 6 3
    [2_z ] [E_{2u} ] [6_z/m_z] [6_z/m_z], [6_z], [2_z/m_z^{\star} ] 6 6 2 3
    [\bar 1 ] [E_{1g} ] [6_z/m_z] [6_z/m_z], [\bar 3_z], [2_z^{\star}/m_z^{\star} ] 6 6 0 6
    1 Reducible [6_z/m_z] [\bar 6_z], [6_z], [\bar 3_z ], [3_z], [m_z^{\star}], [2_z^{\star}], [\bar 1^{\star} ] 12 12 12 6
    [{\bf 6}_{\bi z}{\bf 2}_{\bi x}{\bf 2}_{\bi y}] [6_z] [A_2 ] [6_z2_x2_y] [6_z2_x^{\star}2_y^{\star}] 2 2 2 1
    [3_z2_x] [B_1 ] [6_z2_x2_y ] [6_z^{\star}2_x2_y^{\star}] 2 2 0 1
    [3_z2_y] [B_2 ] [6_z2_x2_y ] [6_z^{\star}2_x^{\star}2_y] 2 2 0 1
    [3_z ] Reducible [6_z2_x2_y] [6_z^{\star}], [3_z2_x^{\star} ], [3_z2_y^{\star} ] 4 4 2 1
    [2_x2_y2_z] [(2_{x{^\prime}}2_{y{^\prime}}2_z ], [2_{x{^\prime}{^\prime}}2_{y{^\prime}{^\prime}}2_z)] [E_2] [2_x2_y2_z] [6_z2_x2_y ] 3 1 0 3
    [2_z ] [E_2] [6_z 2_x2_y] [6_z], [2_x^{\star}2_y^{\star}2_z(3) ] 6 6 2 6
    [2_x] [(2_{x{^\prime}}], [2_{x{^\prime}{^\prime}})] [E_1] [2_x2_y2_z] [6_z2_x2_y], [3_z2_x], [2_x2_y^{\star}2_z^{\star}] 6 2 6 6
    [2_y] [(2_{y{^\prime}}], [2_{y{^\prime}{^\prime}})] [E_1] [2_x2_y2_z] [6_z2_x2_y], [3_z2_y], [2_x^{\star}2_y2_z^{\star} ] 6 2 6 6
    1 [E_1] [6_z2_x2_y] [6_z], [3_z], [2_z^{\star} ], [2_x^{\star}(3)], [2_y^{\star}(3)] 12 12 12 12
    [{\bf 6}_{\bi z}{\bi m}_{\bi x}{\bi m}_{\bi y} ] [6_z] [A_2] [6_zm_xm_y] [6_zm_x^{\star}m_y^{\star} ] 2 2 1 1
    [3_zm_x] [B_2] [6_zm_xm_y] [6_z^{\star}m_xm_y^{\star}] 2 2 1 1
    [3_zm_y] [B_1] [6_zm_xm_y] [6_z^{\star}m_x^{\star}m_y] 2 2 1 1
    [3_z ] Reducible [6_zm_xm_y] [6_z^{\star}], [3_zm_x^{\star} ], [3_zm_y^{\star}] 4 4 1 1
    [m_xm_y2_z] [(m_{x{^\prime}}m_{y{^\prime}}2_z ], [m_{x{^\prime}{^\prime}}m_{y{^\prime}{^\prime}}2_z)] [E_2] [m_xm_y2_z] [6_zm_xm_y] 3 1 1 3
    [m_x] [(m_{x{^\prime}}], [m_{x{^\prime}{^\prime}})] [E_1] [m_xm_y2_z] [6_zm_xm_y], [3_zm_x], [m_xm_y^{\star}2_z^{\star}] 6 2 6 6
    [m_y ] [(m_{y{^\prime}}], [m_{y{^\prime}{^\prime}})] [E_1] [m_xm_y2_z] [6_zm_xm_y], [3_zm_y], [m_x^{\star}m_y2_z^{\star}] 6 2 6 6
    [2_z ] [E_2] [6_zm_xm_y] [6_z], [m_x^{\star}m_y^{\star}2_z(3) ] 6 6 1 6
    1 [E_1] [6_zm_xm_y] [6_z], [3_z], [2_z^{\star} ], [m_x^{\star}(3)], [m_y^{\star}(3)] 12 12 12 12
    [{\bar {\bf 6}}_{\bi z}{\bi m_x}{\bf 2}_{\bi y} ] [\bar 6_z ] [A_2{^\prime} ] [\bar6_zm_x2_y] [\bar6_zm_x^{\star}2_y^{\star}] 2 2 0 1
    [3_zm_x ] [A_2{^\prime}{^\prime}] [\bar6_zm_x2_y] [\bar6_z^{\star}m_x2_y^{\star} ] 2 2 2 1
    [3_z2_y ] [A_1{^\prime}] [\bar6_zm_x2_y] [\bar6_z^{\star}m_x^{\star}2_y ] 2 2 0 1
    [3_z ] Reducible [\bar6_zm_x2_y] [\bar 6_z^{\star} ], [3_zm_x^{\star} ], [3_z2_y^{\star}] 4 4 2 1
    [m_x2_ym_z] [(m_{x{^\prime}}2_{y{^\prime}}m_z ], [m_{x{^\prime}{^\prime}}2_{y{^\prime}{^\prime}}m_z)] [E{^\prime}] [m_x2_ym_z] [\bar6_zm_x2_y] 3 1 3 3
    [m_z ] [E{^\prime}] [\bar6_zm_x2_y] [\bar6_z, m_x^{\star}2_y^{\star}m_z(3) ] 6 6 6 6
    [m_x] [(m_{x{^\prime}}], [m_{x{^\prime}{^\prime}})] [E{^\prime}{^\prime}] [m_y2_ym_z] [\bar6_zm_x2_y], [3_zm_x], [m_x2_y^{\star}m_z^{\star}] 6 2 6 6
    [2_y] [(2_{y{^\prime}}], [2_{y{^\prime}{^\prime}})] [E{^\prime}{^\prime}] [m_x2_ym_z] [\bar6_zm_x2_y], [3_z2_y], [m_x^{\star}2_ym_z^{\star}] 6 2 3 6
    1 [E{^\prime}{^\prime}] [\bar6_zm_x2_y] [\bar 6_z], [3_z], [m_z^{\star} ], [m_x^{\star}(3)], [2_y^{\star}(3)] 12 12 12 12
    [\bar 6_z2_xm_y ] [\bar 6_z ] [A_2{^\prime} ] [\bar6_z2_xm_y] [\bar6_z2_x^{\star}m_y^{\star} ] 2 2 0 1
    [3_zm_y] [A_2{^\prime} ] [\bar6_z2_xm_y] [\bar6_z^{\star}2_x^{\star}m_y] 2 2 2 1
    [3_z2_x] [A_1{^\prime}{^\prime}] [\bar6_z2_xm_y ] [\bar6_z^{\star}2_xm_y^{\star} ] 2 2 0 1
    [3_z] Reducible [\bar6_z2_xm_y ] [\bar 6_z^{\star}], [3_zm_y^{\star} ], [3_z2_x^{\star}] 4 4 2 1
    [2_xm_ym_z] [(2_{x{^\prime}}m_{y{^\prime}}m_z ], [2_{x{^\prime}{^\prime}}m_{y{^\prime}{^\prime}}m_z)] [E{^\prime}] [m_x2_ym_z] [\bar6_z2_xm_y] 3 1 3 3
    [m_z] [E{^\prime}] [\bar6_z2_xm_y ] [\bar6_z], [2_x^{\star}m_y^{\star}m_z(3) ] 6 6 6 6
    [m_y] [(m_{y{^\prime}}], [m_{y{^\prime}{^\prime}})] [E{^\prime}{^\prime}] [m_x2_ym_z] [\bar 6_z2_xm_y], [3_zm_y], [2_x^{\star}m_ym_z^{\star}] 6 2 6 6
    [2_x] [(2_{x{^\prime}}], [2_{x{^\prime}{^\prime}})] [E{^\prime}{^\prime}] [m_x2_ym_z] [\bar 6_z2_xm_y], [3_z2_x], [2_xm_y^{\star}m_z^{\star}] 6 2 3 6
    1 [E{^\prime}{^\prime}] [\bar6_z2_xm_y ] [\bar 6_z], [3_z], [m_z^{\star} ], [m_y^{\star}(3)], [2_x^{\star}(3)] 12 12 12 12
    [{\bf 6}_{\bi z}/{\bi m_zm_xm_y} ] [\bar 6_zm_x2_y] [B_{2u}] [6_z/m_zm_xm_y] [6_z^{\star}/m_zm_xm_y^{\star}] 2 2 0 1
    [\bar 6_z2_xm_y] [B_{1u} ] [6_z/m_zm_xm_y] [6_z^{\star}/m_zm_x^{\star}m_y] 2 2 0 1
    [6_zm_xm_y] [A_{2u} ] [6_z/m_zm_xm_y] [6_z/m_z^{\star}m_xm_y ] 2 2 2 1
    [6_z2_x2_y] [A_{1u} ] [6_z/m_zm_xm_y] [6_z/m_z^{\star}m_x^{\star}m_y^{\star} ] 2 2 0 1
    [6_z/m_z ] [A_{2g} ] [6_z/m_zm_xm_y] [6_z/m_zm_x^{\star}m_y^{\star}] 2 2 0 1
    [\bar 6_z] Reducible [6_z/m_zm_xm_y] [\bar 6_zm_x^{\star}2_y^{\star}], [\bar6_z2_x^{\star}m_y^{\star} ], [6_z^{\star}/m_z ] 4 4 0 1
    [6_z] Reducible [6_z/m_zm_xm_y] [6_zm_x^{\star}m_y^{\star}], [6_z2_x^{\star}2_y^{\star}], [6_z/m_z^{\star}] 4 4 2 1
    [\bar 3_zm_x] [B_{1g} ] [6_z/m_zm_xm_y] [6_z^{\star}/m_z^{\star}m_xm_y^{\star} ] 2 2 0 1
    [\bar 3_zm_y] [B_{2g}] [6_z/m_zm_xm_y] [6_z^{\star}/m_z^{\star}m_x^{\star}m_y ] 2 2 0 1
    [3_zm_x] Reducible [6_z/m_zm_xm_y] [\bar 6_z^{\star}m_x2_y^{\star}], [6_z^{\star}m_xm_y^{\star}], [\bar 3_z^{\star}m_x ] 4 4 2 1
    [3_zm_y] Reducible [6_z/m_zm_xm_y] [\bar 6_z^{\star}2_x^{\star}m_y], [6_z^{\star}m_x^{\star}m_y], [\bar 3_z^{\star}m_y] 4 4 2 1
    [3_z2_x] Reducible [6_z/m_zm_xm_y] [\bar 6_z^{\star}2_xm_y^{\star}], [6_z^{\star}2_x2_y^{\star}], [\bar 3_z^{\star}m_x^{\star} ] 4 4 0 1
    [3_z2_y] Reducible [6_z/m_zm_xm_y] [\bar 6_z^{\star}m_x^{\star}2_y], [6_z^{\star}2_x^{\star}2_y], [\bar 3_z^{\star}m_y^{\star}] 4 4 0 1
    [\bar 3_z] Reducible [6_z/m_zm_xm_y] [6_z^{\star}/m_z^{\star}], [\bar 3_zm_x^{\star} ], [\bar 3_zm_y^{\star}] 4 4 0 1
    [3_z] Reducible [6_z/m_zm_xm_y] [\bar 6_z^{\star}], [6_z^{\star} ], [3_zm_x^{\star}], [3_zm_y^{\star}], [3_z2_x^{\star} ], [3_z2_y^{\star}], [\bar 3_z^{\star}] 8 8 2 1
    [m_xm_ym_z] [(m_{x{^\prime}}m_{y{^\prime}}m_z ], [m_{x{^\prime}{^\prime}}m_{y{^\prime}{^\prime}}m_z)] [E_{2g} ] [m_xm_ym_z ] [6_z/m_zm_xm_y] 3 1 0 3
    [m_xm_y2_z ] [(m_{x{^\prime}}m_{y{^\prime}}2_z ], [m_{x{^\prime}{^\prime}}m_{y{^\prime}{^\prime}}2_z)] [E_{2u}] [m_xm_ym_z] [6_z/m_zm_xm_y], [6_zm_xm_y], [m_xm_ym_z^{\star}] 6 2 2 3
    [2_xm_ym_z] [(2_{x{^\prime}}m_{y{^\prime}}m_z ], [2_{x{^\prime}{^\prime}}m_{y{^\prime}{^\prime}}m_z)] [E_{1u} ] [m_xm_ym_z] [6_z/m_z m_xm_y], [\bar 6_z2_xm_y ], [m_x^{\star}m_ym_z] 6 2 6 3
    [m_x2_ym_z] [(m_{x{^\prime}}2_{y{^\prime}}m_z ], [m_{x{^\prime}{^\prime}}2_{y{^\prime}{^\prime}}m_z)] [E_{1u}] [m_xm_ym_z] [6_z/m_z m_xm_y], [\bar 6_zm_x2_y ], [m_xm_y^{\star}m_z] 6 2 6 3
    [2_x2_y2_z] [(2_{x{^\prime}}2_{y{^\prime}}2_z ], [2_{x{^\prime}{^\prime}}2_{y{^\prime}{^\prime}}2_z)] [E_{2u}] [m_xm_ym_z ] [6_z/m_zm_xm_y], [6_z2_x2_y], [m_x^{\star}m_y^{\star}m_z^{\star}] 6 6 0 3
    [2_z/m_z] [E_{2g}] [6_z/m_zm_xm_y] [6_z/m_z], [m_x^{\star}m_y^{\star}m_z(3) ] 6 6 0 6
    [2_x/m_x] [(2_{x{^\prime}}/m_{x{^\prime}} ], [2_{x{^\prime}{^\prime}}/m_{x{^\prime}{^\prime}})] [E_{1g}] [m_xm_ym_z] [6_z/m_zm_xm_y], [\bar 3_zm_x], [m_xm_y^{\star}m_z^{\star} ] 6 2 0 6
    [2_y/m_y] [(2_{y{^\prime}}/m_{y{^\prime}} ], [2_{y{^\prime}{^\prime}}/m_{y{^\prime}{^\prime}})] [E_{1g}] [m_xm_ym_z ] [6_z/m_zm_xm_y], [\bar 3m_y], [m_x^{\star}m_ym_z^{\star} ] 6 2 0 6
    [m_z] [E_{1u}] [6_z/m_zm_xm_y] [6_z/m_z], [\bar 6_z], [2_x^{\star}m_y^{\star}m_z], [m_x^{\star}2_y^{\star}m_z], [2_z^{\star}/m_z ] 12 12 12 6
    [m_x] [(m_{x{^\prime}}], [m_{x{^\prime}{^\prime}})] Reducible [m_xm_ym_z ] [\bar 6_zm_x2_y], [6_zm_xm_y], [\bar 3_zm_x], [3_zm_x], [m_xm_y^{\star}2_z^{\star}], [m_x2_y^{\star}m_z^{\star}], [ 2_x^{\star}/m_x] 12 4 12 6
    [m_y] [(m_{y{^\prime}}], [m_{y{^\prime}{^\prime}})] Reducible [m_xm_ym_z] [\bar 6_z2_xm_y], [6_zm_xm_y], [\bar 3_zm_y], [3_zm_y], [m_x^{\star}m_y2_z^{\star}], [2_x^{\star}m_ym_z^{\star}], [2_y^{\star}/m_y] 12 4 12 6
    [2_z] [E_{2u}] [6_z/m_zm_xm_y ] [6_z/m_z], [6_z], [m_x^{\star}m_y^{\star}2_z (3) ], [2_x^{\star}2_y^{\star}2_z(3)], [2_z/m_z^{\star}] 12 12 2 6
    [2_x ] [(2_{x{^\prime}}], [2_{x{^\prime}{^\prime}})] Reducible [m_xm_ym_z] [\bar 6_z2_xm_y], [6_z2_x2_y], [\bar 3_zm_x], [3_z2_x], [2_xm_y^{\star}m_z^{\star}], [2_x2_y^{\star}2_z^{\star}], [2_x/m_x^{\star}] 12 4 6 6
    [2_y] [(2_{y{^\prime}}], [2_{y{^\prime}{^\prime}})] Reducible [m_xm_ym_z] [\bar 6_zm_x2_y], [6_z2_x2_y], [\bar 3_zm_y], [3_z2_y], [m_x^{\star}2_ym_z^{\star}], [2_x^{\star}2_y2_z^{\star}], [2_y/m_y^{\star}] 12 4 6 6
    [\bar 1 ] [E_{1g}] [6_z/m_zm_xm_y ] [6_z/m_z], [\bar 3_z], [2_z^{\star}/m_z^{\star}], [2_x^{\star}/m_x^{\star}(3)], [2_y^{\star}/m_y^{\star}(3) ] 12 12 0 12
    1 Reducible [6_z/m_zm_xm_y ] [\bar 6_z], [6_z], [\bar 3_z ], [3_z], [m_z^{\star}], [m_x^{\star}(3)], [m_y^{\star}(3) ], [2_z^{\star}], [2_x^{\star}(3)], [2_y^{\star}(3)], [\bar 1^{\star}] 24 24 24 12
    [{\bf 23}] [3_p] [(3_q], [3_r], [3_s)] T [3_p] 23 4 1 4 4
    [2_x2_y2_z] E 23 23 3 3 0 3
    [2_z] [(2_x], [2_y)] T [2_x2_y2_z] [23], [2_x^{\star}2_y^{\star}2_z ] 6 2 6 6
    1 T [23 ] [3_p(4)], [2_z^{\star}(3)] 12 12 12 12
    [{\bi m}{\bar {\bf 3}} ] 23 [A_u ] [m\bar3] [m^{\star}\bar3^{\star}] 2 2 0 1
    [\bar 3_p] [(\bar 3_q], [\bar3_r], [\bar 3_s) ] [T_g] [\bar3_p] [m\bar3] 4 1 0 4
    [3_p] [(3_q], [3_r], [3_s)] [T_u] [\bar3_p] [m\bar3, 23 ] 8 2 8 4
    [m_xm_ym_z] [E_g ] [m\bar3 ] [m\bar3] 3 3 0 3
    [m_xm_y2_z ] [(2_xm_ym_z], [m_x2_ym_z)] [T_u ] [m_xm_ym_z] [m\bar3, m_xm_ym_z^{\star}] 6 2 6 3
    [2_x2_y2_z] [E_u ] [m\bar3] [m\bar3], [23], [m_x^{\star}m_y^{\star}m_z^{\star} ] 6 6 0 3
    [2_z/m_z] [(2_x/m_x], [2_y/m_y)] [T_g] [m_xm_ym_z] [m\bar3], [m_x^{\star}m_y^{\star}m_z ] 6 2 0 6
    [m_z] [(m_x], [m_y)] [T_u ] [m_xm_ym_z ] [m\bar3], [2_x^{\star}m_y^{\star}m_z ], [m_x^{\star}2_y^{\star}m_z], [2_z^{\star}/m_z] 12 4 12 6
    [2_z] [(2_x], [2_y)] Reducible [m_xm_ym_z] [m\bar3], [23], [m_x^{\star}m_y^{\star}2_z ], [2_x^{\star}2_y^{\star}2_z], [2_z/m_z^{\star}] 12 4 6 6
    [\bar 1] [T_g ] [m\bar3] [\bar3_p(4)], [2_z^{\star}/m_z^{\star}(3) ] 12 12 0 12
    1 [T_u ] [m\bar3] [\bar3_p(4)], [3_p(4)], [m_z^{\star}(3)], [2_z^{\star}(3)], [\bar1^{\star}] 24 24 24 12
    [{\bf 432}] 23 [A_2] 432 [4^{\star}32^{\star}] 2 2 0 1
    [3_p2_{x\bar{y}}] [(3_q2_{x\bar{y}} ], [ 3_r2_{xy}], [3_s2_{xy})] [T_2] [3_p2_{x\bar{y}} ] 432 4 1 0 4
    [3_p] [(3_q], [3_r], [3_s)] [T_1] [3_p2_{x\bar{y}} ] [23], [3_p2_{x\bar{y}}^{\star} ] 8 2 8 4
    [4_z2_x2_{xy}] [(4_x2_y2_{yz}], [4_y2_z2_{xz})] E [4_z2_x2_{xy}] 432 3 1 0 3
    [4_z] [(4_x], [4_y)] [T_1] [4_z2_x2_{xy}] [432], [4_z2_x^{\star}2_{xy}^{\star} ] 6 2 6 3
    [2_x2_y2_z] E 432 [23], [4_z^{\star}2_x2_{xy}^{\star} ] 6 6 0 6
    [2_{x\bar{y}}2_{xy}2_z ] [(2_{y\bar{z}}2_{yz}2_x ], [2_{z\bar{x}}2_{zx}2_y)] [T_2] [4_z2_x2_{xy}] [432], [4_z^{\star}2_x^{\star}2_{xy} ] 6 2 0 6
    [2_z] [(2_x], [2_y)] Reducible [4_z2_x2_{xy}] 23, [4_y2_z2_{xy}], [4_z^{\star} ], [2_{x\bar{y}}^{\star}2_{xy}^{\star}2_z], [2_x^{\star}2_y^{\star}2_z ] 12 4 6 12
    [2_{xy}] [(2_{yz}], [2_{zx} ], [2_{x\bar{y}}], [2_{y\bar{z}}], [2_{z\bar{x}})] [T_1], [T_2 ] [2_{x\bar{y}}2_{xy}2_z ] [432], [3_r2_{xy}], [3_s2_{xy} ], [4_z2_x2_{xy}], [2_{x\bar{y}}2_{xy}^{\star}2_z^{\star}] 12 2 12 12
    1 [T_1], [T_2 ] 432 [3_p(4) ], [4_z(3)], [2_z^{\star}(3) ], [2_{xy}^{\star}(6)] 24 24 24 24
    [{\bar {\bf 4}}{\bf 3}{\bi m}] 23 [A_2 ] [\bar43m] [\bar4^{\star}3m^{\star} ] 2 2 0 1
    [3_pm_{x\bar{y}}] [(3_qm_{x\bar{y}} ], [3_rm_{xy}], [3_sm_{xy})] [T_2] [3_pm_{x\bar{y}} ] [\bar43m ] 4 1 4 4
    [3_p] [(3_q], [3_r], [3_s)] [T_1 ] [3_pm_{x\bar{y}} ] [\bar43m], [23], [3_pm_{x\bar{y}}^{\star} ] 8 2 4 4
    [\bar4_z2_xm_{xy}] [(\bar4_x2_ym_{yz} ], [\bar4_y2_zm_{zx})] E [\bar4_z2_xm_{x\bar{y}}] [\bar43m] 3 1 0 3
    [\bar4_z] [(\bar4_x, \bar4_y)] [T_1 ] [\bar4_z2_xm_{x\bar{y}}] [\bar43m], [\bar4_z2_x^{\star}m_{xy}^{\star} ] 6 2 0 3
    [m_{x\bar{y}}m_{xy}2_z] [(m_{y\bar{z}}m_{yz}2_x ], [m_{z\bar{x}}m_{zx}2_y)] [T_2 ] [\bar4_z2_xm_{x\bar{y}}] [\bar43m], [\bar4_z^{\star}2_x^{\star}m_{xy} ] 6 2 6 6
    [2_x2_y2_z] E [\bar43m ] [23, \bar4_z^{\star}2_xm_{xy}^{\star} ] 6 6 0 6
    [m_{xy}] [(m_{yz}], [m_{zx} ], [m_{x\bar{y}}], [m_{y\bar{z}}], [m_{z\bar{x}})] [T_1, T_2] [m_{x\bar{y}}m_{xy}2_z] [\bar43m], [3_rm_{xy}], [3_sm_{xy}], [\bar4_z2_xm_{xy}], [m_{x\bar{y}}^{\star}m_{xy}2_z^{\star} ] 12 2 12 12
    [2_z] [(2_x], [2_y)] Reducible [\bar4_z2_xm_{xy} ] 23, [\bar4_z^{\star}], [4_z^{\star} ], [m_{x\bar{y}}^{\star}m_{xy}^{\star}2_z], [2_x^{\star}2_y^{\star}2_z ] 12 4 6 12
    1 [T_1], [T_2] [\bar43m ] [3_p(4) ], [\bar4_z(3)], [m_{xy}^{\star}(6)], [2_z^{\star}(3)] 24 24 24 24
    [{\bi m}{\bar {\bf 3}}{\bi m} ] [\bar43m] [A_{2u}] [m\bar3m] [m^{\star}\bar3^{\star}m] 2 2 0 1
    432 [A_{1u}] [m\bar3m] [m^{\star}\bar3^{\star}m^{\star}] 2 2 0 1
    [m\bar3 ] [A_{2g}] [m\bar3m] [m\bar3m^{\star}] 2 2 0 1
    23 Reducible [m\bar3m] [\bar4^{\star}3m^{\star}], [4^{\star}32^{\star} ], [m_z^{\star}\bar3_p ] 4 4 0 1
    [\bar 3_pm_{x\bar{y}}] [(\bar 3_qm_{x\bar{y}} ], [\bar3_rm_{xy}], [\bar 3_sm_{xy})] [T_{2g}] [\bar3_pm_{x\bar{y}}] [m\bar3m] 4 1 0 4
    [3_pm_{x\bar{y}}] [(3_qm_{x\bar{y}} ], [3_rm_{xy}], [3_sm_{xy})] [T_{1u}] [\bar3_pm_{x\bar{y}} ] [m\bar3m ], [\bar43m], [\bar3_p^{\star}m_{x\bar{y}}] 8 2 8 4
    [3_p2_{x\bar{y}}] [(3_q2_{x\bar{y}} ], [3_r2_{xy}], [3_s2_{xy})] [T_{2u}] [\bar3_pm_{x\bar{y}} ] [m\bar3m], [432], [\bar3_p^{\star}m_{x\bar{y}} ] 8 2 0 4
    [\bar 3_p] [(\bar 3_q], [\bar3_r], [\bar 3_s)] [T_{1g}] [\bar3_pm_{x\bar{y}}] [m\bar3m], [m\bar3], [\bar3_pm_{x\bar{y}}^{\star} ] 8 2 0 4
    [3_p] [(3_q], [3_r], [3_s)] Reducible [\bar3_pm_{x\bar{y}}] [\bar43m], [432], [m\bar3 ], [23], [3_pm_{x\bar{y}}^{\star}], [3_p2_{x\bar{y}}^{\star} ], [\bar3_p^{\star}] 16 4 8 4
    [4_z/m_zm_xm_{xy}] [(4_x/m_xm_ym_{yz} ], [4_y/m_ym_zm_{zx})] [E_g ] [4_z/m_zm_xm_{xy}] [m\bar3m] 3 1 0 3
    [\bar4_z2_xm_{xy}] [(\bar4_x2_ym_{yz} ], [\bar4_y2_zm_{zx})] [E_u ] [4_z/m_zm_xm_{xy}] [m\bar3m], [\bar43m], [4_z^{\star}/m_z^{\star}m_x^{\star}m_{xy}] 6 2 0 3
    [\bar4_zm_x2_{xy}] [(\bar4_xm_y2_{yz} ], [\bar4_ym_z2_{zx})] [T_{2u} ] [4_z/m_zm_xm_{xy}] [m\bar3m], [4_z^{\star}/m_z^{\star}m_xm_{xy}^{\star} ] 6 2 0 3
    [4_zm_xm_{xy}] [(4_xm_ym_{yz}], [4_ym_zm_{zx})] [T_{1u} ] [4_z/m_zm_xm_{xy}] [m\bar3m], [4_z/m_z^{\star}m_xm_{xy} ] 6 2 6 3
    [4_z2_x2_{xy}] [(4_x2_y2_{yz}], [4_y2_z2_{zx})] [E_u] [4_z/m_zm_xm_{xy} ] [m\bar3m], [432], [4_z/m_z^{\star}m_x^{\star}m_{xy}^{\star} ] 6 2 0 3
    [4_z/m_z] [(4_x/m_x], [4_y/m_y)] [T_{1g} ] [4_z/m_zm_xm_{xy}] [m\bar3m], [4_z/m_zm_x^{\star}m_{xy}^{\star} ] 6 2 0 3
    [\bar4_z] [(\bar4_x], [\bar4_y)] Reducible [4_z/m_zm_xm_{xy}] [m\bar3m], [\bar43m], [\bar4_z2_x^{\star}m_{xy}^{\star}], [\bar4_zm_x^{\star}2_{xy}^{\star} ], [4_z^{\star}/m_z^{\star}] 12 4 0 3
    [4_z] [(4_x ], [4_y)] Reducible [4_z/m_zm_xm_{xy}] [m\bar3m], [432], [4_zm_x^{\star}m_{xy}^{\star} ], [4_z2_x^{\star}2_{xy}^{\star}], [4_z/m_z^{\star}] 12 4 6 3
    [m_xm_ym_z] [E_g] [m\bar3m ] [m\bar3], [4_z^{\star}/m_zm_xm_{xy}^{\star} ] 6 6 0 6
    [m_{x\bar{y}}m_{xy}m_z] [(m_{y\bar{z}}m_{yz}m_x ], [m_{z\bar{x}}m_{zx}m_y)] [T_{2g} ] [4_z/m_zm_xm_{xy}] [m\bar3m ], [4_z^{\star}/m_zm_xm_{xy}^{\star} ] 6 2 0 6
    [m_xm_y2_z] [(2_xm_ym_z], [m_x2_ym_z)] Reducible [4_z/m_zm_xm_{xy}] [m\bar3], [4_y/m_ym_zm_{zx}], [\bar4_z^{\star}m_x2_{xy}^{\star}], [4_z^{\star}m_xm_{xy}^{\star}], [ m_xm_ym_z^{\star}] 12 4 6 6
    [m_{x\bar{y}}m_{xy}2_z] [(m_{y\bar{z}}m_{yz}2_x ], [m_{z\bar{x}}m_{zx}2_y)] Reducible [4_z/m_zm_xm_{xy}] [m\bar3m], [\bar43m], [\bar4_z^{\star}2_x^{\star}m_{xy}], [4_z^{\star}m_x^{\star}m_{xy}], [m_{x\bar{y}}m_{xy}m_z^{\star}] 12 4 6 6
    [m_{x\bar{y}}2_{xy}m_z] [(m_{y\bar{z}}2_{yz}m_x ], [m_{z\bar{x}}2_{zx}m_y], [ 2_{x\bar{y}}m_{xy}m_z], [2_{y\bar{z}}m_{yz}m_x], [2_{z\bar{x}}m_{zx}m_y)] [T_{1u}], [T_{2u}] [m_{x\bar{y}}m_{xy}m_z ] [m\bar3m(m_{zx})], [m\bar3m(2_{zx}) ], [4_z/m_zm_xm_{xy}], [ m_{x\bar{y}}m_{xy}^{\star}m_z] 12 2 12 6
    [2_x2_y2_z] [E_u] [m\bar3m ] [m\bar3], [23], [\bar4_z^{\star}2_xm_{xy}^{\star} ], [4_z^{\star}2_x2^{\star}_{xy}], [m_x^{\star}m_y^{\star}m_z^{\star} ] 12 12 0 6
    [2_{x\bar{y}}2_{xy}2_z] [(2_{y\bar{z}}2_{yz}2_x ], [2_{z\bar{x}}2_{zx}2_y)] Reducible [4_z/m_zm_xm_{xy}] [m\bar3m], [432], [\bar4_zm_x^{\star}2_{xy} ], [4^{\star}_z2^{\star}_x2_{xy}], [m_{x\bar{y}}^{\star}m_{xy}^{\star}m_z^{\star} ] 12 4 0 6
    [2_z/m_z] [(2_x/m_x, 2_y/m_y)] Reducible [4_z/m_zm_xm_{x\bar{y}}] [m\bar3], [ 4_y/m_ym_zm_{zx}], [4_z^{\star}/m_z], [m_x^{\star}m_y^{\star}m_z], [m_{x\bar{y}}^{\star}m_{xy}^{\star}m_z ] 12 4 0 12
    [2_{xy}/m_{xy}] [(2_{yz}/m_{yz} ], [2_{zx}/m_{zx}], [2_{x\bar{y}}/m_{x\bar{y}}], [2_{y\bar{z}}/m_{y\bar{z}} ], [2_{z\bar{x}}/m_{z\bar{x}})] [T_{1g}], [T_{2g}] [m_{x\bar{y}}m_{xy}m_z] [m\bar3m], [\bar3_rm_{xy} (2)], [4_z/m_zm_xm_{xy}], [m_{x\bar{y}}^{\star}m_{xy}m_z^{\star}] 12 2 0 12
    [m_z] [(m_x], [m_y)] [T_{1u}], [T_{2u} ] [4_z/m_zm_xm_{x\bar{y}} ] [m\bar3], [\bar4_xm_z2_{yz} ], [4_ym_zm_{zx}], [4_z/m_z], [2_x^{\star}m_y^{\star}m_z (2) ], [m_{x\bar{y}}^{\star}2_{xy}^{\star}m_z(2)], [2_z^{\star}/m_z ] 24 8 24 12
    [m_{xy}] [(m_{yz} ], [m_{zx}], [m_{x\bar{y}}], [m_{y\bar{z}}], [m_{z\bar{x}})] [T_{1u}] [m_{x\bar{y}}m_{xy}m_z ] [m\bar3m], [\bar43m ], [\bar4_z2_xm_{xy}], [4_zm_xm_{xy}], [\bar3_rm_{xy} ], [\bar3_sm_{xy}], [3_rm_{xy}], [3_sm_{xy}], [m_{x\bar{y}}^{\star}m_{xy}2_z^{\star} ], [2_{xy}^{\star}/m_{xy}] 24 4 24 12
    [2_z] [(2_x], [2_y)] Reducible [4_z/m_zm_xm_{x\bar{y}} ] [m\bar3], [23], [\bar4_y2_zm_{zx}], [4_y2_z2_{zx}], [\bar4_z^{\star}], [4_z^{\star}], [m_x^{\star}m_y^{\star}2_z], [m_{x\bar{y}}^{\star}m_{xy}^{\star}2_z ], [2_x^{\star}2_y^{\star}2_z ], [2_{x\bar{y}}^{\star}2_{xy}^{\star}2_z ], [2_z/m_z^{\star}] 24 8 6 12
    [2_{xy}] [(2_{yz} ], [ 2_{zx}], [2_{x\bar{y}}], [2_{y\bar{z}}], [2_{z\bar{x}})] [T_{2u}] [m_{x\bar{y}}m_{xy}m_z ] [m\bar3m], [432 ], [\bar3_rm_{xy}], [\bar3_sm_{xy}], [3_r2_{xy}], [3_s2_{xy}], [\bar4_zm_x2_{xy}], [4_z2_x2_{xy}], [m_{x\bar{y}}^{\star}2_{xy}m_z^{\star} ], [2_{x\bar{y}}^{\star}2_{xy}2_z^{\star}], [2_{xy}/m_{xy}^{\star} ] 24 4 12 12
    [\bar 1] [T_{1g}], [T_{2g} ] [m\bar3m ] [\bar3_p(4)], [4_z/m_z(3)], [2_z^{\star}/m_z^{\star}(3)], [2_{xy}^{\star}/m_{xy}^{\star}(6) ] 24 24 0 24
    1 [T_{1u}], [T_{2u} ] [m\bar3m ] [\bar3_p(4)], [\bar4_z(3)], [4_z(3)], [m_z^{\star}(3)], [m_{xy}^{\star}(6) ], [2_z^{\star}(3)], [2_{xy}^{\star}(6)], [ \bar1^{\star} ] 48 48 48 24
    [u =x], y, z, [xy], [yz], [zx], [x\bar{y}], [y\bar{z}], [z\bar{x}], [x{^\prime}], [x{^\prime}{^\prime}], [y{^\prime} ], [y{^\prime}{^\prime}].
  • [F_1]: this point group is a proper subgroup of G given in the first column and expresses the symmetry of the ferroic phase in the first single-domain state [{\bf S}_1]. In accordance with IT A (2005[link]), five groups are given in two orientations (bold and normal type). Subscripts of generators in the group symbol specify their orientation in the Cartesian (rectangular) crystallophysical coordinate system of the group G (see Tables 3.4.2.5[link] and 3.4.2.6[link], and Figs. 3.4.2.3[link]and 3.4.2.4[link]). In the cubic groups, the direction of the body diagonal is denoted by abbreviated symbols: [p\equiv[111]] (all positive), [q\equiv[\bar 1\bar 11]], [r\equiv[1\bar 1\bar 1] ], [s\equiv[\bar 1 1 \bar 1]]. In the hexagonal and trigonal groups, axes x′, y′ and x′′, y′′ of a Cartesian coordinate system are rotated about the z axis through 120 and 240°, respectively, from the crystallophysical Cartesian coordinate axes x and [y ].

    Symmetry groups in parentheses are groups conjugate to [F_1 ] under G (see Section 3.2.3.2[link] ). These are symmetry groups (stabilizers) of some domain states [{\bf S}_k] different from [{\bf S}_1] (for more details see Section 3.4.2.2.3[link]).

  • [\Gamma_{\eta}]: physically irreducible representation of the group G. This specifies the transformation properties of the principal tensor parameter of the phase transition in a continuum description and transformation properties of the primary order parameter [\eta] of the equitranslational phase transitions in the microscopic description. The letters A, B signify one-dimensional representations, and letters E and T two- and three-dimensional irreducible representations, respectively. Two letters T indicate that the symmetry descent [G\subset F_1] can be accomplished by two non-equivalent three-dimensional irreducible representations (see Table 3.1.3.2[link] ). `Reducible' denotes a reducible representation of G. In this case, there are always several non-equivalent reducible representations inducing the same descent [G\subset F_1 ] [for more detailed information see the software GI[\star ]KoBo-1 and Kopský (2001[link])].

    Knowledge of [\Gamma_{\eta}] enables one to determine for all ferroic transitions property tensors and their components that are different in all principal domain states, and, for equitranslational transitions only, microscopic displacements and/or ordering of atoms and molecules that are different in different basic (microscopic) domain states (for details see Section 3.1.3[link] , especially Table 3.1.3.1[link] , and Section 3.1.2[link] ).

  • [{N_G} {(F}{_1)}]: the normalizer of [F_1] in G (defined in Section 3.2.3.2.4[link] ) determines subgroups conjugate to [F_1] in G and specifies which domain states have the same symmetry (stabilizer in G). The number [n_F] of subgroups conjugate to [F_1 ] in G is [n_F =] [[G:N_G(F_{1})] =] [|G|:|N_G(F_{1})|] [see equation (3.4.2.36)[link]] and the number [d_F] of principal domain states with the same symmetry is [d_F =] [[N_G(F_{1}):F_1] =] [|N_G(F_{1})|:|F_1|] [see equation (3.4.2.35)[link]]. There are three possible cases:

    (i) [N_G(F_1)=G ]. There are no subgroups conjugate to [F_1] and the symmetry group [F_i] (stabilizer of [{\bf S}_i] in G) of all principal domain states [{\bf S}_1,{\bf S}_2,\ldots,{\bf S}_n] is equal to [F_i] for all [i=1,2,\ldots,n]; hence domain states cannot be distinguished by their symmetry. The group [F_1] is a normal subgroup of [G, F_1\triangleleft G] (see Section 3.2.3.2[link] ). This is always the case if there are just two single-domain states [{\bf S}_1], [{\bf S}_2], i.e. if the index of [F_1] in G equals two, [[G:F_1]=] [|G|:|F_1|=2].

    (ii) [N_G(F_1)=F_1]. Then any two domain states [{\bf S}_i], [{\bf S}_k] have different symmetry groups (stabilizers), [{\bf S}_i\neq{\bf S}_k\Leftrightarrow F_i\neq F_k ], i.e. there is a one-to-one correspondence between single-domain states and their symmetries, [{\bf S}_i\Leftrightarrow F_i]. In this case, principal domain states [{\bf S}_i] can be specified by their symmetries [F_i, i=1,2,\ldots,n]. The number [n_F] of different groups conjugate to [F_1] is equal to the index [[G:F_1]=|G|:|F_1|=n ].

    (iii) [F_1\subset N_G(F_1)\subset G]. Some, but not all, domain states [{\bf S}_i], [{\bf S}_k] have identical symmetry groups (stabilizers) [F_i=F_k]. The number [d_F] of domain states with the same symmetry group is [d_F] = [[N_G(F_1):F_1]] = [|N_G(F_1)|:|F_1| ] [see equation (3.4.2.35[link])], [1 \,\lt\, d_F \,\lt\, n ]. The number [n_F] of different groups conjugate to [F_1] is equal to the index [n_F=] [[G:N_G(F_1)] =] [|G|:|N_G(F_1)|] [see equation (3.4.2.36[link])] and in this case [1 \,\lt\, n_F \,\lt\, n]. It always holds that [n_Fd_F=n] [see equation (3.4.2.37[link])].

  • [{K}_G(F_1,g_{1{j}})]: twinning group of a domain pair ([{\bf S}_1], [{\bf S}_j]). This group is defined in Section 3.4.3.2[link]. It can be considered a colour (polychromatic) group involving c colours, where [c=[K_{1j}:F_1]], and is, therefore, defined by two groups [K_{1j}] and [F_1], and its full symbol is [K_{1j}(F_1)]. In this column only [K_{1j}] is given, since [F_1] appears in the second column of the table.

    If the group symbol of [K_{1j}] contains generators with the star symbol, [^{\star}], which signifies transposing operations of the domain pair ([{\bf S}_1], [{\bf S}_j]), then the symbol [K_{1j}(F_1)] denotes a dichromatic (`black-and-white') group signifying a completely transposable domain pair. In this special case, just the symbol [K_{1j}] containing stars [^{\star}] specifies the group [F_1 ] unequivocally.

    The number in parentheses after the group symbol of [K_{1j}] is equal to the number of twinning groups [K_{1k}] equivalent with [K_{1j}].

    In the continuum description, a twinning group is significant in at least in two instances:

    • (1) A twinning group [K_{1j}(F_1)] specifies the distinction of two domain states [{\bf S}_1] and [{\bf S}_j=g_{1j}{\bf S}_1 ], where [g_{1j} \in G] (see Sections 3.4.3.2[link] and 3.4.3.4[link]).

    • (2) A twinning group [K_{1j}(F_1)] may assist in signifying classes of equivalent domain pairs (orbits of domain pairs). In most cases, to a twinning group [F_{1j}] there corresponds just one class of equivalent domain pairs (an orbit) G([{\bf S}_1,{\bf S}_j ]); then a twinning group can represent this class of equivalent domain pairs. Nevertheless, in some cases two or more classes of equivalent domain pairs have a common twinning group. Then one has to add a switching operation [g_{1j}] to the twinning group, [K_{1j}(F_1, g_{1j})] (see the end of Section 3.4.3.2[link]). In this way, classes of equivalent domain pairs G([{\bf S}_1], [{\bf S}_j]) are denoted in synoptic Tables 3.4.2.7[link] and 3.4.3.6[link].

    Twinning groups given in column [K_{1j}] thus specify all G-orbits of domain pairs. The number of G-orbits and representative domain pairs for each orbit are determined by double cosets of group [F_1] (see Section 3.4.3.2[link]). Representative domain pairs from each orbit of domain pairs are further analysed in synoptic Table 3.4.3.4[link] (non-ferroelastic domain pairs) and in synoptic Table 3.4.3.6[link] (ferroelastic domain pairs).

    The set of the twinning groups [K_{1j}] given in this column is analogous to the concept of a complete twin defined as `an edifice comprising in addition to an original crystal (domain state [{\bf S}_1]) as many twinned crystals (domain states [{\bf S}_j]) as there are possible twin laws' (see Curien & Le Corre, 1958[link]). If a traditional definition of a twin law [`a geometrical relationship between two crystal components of a twin', see Section 3.3.2[link] and Koch (2004[link]); Curien & Le Corre (1958[link])] is applied sensu stricto to domain twins then one gets the following correspondence:

    • (i) a twin law of a non-ferroelastic domain twin is specified by the twinning group [K_{1j}] (see Section 3.4.3.3[link] and Table 3.4.3.4[link]);

    • (ii) two twin laws of two compatible ferroelastic domain twins, resulting from one ferroelastic single-domain pair [\{{\bf S}_1,{\bf S}_j\} ], are specified by two layer groups [\overline{\sf {J}}_{1j}] associated with the twinning group [K_{1j}] of this ferroelastic single-domain pair [\{({\bf S}_1,{\bf S}_j)\}] (see Section 3.4.3.4[link] and Table 3.4.3.6[link]).

  • n: number of principal single-domain states, the finest sub­division of domain states in a continuum description, [n=] [[G:F_1]=|G|:|F_1|] [see equation (3.4.2.11[link])].

  • [{d_{F}}]: number of principal domain states with the same symmetry group (stabilizer), [d_F=[N_G(F_1):F_1]=|N_G(F_1)|:|F_1| ] [see equation (3.4.2.35[link])]. If [d_F>1], then the group [F_1] does not specify the first single-domain state [{\bf S}_1]. The number [n_F] of subgroups conjugate with [F_1 ] is [n_F=n:d_F].

  • [{ n_e}]: number of ferroelectric single-domain states, [n_e=] [[G:C_1]=|G|:|C_1|], where [C_1] is the stabilizer (in G) of the spontaneous polarization in the first domain state [{\bf S}_1 ] [see equation (3.4.2.32)[link]]. The number [d_e] of principal domain states compatible with one ferroelectric domain state (degeneracy of ferroelectric domain states) equals [d_e=[C_1:F_1]=|C_1|:|F_1|] [see equation (3.4.2.33[link])].

    Aizu's classification of ferroelectric phases (Aizu, 1969[link]; see Table 3.4.2.3[link]): [n_e=n], fully ferroelectric; [1 \,\lt\, n_e \,\lt\, n], partially ferroelectric; [n_e=1], non-ferroelectric, the parent phase is polar and the spontaneous polarization in the ferroic phase is the same as in the parent phase; [n_e=0], non-ferroelectric, parent phase is non-polar.

  • [{ n_a}]: number of ferroelastic single-domain states, [n_a=] [[G:A_1]] [=|G|:|A_1|], where [A_1] is the stabilizer (in G) of the spontaneous strain in the first domain state [{\bf S}_1 ] [see equation (3.4.2.28[link])]. The number [d_a] of principal domain states compatible with one ferroelastic domain state (degeneracy of ferroelastic domain states) is given by [d_a=[A_1:F_1]=|A_1|:|F_1|] [see equation (3.4.2.29[link])].

    Aizu's classification of ferroelastic phases (Aizu, 1969[link]; see Table 3.4.2.3[link]): [n_a=n], fully ferroelastic; [1 \,\lt\, n_a \,\lt\, n ], partially ferroelastic; [n_e=1], non-ferroelastic.

Example 3.4.2.5. Orthorhombic phase of perovskite crystals.  The parent phase has symmetry [G=m\bar3m] and the symmetry of the ferroic orthorhombic phase is [F_1=m_{x\bar y}2_{xy}m_z]. In Table 3.4.2.7[link], we find that [n=n_e], i.e. the phase is fully ferroelectric. Then we can associate with each principal domain state a spontaneous polarization. In column [K_{1j}] there are four twinning groups. As explained in Section 3.4.3[link], these groups represent four `twin laws' that can be characterized by the angle between the spontaneous polarization in single-domain state [{\bf S}_1] and [{\bf S}_j], [j=2,3,4,5]. If we choose [{\bf P}_{(s)}^{(1)}] along the direction [110] ([F_1] does not specify unambiguously this direction, since [d_F=2]!), then the angles between [{\bf P}_{(s)}^{(1)} ] and [{\bf P}_{(s)}^{(j)}], representing the `twin law' for these four twinning groups [m\bar3m(m_{zx})], [m\bar3m(2_{zx})], [4_z/m_zm_xm_{xy}], [m_{x\bar{y}}m_{xy}^{\star}m_z], are, respectively, 60, 120, 90 and 180°.

References

IEEE Standards on Piezoelectricity (1987). IEEE Std 176–987. New York: The Institute of Electrical and Electronics Engineers, Inc.
International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, 5th ed., edited by Th. Hahn. Heidelberg: Springer.
Aizu, K. (1969). Possible species of `ferroelastic' crystals and of simultaneously ferroelectric and ferroelastic crystals. J. Phys. Soc. Jpn, 27, 387–396.
Aizu, K. (1970a). Possible species of ferromagnetic, ferroelectric and ferroelastic crystals. Phys. Rev. B, 2, 754–772.
Curien, H. & Le Corre, Y. (1958). Notation des macles à l'aide du symbolisme des groupes de couleurs de Chubnikov. Bull. Soc. Fr. Mineral. Cristallogr. 81, 126–132.
Janovec, V. (2012). Non-ferromagnetic species of ferroic crystals. http://palata.fzu.cz/species/ .
Koch, E. (2004). Twinning. In International Tables for Crystallography, Vol. C, Mathematical, Physical and Chemical Tables, 3rd ed., edited by E. Prince, ch. 1.3. Dordrecht: Kluwer Academic Publishers.
Kopský, V. (2001). Tensor parameters of ferroic phase transitions I. Theory and tables. Phase Transit. 73, 1–422.
Nye, J. F. (1985). Physical Properties of Crystals. Oxford: Clarendon Press.
Shuvalov, L. A. (1988). Editor. Modern Crystallography IV. Physical Properties of Crystals. Berlin: Springer.
Sirotin, Yu. I. & Shaskolskaya, M. P. (1982). Fundamentals of Crystal Physics. Moscow: Mir.








































to end of page
to top of page