# Reverse lookup (crystod-group --parent G --child H); testsuite section 19
# Counterpart of the reverse search of ISOSUBGROUP (ISOTROPY Software Suite,
# https://iso.byu.edu): every stratum of every irrep at the special k points of
# G whose isotropy subgroup has type H, with the cell size, the index and the
# conventional basis and origin of the subgroup in the parent setting.
# The first run for a parent builds the table of all strata (one progress line
# per k point on stderr) and caches it as
# $CRYSTOD_CACHE_DIR/isotropy/<number>_<version>.json.gz
# (default ~/.cache/crystod/isotropy/...); later runs read the file.
# Several rows of one type are different embeddings of H in G (orientation or
# origin of the subgroup cell). --size, --index and --kpoint filter the rows;
# --no-cache rebuilds the table (a table built from other sources is rebuilt
# too). In a parent with improper operations, a row of the enantiomorphic
# partner type counts as a match (with a note); in a Sohncke parent such as
# P4_222 it does not.
# --coupled also searches the pairs of irreps (same k point first, then
# different k points) and lists the coupled strata of type H that need both
# irreps: the Pnma tilt system of the perovskites (size 4) is R4+ (0,a,a) +
# M3+ (d;0;0), the hybrid improper ferroelectric Cmc2_1 of I4/mmm is X2+ (0;a)
# + X3- (0;c). --secondary adds the secondary order parameters of every row.
# The progress lines of the coupled search (stderr) appear once it has run for
# a few seconds.

$ crystod-group --parent Pm-3m --child I4/mcm
building the isotropy table of Pm-3m: GM (1 of 4) ...
building the isotropy table of Pm-3m: R (2 of 4) ...
building the isotropy table of Pm-3m: X (3 of 4) ...
building the isotropy table of Pm-3m: M (4 of 4) ...

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

* Subgroup *
I4/mcm (No. 140)

* Isotropy table *
40 irreps at the special k points GM, R, X, M; 243 strata
cache: ~/.cache/crystod/isotropy/221_0.4.5.json.gz (written)

* Isotropy subgroups of type I4/mcm *
irrep  k  direction      size  index  conventional basis         origin
R3+    R  (0,a)          2     6      (0,-1,1),(0,1,1),(-2,0,0)  (0,1/2,1/2)
R4+    R  (0,0,a)        2     6      (-1,0,1),(1,0,1),(0,2,0)   (0,0,0)
R3-    R  (a,0)          2     6      (0,-1,1),(0,1,1),(-2,0,0)  (1/2,0,0)
R5-    R  (0,0,a)        2     6      (-1,0,1),(1,0,1),(0,2,0)   (1/2,1/2,1/2)
M5+    M  (0,0;0,a;a,0)  4     12     (2,0,0),(0,2,0),(0,0,2)    (0,0,0)
M5+    M  (0,0;a,0;0,a)  4     12     (2,0,0),(0,2,0),(0,0,2)    (1/2,1/2,1/2)
M1-    M  (0;a;a)        4     12     (2,0,0),(0,2,0),(0,0,2)    (0,0,1/2)
M2-    M  (0;a;a)        4     12     (2,0,0),(0,2,0),(0,0,2)    (0,0,1/2)
M3-    M  (0;a;a)        4     12     (2,0,0),(0,2,0),(0,0,2)    (1/2,1/2,0)
M4-    M  (0;a;a)        4     12     (2,0,0),(0,2,0),(0,0,2)    (1/2,1/2,0)

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 --child R-3c

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

* Subgroup *
R-3c (No. 167)

* Isotropy table *
40 irreps at the special k points GM, R, X, M; 243 strata
cache: ~/.cache/crystod/isotropy/221_0.4.5.json.gz (read)

* Isotropy subgroups of type R-3c *
irrep  k  direction       size  index  conventional basis         origin
R4+    R  (a,a,a)         2     8      (1,0,-1),(-1,1,0),(2,2,2)  (0,0,0)
R5-    R  (a,a,a)         2     8      (1,0,-1),(-1,1,0),(2,2,2)  (1/2,1/2,1/2)
X5+    X  (a,-a;a,a;a,a)  8     32     (2,0,-2),(0,2,2),(2,-2,2)  (0,0,0)
X5-    X  (a,a;a,a;a,a)   8     32     (2,-2,0),(0,2,-2),(2,2,2)  (1/2,1/2,1/2)

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 --child Pnma

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

* Subgroup *
Pnma (No. 62)

* Isotropy table *
40 irreps at the special k points GM, R, X, M; 243 strata
cache: ~/.cache/crystod/isotropy/221_0.4.5.json.gz (read)

* Isotropy subgroups of type Pnma *
irrep  k  direction      size  index  conventional basis        origin
X5+    X  (0,a;0,b;c,0)  8     48     (0,2,0),(0,0,2),(2,0,0)   (0,0,0)
X5-    X  (0,a;0,b;c,0)  8     48     (0,0,-2),(0,2,0),(2,0,0)  (1/2,1/2,1/2)

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 --child Pnma --size 4

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

* Subgroup *
Pnma (No. 62)

* Isotropy table *
40 irreps at the special k points GM, R, X, M; 243 strata
cache: ~/.cache/crystod/isotropy/221_0.4.5.json.gz (read)
selected: size 4

* Isotropy subgroups of type Pnma *
no single-irrep stratum of type Pnma (No. 62) with size 4;
the subgroup may need two coupled irreps (--coupled)

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 --child 140 --index 12 --kpoint M

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

* Subgroup *
I4/mcm (No. 140)

* Isotropy table *
40 irreps at the special k points GM, R, X, M; 243 strata
cache: ~/.cache/crystod/isotropy/221_0.4.5.json.gz (read)
selected: index 12, k point M

* Isotropy subgroups of type I4/mcm *
irrep  k  direction      size  index  conventional basis       origin
M5+    M  (0,0;0,a;a,0)  4     12     (2,0,0),(0,2,0),(0,0,2)  (0,0,0)
M5+    M  (0,0;a,0;0,a)  4     12     (2,0,0),(0,2,0),(0,0,2)  (1/2,1/2,1/2)
M1-    M  (0;a;a)        4     12     (2,0,0),(0,2,0),(0,0,2)  (0,0,1/2)
M2-    M  (0;a;a)        4     12     (2,0,0),(0,2,0),(0,0,2)  (0,0,1/2)
M3-    M  (0;a;a)        4     12     (2,0,0),(0,2,0),(0,0,2)  (1/2,1/2,0)
M4-    M  (0;a;a)        4     12     (2,0,0),(0,2,0),(0,0,2)  (1/2,1/2,0)

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 P4_2/mcm --child P4_322
building the isotropy table of P4_2/mcm: GM (1 of 6) ...
building the isotropy table of P4_2/mcm: M (2 of 6) ...
building the isotropy table of P4_2/mcm: X (3 of 6) ...
building the isotropy table of P4_2/mcm: A (4 of 6) ...
building the isotropy table of P4_2/mcm: R (5 of 6) ...
building the isotropy table of P4_2/mcm: Z (6 of 6) ...

* Supergroup *
P4_2/mcm (No. 132)

* Subgroup *
P4_322 (No. 95)

* Isotropy table *
38 irreps at the special k points GM, M, X, A, R, Z; 108 strata
cache: ~/.cache/crystod/isotropy/132_0.4.5.json.gz (written)

* Isotropy subgroups of type P4_322 *
irrep  k  direction  size  index  conventional basis       origin
Z3     Z  (0,a)      2     4      (1,0,0),(0,1,0),(0,0,2)  (0,0,3/4)
Z4     Z  (0,a)      2     4      (1,0,0),(0,1,0),(0,0,2)  (0,0,1/4)

note: 95 <-> 91 (P4_322 <-> P4_122) are enantiomorphic partner types: an
improper operation of the parent maps the order parameter of such a stratum to
another one of the same stratum whose stabilizer is the partner. The rows
Z3(0,a) at Z, Z4(0,a) at Z are listed with a stabilizer of type P4_122
(No. 91).

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 P4_222 --child P4_122
building the isotropy table of P4_222: GM (1 of 6) ...
building the isotropy table of P4_222: M (2 of 6) ...
building the isotropy table of P4_222: X (3 of 6) ...
building the isotropy table of P4_222: A (4 of 6) ...
building the isotropy table of P4_222: R (5 of 6) ...
building the isotropy table of P4_222: Z (6 of 6) ...

* Supergroup *
P4_222 (No. 93)

* Subgroup *
P4_122 (No. 91)

* Isotropy table *
28 irreps at the special k points GM, M, X, A, R, Z; 52 strata
cache: ~/.cache/crystod/isotropy/93_0.4.5.json.gz (written)

* Isotropy subgroups of type P4_122 *
irrep  k  direction  size  index  conventional basis       origin
Z1     Z  (a)        2     2      (1,0,0),(0,1,0),(0,0,2)  (0,0,1/2)
Z3     Z  (a)        2     2      (1,0,0),(0,1,0),(0,0,2)  (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 I4/mmm --child Cmc2_1
building the isotropy table of I4/mmm: GM (1 of 5) ...
building the isotropy table of I4/mmm: X (2 of 5) ...
building the isotropy table of I4/mmm: M (3 of 5) ...
building the isotropy table of I4/mmm: N (4 of 5) ...
building the isotropy table of I4/mmm: P (5 of 5) ...

* Supergroup *
I4/mmm (No. 139)

* Subgroup *
Cmc2_1 (No. 36)

* Isotropy table *
37 irreps at the special k points GM, X, M, N, P; 132 strata
cache: ~/.cache/crystod/isotropy/139_0.4.5.json.gz (written)

* Isotropy subgroups of type Cmc2_1 *
no single-irrep stratum of type Cmc2_1 (No. 36);
the subgroup may need two coupled irreps (--coupled)

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 --child Pnma --size 4 --coupled
searching the coupled irrep pairs of Pm-3m: R + X (8 of 10) ...
searching the coupled irrep pairs of Pm-3m: R + M (9 of 10) ...
searching the coupled irrep pairs of Pm-3m: X + M (10 of 10) ...

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

* Subgroup *
Pnma (No. 62)

* Isotropy table *
40 irreps at the special k points GM, R, X, M; 243 strata
cache: ~/.cache/crystod/isotropy/221_0.4.5.json.gz (read)
selected: size 4

* Isotropy subgroups of type Pnma *
no single-irrep stratum of type Pnma (No. 62) with size 4;
the subgroup may need two coupled irreps (see below)

* Coupled isotropy subgroups of type Pnma *
40 irreps pass the point-group filter; 780 pairs searched
(180 at the same k point, 600 at different k points)
irrep1  dir1           irrep2  dir2           size  index  conventional basis           origin
R4+     (0,a,a)        X5+     (0,0;d,d;0,0)  4     24     (-1,-1,0),(0,0,-2),(1,-1,0)  (0,0,0)
R4+     (0,a,a)        X5-     (0,0;d,d;0,0)  4     24     (0,0,2),(-1,-1,0),(1,-1,0)   (0,1/2,1/2)
R5+     (0,a,-a)       X5+     (0,0;d,d;0,0)  4     24     (-1,-1,0),(0,0,-2),(1,-1,0)  (0,0,0)
R5+     (0,a,-a)       X5-     (0,0;d,d;0,0)  4     24     (0,0,2),(-1,-1,0),(1,-1,0)   (0,1/2,1/2)
R4-     (0,a,a)        X5+     (0,0;d,d;0,0)  4     24     (0,0,-2),(1,-1,0),(-1,-1,0)  (0,1/2,0)
R4-     (0,a,a)        X5-     (0,0;d,d;0,0)  4     24     (1,-1,0),(0,0,2),(-1,-1,0)   (0,0,1/2)
R5-     (0,a,-a)       X5+     (0,0;d,d;0,0)  4     24     (0,0,-2),(1,-1,0),(-1,-1,0)  (0,1/2,0)
R5-     (0,a,-a)       X5-     (0,0;d,d;0,0)  4     24     (1,-1,0),(0,0,2),(-1,-1,0)   (0,0,1/2)
R4+     (0,a,a)        M2+     (d;0;0)        4     24     (-1,-1,0),(0,0,-2),(1,-1,0)  (0,0,0)
R4+     (0,a,a)        M3+     (d;0;0)        4     24     (-1,-1,0),(0,0,-2),(1,-1,0)  (0,0,0)
R4+     (0,a,a)        M2-     (d;0;0)        4     24     (0,0,2),(-1,-1,0),(1,-1,0)   (0,1/2,1/2)
R4+     (0,a,a)        M3-     (d;0;0)        4     24     (0,0,2),(-1,-1,0),(1,-1,0)   (0,1/2,1/2)
R5+     (0,a,a)        M2+     (d;0;0)        4     24     (1,-1,0),(0,0,2),(-1,-1,0)   (0,0,0)
R5+     (0,a,a)        M3+     (d;0;0)        4     24     (1,-1,0),(0,0,2),(-1,-1,0)   (0,0,0)
R5+     (0,a,a)        M2-     (d;0;0)        4     24     (0,0,-2),(1,-1,0),(-1,-1,0)  (0,1/2,1/2)
R5+     (0,a,a)        M3-     (d;0;0)        4     24     (0,0,-2),(1,-1,0),(-1,-1,0)  (0,1/2,1/2)
R4-     (0,a,a)        M2+     (d;0;0)        4     24     (1,-1,0),(0,0,2),(-1,-1,0)   (0,0,1/2)
R4-     (0,a,a)        M3+     (d;0;0)        4     24     (1,-1,0),(0,0,2),(-1,-1,0)   (0,0,1/2)
R4-     (0,a,a)        M2-     (d;0;0)        4     24     (0,0,-2),(1,-1,0),(-1,-1,0)  (0,1/2,0)
R4-     (0,a,a)        M3-     (d;0;0)        4     24     (0,0,-2),(1,-1,0),(-1,-1,0)  (0,1/2,0)
R5-     (0,a,a)        M2+     (d;0;0)        4     24     (-1,-1,0),(0,0,-2),(1,-1,0)  (0,0,1/2)
R5-     (0,a,a)        M3+     (d;0;0)        4     24     (-1,-1,0),(0,0,-2),(1,-1,0)  (0,0,1/2)
R5-     (0,a,a)        M2-     (d;0;0)        4     24     (0,0,2),(-1,-1,0),(1,-1,0)   (0,1/2,0)
R5-     (0,a,a)        M3-     (d;0;0)        4     24     (0,0,2),(-1,-1,0),(1,-1,0)   (0,1/2,0)
X5+     (0,0;0,0;a,a)  M2+     (0;g;0)        4     24     (0,-1,-1),(2,0,0),(0,-1,1)   (0,0,0)
X5+     (0,0;0,0;a,a)  M3+     (0;g;0)        4     24     (0,-1,-1),(2,0,0),(0,-1,1)   (0,0,0)
X5+     (0,0;0,0;a,a)  M2-     (0;g;0)        4     24     (2,0,0),(0,1,-1),(0,1,1)     (0,1/2,0)
X5+     (0,0;0,0;a,a)  M3-     (0;g;0)        4     24     (2,0,0),(0,1,-1),(0,1,1)     (0,1/2,0)
X5-     (0,0;0,0;a,a)  M2+     (0;g;0)        4     24     (0,-1,1),(-2,0,0),(0,-1,-1)  (1/2,0,0)
X5-     (0,0;0,0;a,a)  M3+     (0;g;0)        4     24     (0,-1,1),(-2,0,0),(0,-1,-1)  (1/2,0,0)
X5-     (0,0;0,0;a,a)  M2-     (0;g;0)        4     24     (2,0,0),(0,1,1),(0,-1,1)     (1/2,0,1/2)
X5-     (0,0;0,0;a,a)  M3-     (0;g;0)        4     24     (2,0,0),(0,1,1),(0,-1,1)     (1/2,0,1/2)
every row needs both irreps: neither irrep alone has this isotropy subgroup

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 --child Cmc2_1 --coupled
searching the coupled irrep pairs of I4/mmm: M + P (14 of 15) ...
searching the coupled irrep pairs of I4/mmm: N + P (15 of 15) ...

* Supergroup *
I4/mmm (No. 139)

* Subgroup *
Cmc2_1 (No. 36)

* Isotropy table *
37 irreps at the special k points GM, X, M, N, P; 132 strata
cache: ~/.cache/crystod/isotropy/139_0.4.5.json.gz (read)

* Isotropy subgroups of type Cmc2_1 *
no single-irrep stratum of type Cmc2_1 (No. 36);
the subgroup may need two coupled irreps (see below)

* Coupled isotropy subgroups of type Cmc2_1 *
37 irreps pass the point-group filter; 666 pairs searched
(134 at the same k point, 532 at different k points)
irrep1  dir1        irrep2  dir2                 size  index  conventional basis           origin
X2+     (0;a)       X3-     (0;c)                2     8      (0,0,1),(-1,-1,0),(1,-1,0)   (3/4,3/4,1/2)
X3+     (0;a)       X2-     (0;c)                2     8      (-1,-1,0),(0,0,1),(-1,1,0)   (1/2,1/2,1/4)
M5+     (a,-a)      M5-     (c,c)                2     8      (-1,1,0),(1,1,0),(0,0,-1)    (3/4,3/4,0)
GM5-    (a,a)       X2+     (c;0)                2     8      (0,0,-1),(1,-1,0),(-1,-1,0)  (1/4,3/4,1/2)
GM5-    (a,a)       X3+     (c;0)                2     8      (-1,1,0),(0,0,1),(1,1,0)     (1/2,1/2,1/4)
GM5-    (a,a)       X2-     (c;0)                2     8      (-1,1,0),(0,0,1),(1,1,0)     (1/2,1/2,1/4)
GM5-    (a,a)       X3-     (c;0)                2     8      (0,0,-1),(1,-1,0),(-1,-1,0)  (1/4,3/4,1/2)
GM2-    (a)         M5+     (b,b)                2     8      (1,1,0),(-1,1,0),(0,0,1)     (1/4,3/4,0)
GM2-    (a)         M5-     (b,b)                2     8      (-1,1,0),(1,1,0),(0,0,-1)    (3/4,3/4,0)
GM3-    (a)         M5+     (b,b)                2     8      (1,1,0),(-1,1,0),(0,0,1)     (1/4,3/4,0)
GM3-    (a)         M5-     (b,b)                2     8      (-1,1,0),(1,1,0),(0,0,-1)    (3/4,3/4,0)
X1+     (a;a)       P5      (0.609c,c;0.609d,d)  8     32     (-2,0,0),(0,2,0),(0,0,-2)    (0,0,0)
X2+     (a;a)       P5      (0.609c,c;0.609d,d)  8     32     (-2,0,0),(0,2,0),(0,0,-2)    (0,0,0)
X1-     (a;a)       P5      (0.609c,c;0.609d,d)  8     32     (0,2,0),(-2,0,0),(0,0,2)     (1/2,1/2,0)
X2-     (a;a)       P5      (0.609c,c;0.609d,d)  8     32     (0,2,0),(-2,0,0),(0,0,2)     (1/2,1/2,0)
N1+     (a;a;a;a)   P5      (e,f;-f,e)           16    64     (-2,2,0),(2,2,0),(0,0,-2)    (1/2,1/2,0)
N2+     (a;-a;a;a)  P5      (e,f;-f,e)           16    64     (-2,2,0),(2,2,0),(0,0,-2)    (1/2,1/2,0)
N1-     (a;a;a;a)   P5      (e,f;-f,e)           16    64     (-2,2,0),(2,2,0),(0,0,-2)    (1/2,1/2,0)
N2-     (a;-a;a;a)  P5      (e,f;-f,e)           16    64     (-2,2,0),(2,2,0),(0,0,-2)    (1/2,1/2,0)
every row needs both irreps: neither irrep alone has this isotropy subgroup

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 Cmcm --child Pna2_1 --coupled --kpoint Y --secondary
building the isotropy table of Cmcm: GM (1 of 6) ...
building the isotropy table of Cmcm: Y (2 of 6) ...
building the isotropy table of Cmcm: Z (3 of 6) ...
building the isotropy table of Cmcm: R (4 of 6) ...
building the isotropy table of Cmcm: S (5 of 6) ...
building the isotropy table of Cmcm: T (6 of 6) ...

* Supergroup *
Cmcm (No. 63)

* Subgroup *
Pna2_1 (No. 33)

* Isotropy table *
25 irreps at the special k points GM, Y, Z, R, S, T; 56 strata
cache: ~/.cache/crystod/isotropy/63_0.4.5.json.gz (written)
selected: k point Y

* Isotropy subgroups of type Pna2_1 *
no single-irrep stratum of type Pna2_1 (No. 33) with k point Y;
the subgroup may need two coupled irreps (see below)

* Coupled isotropy subgroups of type Pna2_1 *
8 irreps pass the point-group filter; 28 pairs searched
(28 at the same k point, 0 at different k points)
irrep1  dir1  irrep2  dir2  size  index  conventional basis          origin
Y2+     (a)   Y1-     (b)   2     4      (0,-1,0),(-1,0,0),(0,0,-1)  (0,0,0)
Y4+     (a)   Y2-     (b)   2     4      (0,0,-1),(0,-1,0),(-1,0,0)  (0,3/4,0)
every row needs both irreps: neither irrep alone has this isotropy subgroup

* Secondary order parameters *
H = Pna2_1 (33), index 4: Y2+(a) Y1-(b)
irrep  k            dim  n_free  direction  coupling            type
GM1+   (0,0,0)      1    1       (a)        Q1^2 eta, Q2^2 eta  strain
GM2-   (0,0,0)      1    1       (a)        Q1^1 Q2^1 eta       polar
Y2+    (1/2,1/2,0)  1    1       (a)        -                   primary
Y1-    (1/2,1/2,0)  1    1       (a)        -                   primary
coupling: lowest invariant Q^m eta (Q primary, eta this irrep) that does not vanish on the fixed space of H, total degree <= 4

* Secondary order parameters *
H = Pna2_1 (33), index 4: Y4+(a) Y2-(b)
irrep  k            dim  n_free  direction  coupling            type
GM1+   (0,0,0)      1    1       (a)        Q1^2 eta, Q2^2 eta  strain
GM3-   (0,0,0)      1    1       (a)        Q1^1 Q2^1 eta       polar
Y4+    (1/2,1/2,0)  1    1       (a)        -                   primary
Y2-    (1/2,1/2,0)  1    1       (a)        -                   primary
coupling: lowest invariant Q^m eta (Q primary, eta this irrep) that does not vanish on the fixed space of H, total degree <= 4

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).
