# Isotropy subgroups (crystod-group --parent); testsuite section 17
# Validated against ISOSUBGROUP (ISOTROPY Software Suite, https://iso.byu.edu)
# Order-parameter components: arms separated by ";", components within one arm by ","
# --kpoint K (a name such as GM, or three primitive coordinates) in place of --irrep
# lists every irrep of that k point in one table

$ crystod-group --parent Pm-3m --irrep GM4- --order-parameter 0 0 a

* Supergroup *
Pm-3m (No. 221)

* Irrep *
GM4-: order parameter dimension 3
  Landau condition: satisfied (no cubic invariant)
  Lifshitz condition: satisfied (no Lifshitz invariant)
  -> a continuous transition is allowed

* Isotropy subgroup *
GM4-(0,0,a) -> P4mm (No. 99)
cell size 1, index 6
sublattice basis (parent primitive units): (1,0,0), (0,1,0), (0,0,1)
conventional basis (parent conventional units): (0,0,1), (1,0,0), (0,1,0)
origin: (0,0,0)

Conventions and validation: ISOSUBGROUP (https://iso.byu.edu):
H. T. Stokes, S. van Orden and B. J. Campbell, "Tool for Generating
Isotropy Subgroups of Crystallographic Space Groups",
J. Appl. Cryst. 49, 1849-1853 (2016).

$ crystod-group --parent Pm-3m --irrep GM4-

* Supergroup *
Pm-3m (No. 221)

* Irrep *
GM4-: order parameter dimension 3

* Order parameter directions and isotropy subgroups *
irrep                subgroup           size  index
GM4-(0,0,a)          99 P4mm            1     6    
GM4-(a,a,a)          160 R3m            1     8    
GM4-(0,a,a)          38 Amm2            1     12   
GM4-(0,a,b)          6 Pm               1     24   
GM4-(a,a,b)          8 Cm               1     24   
GM4-(a,b,c)          1 P1               1     48   

Conventions and validation: ISOSUBGROUP (https://iso.byu.edu):
H. T. Stokes, S. van Orden and B. J. Campbell, "Tool for Generating
Isotropy Subgroups of Crystallographic Space Groups",
J. Appl. Cryst. 49, 1849-1853 (2016).

$ crystod-group --parent Pm-3m --kpoint GM

* Supergroup *
Pm-3m (No. 221)

* Kpoint *
GM
k = (0, 0, 0) in the primitive basis, star of 1 arm(s)

* Order parameter directions and isotropy subgroups *
irrep                subgroup           size  index
GM1+(a)              221 Pm-3m          1     1    
GM2+(a)              200 Pm-3           1     2    
GM3+(a,0)            123 P4/mmm         1     3    
GM3+(a,b)            47 Pmmm            1     6    
GM4+(0,0,a)          83 P4/m            1     6    
GM4+(a,a,a)          148 R-3            1     8    
GM4+(0,a,a)          12 C2/m            1     12   
GM4+(a,b,c)          2 P-1              1     24   
GM5+(a,a,a)          166 R-3m           1     4    
GM5+(0,0,a)          65 Cmmm            1     6    
GM5+(a,a,b)          12 C2/m            1     12   
GM5+(a,b,c)          2 P-1              1     24   
GM1-(a)              207 P432           1     2    
GM2-(a)              215 P-43m          1     2    
GM3-(a,0)            89 P422            1     6    
GM3-(0,a)            111 P-42m          1     6    
GM3-(a,b)            16 P222            1     12   
GM4-(0,0,a)          99 P4mm            1     6    
GM4-(a,a,a)          160 R3m            1     8    
GM4-(0,a,a)          38 Amm2            1     12   
GM4-(0,a,b)          6 Pm               1     24   
GM4-(a,a,b)          8 Cm               1     24   
GM4-(a,b,c)          1 P1               1     48   
GM5-(0,0,a)          115 P-4m2          1     6    
GM5-(a,a,a)          155 R32            1     8    
GM5-(0,a,a)          38 Amm2            1     12   
GM5-(a,a,b)          5 C2               1     24   
GM5-(0,a,b)          6 Pm               1     24   
GM5-(a,b,c)          1 P1               1     48   

Conventions and validation: ISOSUBGROUP (https://iso.byu.edu):
H. T. Stokes, S. van Orden and B. J. Campbell, "Tool for Generating
Isotropy Subgroups of Crystallographic Space Groups",
J. Appl. Cryst. 49, 1849-1853 (2016).

$ crystod-group --parent Pm-3m --irrep R4+

* Supergroup *
Pm-3m (No. 221)

* Irrep *
R4+: order parameter dimension 3

* Order parameter directions and isotropy subgroups *
irrep                subgroup           size  index
R4+(0,0,a)           140 I4/mcm         2     6    
R4+(a,a,a)           167 R-3c           2     8    
R4+(0,a,a)           74 Imma            2     12   
R4+(0,a,b)           12 C2/m            2     24   
R4+(a,a,b)           15 C2/c            2     24   
R4+(a,b,c)           2 P-1              2     48   

Conventions and validation: ISOSUBGROUP (https://iso.byu.edu):
H. T. Stokes, S. van Orden and B. J. Campbell, "Tool for Generating
Isotropy Subgroups of Crystallographic Space Groups",
J. Appl. Cryst. 49, 1849-1853 (2016).

$ crystod-group --parent Pm-3m --irrep M3+

* Supergroup *
Pm-3m (No. 221)

* Irrep *
M3+: order parameter dimension 3 (star of 3 arm(s) x small dim 1)

* Order parameter directions and isotropy subgroups *
irrep                subgroup           size  index
M3+(0;0;a)           127 P4/mbm         2     6    
M3+(a;a;a)           204 Im-3           4     8    
M3+(0;a;a)           139 I4/mmm         4     12   
M3+(a;b;c)           71 Immm            4     24   

Conventions and validation: ISOSUBGROUP (https://iso.byu.edu):
H. T. Stokes, S. van Orden and B. J. Campbell, "Tool for Generating
Isotropy Subgroups of Crystallographic Space Groups",
J. Appl. Cryst. 49, 1849-1853 (2016).

$ crystod-group --parent Pm-3m --irrep X5+

* Supergroup *
Pm-3m (No. 221)

* Irrep *
X5+: order parameter dimension 6 (star of 3 arm(s) x small dim 2)

* Order parameter directions and isotropy subgroups *
irrep                subgroup           size  index
X5+(0,0;0,0;0,a)     51 Pmma            2     12   
X5+(0,0;0,0;a,a)     63 Cmcm            2     12   
X5+(0,0;a,0;0,a)     127 P4/mbm         4     12   
X5+(0,0;0,a;a,0)     129 P4/nmm         4     12   
X5+(0,a;0,a;0,a)     205 Pa-3           8     16   
X5+(a,a;a,a;a,a)     166 R-3m           8     32   
X5+(a,-a;a,a;a,a)    167 R-3c           8     32   
X5+(0,0;0,0;a,b)     11 P2_1/m          2     24   
X5+(0,0;a,0;0,b)     55 Pbam            4     24   
X5+(0,0;0,a;0,b)     57 Pbcm            4     24   
X5+(0,0;0,a;b,0)     59 Pmmn            4     24   
X5+(0,0;a,b;b,a)     12 C2/m            4     48   
X5+(0,a;b,b;a,0)     63 Cmcm            8     48   
X5+(0,a;a,0;b,b)     64 Cmce            8     48   
X5+(a,b;a,b;a,b)     148 R-3            8     64   
X5+(0,0;0,a;b,c)     11 P2_1/m          4     48   
X5+(0,0;a,0;b,c)     14 P2_1/c          4     48   
X5+(0,a;0,b;0,c)     61 Pbca            8     48   
X5+(0,a;0,b;c,0)     62 Pnma            8     48   
X5+(a,a;b,c;c,b)     12 C2/m            8     96   
X5+(a,-a;b,c;c,b)    15 C2/c            8     96   
X5+(0,0;a,b;c,d)     2 P-1              4     96   
X5+(0,a;b,c;d,0)     11 P2_1/m          8     96   
X5+(0,a;0,b;c,d)     14 P2_1/c          8     96   
X5+(0,a;b,0;c,d)     14 P2_1/c          8     96   
X5+(a,b;c,d;e,f)     2 P-1              8     192  

Conventions and validation: ISOSUBGROUP (https://iso.byu.edu):
H. T. Stokes, S. van Orden and B. J. Campbell, "Tool for Generating
Isotropy Subgroups of Crystallographic Space Groups",
J. Appl. Cryst. 49, 1849-1853 (2016).

$ crystod-group --parent P6_3/mmc --irrep GM2-

* Supergroup *
P6_3/mmc (No. 194)

* Irrep *
GM2-: order parameter dimension 1

* Order parameter directions and isotropy subgroups *
irrep                subgroup           size  index
GM2-(a)              186 P6_3mc         1     2    

Conventions and validation: ISOSUBGROUP (https://iso.byu.edu):
H. T. Stokes, S. van Orden and B. J. Campbell, "Tool for Generating
Isotropy Subgroups of Crystallographic Space Groups",
J. Appl. Cryst. 49, 1849-1853 (2016).

$ crystod-group --parent Ia-3d --irrep P2

* Supergroup *
Ia-3d (No. 230)

* Irrep *
P1P2: order parameter dimension 8 (star of 2 arm(s) x small dim 2 x 2; complex-type irrep -> physically irreducible real form)

* Order parameter directions and isotropy subgroups *
irrep                                               subgroup           size  index
P1P2(a,a,b,b;-a,a,-b,b)                             23 I222            4     48   
P1P2(a,-a,b,-b;a,a,b,b)                             24 I2_12_12_1      4     48   
P1P2(0,0,0,0;0,a,0,b)                               82 I-4             4     48   
P1P2(2a,2b,2c,2d;a+b-c+d,a+b+c-d,a-b+c+d,-a+b+c+d)  2 P-1              4     96   
P1P2(0,a,0,b;c,d,d,-c)                              5 C2               4     96   
P1P2(a,b,c,d;-a,b,-c,d)                             5 C2               4     96   
P1P2(a,b,c,d;e,f,g,h)                               1 P1               4     192  

Conventions and validation: ISOSUBGROUP (https://iso.byu.edu):
H. T. Stokes, S. van Orden and B. J. Campbell, "Tool for Generating
Isotropy Subgroups of Crystallographic Space Groups",
J. Appl. Cryst. 49, 1849-1853 (2016).

$ crystod-group --parent I4/mmm --irrep X3- X2+

* Supergroup *
I4/mmm (No. 139)

* Coupled irreps *
X3-: order parameter dimension 2 (star of 2 arm(s) x small dim 1)
X2+: order parameter dimension 2 (star of 2 arm(s) x small dim 1)
coupled order parameter dimension 4 (2 + 2)

* Order parameter directions and isotropy subgroups (X3- alone) *
irrep                subgroup           size  index
X3-(0;a)             63 Cmcm            2     4    
X3-(a;a)             136 P4_2/mnm       4     4    
X3-(a;b)             58 Pnnm            4     8    

* Order parameter directions and isotropy subgroups (X2+ alone) *
irrep                subgroup           size  index
X2+(0;c)             64 Cmce            2     4    
X2+(c;c)             127 P4/mbm         4     4    
X2+(c;d)             55 Pbam            4     8    

* Order parameter directions and isotropy subgroups (coupled) *
irrep                subgroup           size  index
X3-(0;a) X2+(0;c)    36 Cmc2_1          2     8    
X3-(0;a) X2+(c;0)    62 Pnma            4     8    
X3-(a;a) X2+(c;c)    38 Amm2            4     16   
X3-(0;a) X2+(c;d)    26 Pmc2_1          4     16   
X3-(a;b) X2+(0;c)    31 Pmn2_1          4     16   
X3-(a;b) X2+(c;d)    6 Pm               4     32   

Every irrep of the coupled table condenses with a nonzero
amplitude; every irrep keeps its own independent free
parameters (X3-: a, b; X2+: c, d).

Conventions and validation: ISOSUBGROUP (https://iso.byu.edu):
H. T. Stokes, S. van Orden and B. J. Campbell, "Tool for Generating
Isotropy Subgroups of Crystallographic Space Groups",
J. Appl. Cryst. 49, 1849-1853 (2016).

$ crystod-group --parent I4/mmm --irrep X3- X2+ --order-parameter 0 a 0 c

* Supergroup *
I4/mmm (No. 139)

* Coupled irreps *
X3-: order parameter dimension 2 (star of 2 arm(s) x small dim 1)
  Landau condition: satisfied (no cubic invariant)
  Lifshitz condition: satisfied (no Lifshitz invariant)
  -> a continuous transition is allowed
X2+: order parameter dimension 2 (star of 2 arm(s) x small dim 1)
  Landau condition: satisfied (no cubic invariant)
  Lifshitz condition: satisfied (no Lifshitz invariant)
  -> a continuous transition is allowed
coupled order parameter dimension 4 (2 + 2)

* Isotropy subgroup *
X3-(0;a) X2+(0;c) -> Cmc2_1 (No. 36)
cell size 2, index 8
sublattice basis (parent primitive units): (2,0,0), (1,1,0), (0,0,1)
conventional basis (parent conventional units): (0,0,1), (-1,-1,0), (1,-1,0)
origin: (-1/4,-1/4,-1/2)

Conventions and validation: ISOSUBGROUP (https://iso.byu.edu):
H. T. Stokes, S. van Orden and B. J. Campbell, "Tool for Generating
Isotropy Subgroups of Crystallographic Space Groups",
J. Appl. Cryst. 49, 1849-1853 (2016).

$ crystod-group --parent Pm-3m --irrep R4+ M3+

* Supergroup *
Pm-3m (No. 221)

* Coupled irreps *
R4+: order parameter dimension 3
M3+: order parameter dimension 3 (star of 3 arm(s) x small dim 1)
coupled order parameter dimension 6 (3 + 3)

* Order parameter directions and isotropy subgroups (R4+ alone) *
irrep                subgroup           size  index
R4+(0,0,a)           140 I4/mcm         2     6    
R4+(a,a,a)           167 R-3c           2     8    
R4+(0,a,a)           74 Imma            2     12   
R4+(0,a,b)           12 C2/m            2     24   
R4+(a,a,b)           15 C2/c            2     24   
R4+(a,b,c)           2 P-1              2     48   

* Order parameter directions and isotropy subgroups (M3+ alone) *
irrep                subgroup           size  index
M3+(0;0;d)           127 P4/mbm         2     6    
M3+(d;d;d)           204 Im-3           4     8    
M3+(0;d;d)           139 I4/mmm         4     12   
M3+(d;e;f)           71 Immm            4     24   

* Order parameter directions and isotropy subgroups (coupled) *
irrep                  subgroup           size  index
R4+(0,0,a) M3+(0;0;d)  127 P4/mbm         4     12   
R4+(0,a,a) M3+(d;0;0)  62 Pnma            4     24   
R4+(0,0,a) M3+(0;d;0)  63 Cmcm            4     24   
R4+(0,0,a) M3+(d;d;0)  137 P4_2/nmc       8     24   
R4+(0,a,a) M3+(0;d;d)  63 Cmcm            8     48   
R4+(a,a,a) M3+(d;d;d)  148 R-3            8     64   
R4+(0,a,b) M3+(d;0;0)  11 P2_1/m          4     48   
R4+(0,a,b) M3+(0;0;d)  12 C2/m            4     48   
R4+(a,a,b) M3+(0;0;d)  14 P2_1/c          4     48   
R4+(a,a,b) M3+(d;d;0)  15 C2/c            8     96   
R4+(0,0,a) M3+(d;e;f)  59 Pmmn            8     48   
R4+(a,b,c) M3+(0;0;d)  2 P-1              4     96   
R4+(0,a,b) M3+(d;e;f)  11 P2_1/m          8     96   
R4+(a,b,c) M3+(d;e;f)  2 P-1              8     192  

Every irrep of the coupled table condenses with a nonzero
amplitude; every irrep keeps its own independent free
parameters (R4+: a, b, c; M3+: d, e, f).

Conventions and validation: ISOSUBGROUP (https://iso.byu.edu):
H. T. Stokes, S. van Orden and B. J. Campbell, "Tool for Generating
Isotropy Subgroups of Crystallographic Space Groups",
J. Appl. Cryst. 49, 1849-1853 (2016).

