hikari.symmetry

Module containing symmetry operations and point groups to be used by other objects.

Submodules

• hikari.symmetry.catalog
• hikari.symmetry.group
• hikari.symmetry.hall
• hikari.symmetry.operations

Attributes

PG	Since hikari's groups do not carry information about lattice translations,
SG	Since hikari's groups do not carry information about lattice translations,

Classes

Operation	Class storing information about symmetry operations, with clear string
BoundedOperation	A subclass of Operation where all three elements of the translation
PointOperation	A subclass of BoundedOperation, asserts translation vector = [0,0,0]
Group	Base immutable class containing information about symmetry groups.
HallSymbol	Parse, interpret, and convert Hall symbol to hikari.symmetry.Group
GroupCatalog	Manage generating and mappings of point and space groups.

Package Contents

class hikari.symmetry.Operation(transformation, translation=np.array([0, 0, 0]))

Class storing information about symmetry operations, with clear string representation and intuitive syntax for combining / utilising operations: “=” to compare two symmetry operations for logical equivalence, “*” to combine two symmetry operations or transform a vector, “** n” to apply symmetry operation n times, “% n” to restrict symmetry operation to n unit cells. Some of the functions may work incorrectly for rhombohedral unit cells #TODO

class Type

Bases: enum.Enum

Enumerator class storing information about type of operation

rotoinversion = 4

identity = 3

reflection = 2

rotation = 1

inversion = 0

rototranslation = -1

transflection = -2

translation = -3

property tf: numpy.ndarray

Return type:

numpy.ndarray

property tl: numpy.ndarray

Return type:

numpy.ndarray

__eq__(other)

Parameters:

other (Operation)

Return type:

bool

__mul__(other)

Parameters:

other (Operation)

Return type:

Operation

__pow__(power, modulo=None)

Parameters:

power (int)

Return type:

Operation

__repr__()

Return type:

str

__str__()

Return type:

str

__hash__()

Return type:

int

classmethod from_code(code)

Create new symmetry operation using symmetry code “x’,y’,z’” containing transformation of each individual coordinate from x,y,z to x’,y’,z’.

Parameters:

code (str) – string representing new coordinates after operation

Returns:

Symmetry operation generated from given coordinate triplet code

Return type:

Operation

classmethod from_matrix(matrix)

Create new symmetry operation using augmented 4x4 transformation matrix

Parameters:

matrix (numpy.ndarray) – augmented 4x4 matrix

Returns:

Symmetry operation generated based on augmented matrix

Return type:

Operation

classmethod from_pair(matrix, vector)

Create new symmetry operation using point transformation 3x3 matrix and 3-length translation vector. Alias for standard creation method.

Parameters:

• matrix (numpy.ndarray) – 3x3 point transformation matrix

• vector (numpy.ndarray) – 3-length translation vector

Returns:

Symmetry operation generated based on matrix - vector pair

Return type:

Operation

static _row_to_str(xyz, r)

Convert xyz: 3-el. list and r - number to single element of code triplet

Parameters:

• xyz (numpy.ndarray[int]) – 3-element list of coordinates, e.g.: [1,-1,0]

• r (float) – translation applied to the row

Returns:

string representing change of coordinate

Return type:

str

static _project(vector, onto)

Return projection of np.ndarray “vector” to np.ndarray “onto”

Parameters:

• vector (numpy.ndarray)

• onto (numpy.ndarray)

Return type:

numpy.ndarray

property code: str

Return type:

str

property matrix: numpy.ndarray

Augmented 4 x 4 transformation matrix with float-type values

Return type:

numpy.ndarray

property det: int

Determinant of 3x3 transformation part of operation’s matrix

Return type:

int

property typ: Type

Crystallographic type of this symmetry operation (see Type)

Return type:

Type

property name: str

‘m’, ‘3’ or ‘2_1’

Type:

Short name of symmetry operation, e.g.

Return type:

str

property fold: int

Number of times operation must be repeated to become identity, inversion or translation: n for n-fold axes, 2 for reflections, 1 for other (max 6)

Return type:

int

property order: int

Number of times operation has to be repeated to become a translation, e.g.: n for all n-fold axes, 2 for other (max 6)

Return type:

int

property glide: numpy.ndarray

Part of the translation vector stemming from operations’ glide

Return type:

numpy.ndarray

property glide_fold: int

Number of types glide component of the operation must be repeated in order to contain only integer values, e.g.: 3 for “6_2”, 4 for “d”

Return type:

int

property origin: numpy.ndarray

Selected point that remains on the symmetry element after operation Loosely based on solving https://math.stackexchange.com/q/1054481/.

Return type:

numpy.ndarray

property reciprocal: PointOperation

Relevant symmetry operation in its respective reciprocal space

Return type:

PointOperation

property trace: int

Trace of 3x3 transformation part of operation’s matrix

Return type:

int

property translational: bool

True if operation has any glide component, False otherwise

Return type:

bool

property invariants: list[numpy.ndarray]

List of directions not affected by this symmetry operation

Return type:

list[numpy.ndarray]

property orientation: numpy.ndarray | None

Direction of symmetry element (if it can be defined) else None

Return type:

Union[numpy.ndarray, None]

property sense: str

“+” or “-”, the “sense” of rotation, as given in ITC A, 11.1.2

Return type:

str

property _hm_span

property _hm_glide

property hm_symbol: str

Return type:

str

property bounded: BoundedOperation

Instance of self that always collapses down to Coset representative

Return type:

BoundedOperation

property unbounded: Operation

Instance of self that can express translations beyond the unit cube

Return type:

Operation

property is_bounded: bool

True if self is a coset representatives i.e. 0 <= tl < 1 element-wise

Return type:

bool

at(point)

Transform operation as little as possible so that its symmetry element contains “point”. To be used after “into” if used together. Based on “ITC 5.2.1. Transformations”.

Parameters:

point (numpy.ndarray) – Target coordinates of point which should lie in element

Returns:

New symmetry operation which contains “point” in its element

Return type:

Operation

into(direction, hexagonal=False)

Rotate operation so that its orientation changes to “direction”, while preserving fractional glide. To be used before respective “at” method. Will most likely not work for unimplemented rhombohedral unit cells.

Parameters:

• direction (numpy.ndarray) – Target orientation for element of symmetry operation

• hexagonal (bool) – True if operation is defined in hexagonal coordinates

Returns:

New symmetry operation whose orientation is “direction”

Return type:

Operation

transform(other)

Transform a column containing rows of coordinate points

Parameters:

other (numpy.ndarray) – A vertical numpy array of coordinate triplets kept in rows

Returns:

Same-shaped array of coordinate triplets transformed by self

Return type:

numpy.ndarray

extincts(hkl)

Return boolean array with truth whenever reflection should be extinct

Parameters:

hkl (numpy.ndarray) – An array containing one hkl or multiple hkls in columns

Returns:

array of booleans, where extinct reflections are marked as True

Return type:

numpy.ndarray

distance_to_point(point)

Parameters:

point (Sequence[Union[int, float]])

Return type:

float

class hikari.symmetry.BoundedOperation(transformation, translation=np.array([0, 0, 0]))

Bases: Operation

A subclass of Operation where all three elements of the translation vector are bounded between 0 (inclusive) and 1 (exclusive). This class is suitable for handling coset representatives of space groups, since upon binding operations related by unit translation become equivalent.

property tl: numpy.ndarray

Return type:

numpy.ndarray

class hikari.symmetry.PointOperation(transformation, translation=np.array([0, 0, 0]))

Bases: BoundedOperation

A subclass of BoundedOperation, asserts translation vector = [0,0,0]

property tl: numpy.ndarray

Return type:

numpy.ndarray

class hikari.symmetry.Group(*generators)

Base immutable class containing information about symmetry groups. It stores information for point and space groups and, among others, allows for iteration over its hikari.symmetry.BoundedOperation elements.

Parameters:

generators (hikari.symmetry.operations.BoundedOperation)

class System

Bases: enum.Enum

Enumerator class with information about associated crystal system

triclinic = 0

monoclinic = 1

orthorhombic = 2

trigonal = 3

tetragonal = 4

cubic = 5

hexagonal = 6

property directions: list[numpy.ndarray]

Return type:

list[numpy.ndarray]

BRAVAIS_PRIORITY_RULES = 'A+B+C=F>R>I>C>B>A>H>P'

AXIS_PRIORITY_RULES = '6>61>62>63>64>65>-6>4>41>42>43>-4>-3>3>31>32>2>21'

PLANE_PRIORITY_RULES = 'm>a+b=e>a+c=e>b+c=e>a>b>c>n>d'

__generators = []

__operations

name

number = 0

classmethod from_generators_operations(generators, operations)

Generate group using already complete list of generators and operators. Does not check if operations are correct or complete for efficiency! :param generators: A complete list of group generators :param operations: A complete list of group operations :return: Symmetry group with given generators and operators.

Parameters:

• generators (list[hikari.symmetry.operations.BoundedOperation])

• operations (list[hikari.symmetry.operations.BoundedOperation])

Return type:

Group

classmethod from_hall_symbol(hall_symbol)

Parameters:

hall_symbol (Union[str, hikari.symmetry.hall.HallSymbol])

Return type:

Group

__eq__(other)

Parameters:

other (Group)

Return type:

bool

__lt__(other)

Parameters:

other (Group)

Return type:

bool

__gt__(other)

Parameters:

other (Group)

Return type:

bool

__le__(other)

Parameters:

other (Group)

Return type:

bool

__ge__(other)

Parameters:

other (Group)

Return type:

bool

__repr__()

Return type:

str

__str__()

Return type:

str

__hash__()

Return type:

int

property auto_generated_name: str

Name of the group generated automatically. Use only as approx.

Return type:

str

property centering_symbol: str

Return type:

str

property generators: list[hikari.symmetry.operations.BoundedOperation]

Return type:

list[hikari.symmetry.operations.BoundedOperation]

property operations: list[hikari.symmetry.operations.BoundedOperation]

Return type:

list[hikari.symmetry.operations.BoundedOperation]

property order: int

Return type:

int

property is_centrosymmetric: bool

True if group has centre of symmetry; False otherwise.

Return type:

bool

property is_enantiogenic: bool

True if determinant of any operation in group is negative.

Return type:

bool

property is_sohncke: bool

True if determinant of all operations in group are positive.

Return type:

bool

property is_achiral

property is_chiral

property is_symmorphic: bool

Return type:

bool

property is_polar: bool

Return type:

bool

property system: System

Predicted crystal system associated with this group

Return type:

System

lauefy()

Returns:

New PointGroup with centre of symmetry added to generators.

Return type:

Group

reciprocate()

Return type:

Group

transform(m)

Transform the group using 4x4 matrix. For reference, see bilbao resources or IUCr pamphlet no. 22.

Example:

Return type:

Group

>>> import numpy
>>> from hikari.symmetry import SG
>>> matrix = numpy.array([(1,0,1,0),(0,1,0,0),(-1,0,0,0),(0,0,0,1)])
>>> SG['P21/c'].transform(matrix).auto_generated_name
P 21/n

Parameters:

m (np.ndarray) – A 4x4 array containing information about new base and origin.

Returns:

Group with new, transformed basis and origin.

Return type:

Group

class hikari.symmetry.HallSymbol(hall_symbol)

Parse, interpret, and convert Hall symbol to hikari.symmetry.Group

Parameters:

hall_symbol (str)

exception HallSymbolException

Bases: Exception

Exception raised when group can’t be generated from the symbol

REGEX

This is a regex which matches every possible lowercase Hall symbol for classical 3D space groups. It matches up to 17 groups in total as follows:

1. Centrosymmetry: [-]?

2. Lattice symbol: [pabcirstf] (required)

3. Generator 1 inversion component: [-]?

4. Generator 1 rotation fold: d (required)

5. Generator 1 rotation direction: [*xyz]?

6. Generator 1 translation components: [12345abcnuvwd]*

7. Generator 2 inversion component: [-]?

8. Generator 2 rotation fold: d

9. Generator 2 rotation direction: [‘“xyz]?

10. Generator 2 translation components: [12345abcnuvwd]*

11. Generator 3 inversion component: [-]?

12. Generator 3 rotation fold: d

13. Generator 3 translation components: [12345abcnuvwd]*

14. Generator 4 rotation fold: d

15. Generator 4 translation components: [12345abcnuvwd]*

16. Origin shift constituent 1 expressed as count of 1/12 shifts: d

17. Origin shift constituent 2 expressed as count of 1/12 shifts: d

18. Origin shift constituent 3 expressed as count of 1/12 shifts: d

DIRECTION_SYMBOLS = ("'", '"', '*')

TRANSLATION_SYMBOLS = '12345abcnuvwd'

LATTICE_GENERATORS

UNIVERSAL_MATRICES

PRINCIPAL_ROTATIONS

FACE_X_DIAGONAL_ROTATIONS

FACE_Y_DIAGONAL_ROTATIONS

FACE_Z_DIAGONAL_ROTATIONS

BODY_DIAGONAL_ROTATIONS

STATIC_TRANSLATIONS

DYNAMIC_TRANSLATIONS

property symbol: str

Return type:

str

property elements: None | Match

Return self.symbol elements indexed as in self.HALL_REGEX doc

Return type:

Union[None, Match]

property generators: List[hikari.symmetry.operations.BoundedOperation]

Return type:

List[hikari.symmetry.operations.BoundedOperation]

class hikari.symmetry.GroupCatalog(table)

Manage generating and mappings of point and space groups. Relies on a built-in pandas DataFrame table to store all the information. Individual columns are named & generated based on GroupCatalogKey data.

For

Some notes on the uniqueness of columns pairwise for accessing:

• Column Hall has 3 groups appear twice due to inconsistency of HM names: c_2_2_-1ac, a_2_2_-1ab, and b_2_2_-1ab.

• There are no overlaps between HM and Hall column names

• There are no overlaps between HM_short and Hall column names

• There are no overlaps between HM_simple and Hall column names

• There are no overlaps between HM_simple and HM column names

• There are 345 overlaps between HM_simple and HM_short column names

Parameters:

table (pandas.DataFrame)

KEYS: list[GroupCatalogKey]

REST_COL_FORMAT

table: pandas.DataFrame

__eq__(other)

__len__()

Return type:

int

classmethod from_json(text)

Load from a json-formatted string

Parameters:

text (str)

Return type:

GroupCatalog

to_json(json_path)

Parameters:

json_path (hikari.utility.typing.PathLike)

Return type:

None

to_rest_table(txt_path)

Generate a .txt file with ReST table containing GroupCatalog elements.

Parameters:

txt_path (hikari.utility.typing.PathLike)

Return type:

None

property accessors: list[GroupCatalogKey]

Lists `cls.KEYS whose accessor priority is not 0 in decreasing order

Return type:

list[GroupCatalogKey]

property standard: GroupCatalog

A subset of current catalog with standard-setting groups only

Return type:

GroupCatalog

keys()

Return type:

list[str]

values()

Return type:

list[hikari.symmetry.group.Group]

items()

Return type:

list[tuple[Union[int, str], hikari.symmetry.group.Group]]

_get_by_key(key)

Iterate over accessors; whenever key is in accessor, return matching group

Parameters:

key (Union[str, int])

Return type:

pandas.DataFrame

_get_by_kwargs(**kwargs)

Return the first group that matches all queries specified in kwargs

Return type:

pandas.DataFrame

get(key=None, **kwargs)

Get first Group matching provided anonymous&known accessors or None

Parameters:

key (Union[str, int])

Return type:

Union[hikari.symmetry.group.Group, None]

__getitem__(item)

Get first Group matching provided anonymous accessor or raise

Parameters:

item (Union[str, int])

Return type:

hikari.symmetry.group.Group

hikari.symmetry.PG

Since hikari’s groups do not carry information about lattice translations, hikari does not differentiate between point groups and space groups. As a result, all groups are instances of the same class Group, and are stored in almost identical `GroupCatalog`s.

This pre-defined GroupCatalog PG holds and provides access to all pre-defined point groups available in hikari. The point groups are named and generated based on a wsv file in resources. Individual groups can be accessed in two different ways:

• using a dict-like anonymous accessor: PG[32], PG[‘m-3m’];

• using get() method: PG.get(number=32), PG.get(HM_short=’m-3m’);

The get() method can use a combination of various keywords to unambiguously localize the correct space group. The following keyword arguments can be used:

• n_c - unique “number:setting” string identifying each group

• number - index of the point group as given in ICT A (integer)*

• setting - arbitrary string declaring group setting

• HM - Full underscore-delimited Hermann-Mauguin symbol*

• HM_short - Hermann-Mauguin symbol with underscores removed*

• HM_simple - Short Hermann-Mauguin symbol without setting information*

• HM_numbered - “number: Short Hermann-Mauguin symbol” string

• Hall - Full Hall symbol*

• standard - True for groups in standard setting only

The keywords marked with “*” are called “accessors” and can be used to get the group using bracket notation get[accessor]. If get finds ambiguity, If get finds ambiguity, for example PG[‘2/m’] matches all settings of point group #5: 1_2/m_1 (std), 1_1_2/m, and 2/m_1_1, the user is warned and the first group in std setting, if possible, is returned.

The following table lists all possible values of selected point Group keywords. Please mind that in the raw docstring all `` should be ignored.

n_c	Hall	HM	HM_short	HM_simple
1	1	1	1	1
2	-1	-1	-1	-1
3:b	2y	1_2_1	121	2
3:c	2	1_1_2	112	2
3:a	2x	2_1_1	211	2
4:b	-2y	1_m_1	1m1	m
4:c	-2	1_1_m	11m	m
4:a	-2x	m_1_1	m11	m
5:b	-2y	1_2/m_1	12/m1	2/m
5:c	-2	1_1_2/m	112/m	2/m
5:a	-2x	2/m_1_1	2/m11	2/m
6	2_2	2_2_2	222	222
7	2_-2	m_m_2	mm2	mm2
8	-2_2	m_m_m	mmm	mmm
9	4	4	4	4
10	-4	-4	-4	-4
11	-4	4/m	4/m	4/m
12	4_2	4_2_2	422	422
13	4_-2	4_m_m	4mm	4mm
14	-4_2	-4_2_m	-42m	-42m
15	-4_2	4/m_m_m	4/mmm	4/mmm
16	3	3	3	3
17	-3	-3	-3	-3
18	3_2	3_1_2	312	312
19	3_-2”	3_m_1	3m1	3m1
20	-3_2	-3_1_m	-31m	-31m
21	6	6	6	6
22	-6	-6	-6	-6
23	-6	6/m	6/m	6/m
24	6_2	6_2_2	622	622
25	6_-2	6_m_m	6mm	6mm
26	-6_2	-6_m_2	-6m2	-6m2
27	-6_2	6/m_m_m	6/mmm	6/mmm
28	2_2_3	2_3	23	23
29	-2_2_3	m_-3	m-3	m-3
30	4_2_3	4_3_2	432	432
31	-4_2_3	-4_3_m	-43m	-43m
32	-4_2_3	m_-3_m	m-3m	m-3m

hikari.symmetry.SG

Since hikari’s groups do not carry information about lattice translations, hikari does not differentiate between point groups and space groups. As a result, all groups are instances of the same class Group, and are stored in almost identical `GroupCatalog`s.

This pre-defined SpaceGroupCatalog SG holds and provides access to all pre-defined space groups available in hikari. The space groups are named and generated based on a wsv file in resources. Individual groups can be accessed in two different ways:

• using a dict-like anonymous accessor: PG[62], PG[‘Pnma’];

• using get() method: PG.get(number=62), PG.get(HM_short=’Pnma’);

The get() method can use a combination of various keywords to unambiguously localize the correct space group. The following keyword arguments can be used:

• n_c - unique “number:setting” string identifying each group

• number - index of the point group as given in ICT A (integer)*

• setting - arbitrary string declaring group setting

• HM - Full underscore-delimited Hermann-Mauguin symbol*

• HM_short - Hermann-Mauguin symbol with underscores removed*

• HM_simple - Short Hermann-Mauguin symbol without setting information*

• HM_numbered - “number: Short Hermann-Mauguin symbol” string

• Hall - Full Hall symbol*

• standard - True for groups in standard setting only

The keywords marked with “*” are called “accessors” and can be used to get the group using bracket notation get[accessor]. If get finds ambiguity, for example SG[‘P2/m’] matches all settings of space group #10: P_1_2/m_1 (std), P_1_1_2/m, and P_2/m_1_1, the user is warned and the first group in std setting, if possible, is returned.

The following table lists all possible values of selected space Group keywords. Please mind that in the raw docstring all `` should be ignored.

n_c	Hall	HM	HM_short	HM_simple
1	p_1	P_1	P1	P1
2	-p_1	P_-1	P-1	P-1
3:b	p_2y	P_1_2_1	P121	P2
3:c	p_2	P_1_1_2	P112	P2
3:a	p_2x	P_2_1_1	P211	P2
4:b	p_2yb	P_1_21_1	P1211	P21
4:c	p_2c	P_1_1_21	P1121	P21
4:a	p_2xa	P_21_1_1	P2111	P21
5:b1	c_2y	C_1_2_1	C121	C2
5:b2	a_2y	A_1_2_1	A121	A2
5:b3	i_2y	I_1_2_1	I121	I2
5:c1	a_2	A_1_1_2	A112	A2
5:c2	b_2	B_1_1_2	B112	B2
5:c3	i_2	I_1_1_2	I112	I2
5:a1	b_2x	B_2_1_1	B211	B2
5:a2	c_2x	C_2_1_1	C211	C2
5:a3	i_2x	I_2_1_1	I211	I2
6:b	p_-2y	P_1_m_1	P1m1	Pm
6:c	p_-2	P_1_1_m	P11m	Pm
6:a	p_-2x	P_m_1_1	Pm11	Pm
7:b1	p_-2yc	P_1_c_1	P1c1	Pc
7:b2	p_-2yac	P_1_n_1	P1n1	Pn
7:b3	p_-2ya	P_1_a_1	P1a1	Pa
7:c1	p_-2a	P_1_1_a	P11a	Pa
7:c2	p_-2ab	P_1_1_n	P11n	Pn
7:c3	p_-2b	P_1_1_b	P11b	Pb
7:a1	p_-2xb	P_b_1_1	Pb11	Pb
7:a2	p_-2xbc	P_n_1_1	Pn11	Pn
7:a3	p_-2xc	P_c_1_1	Pc11	Pc
8:b1	c_-2y	C_1_m_1	C1m1	Cm
8:b2	a_-2y	A_1_m_1	A1m1	Am
8:b3	i_-2y	I_1_m_1	I1m1	Im
8:c1	a_-2	A_1_1_m	A11m	Am
8:c2	b_-2	B_1_1_m	B11m	Bm
8:c3	i_-2	I_1_1_m	I11m	Im
8:a1	b_-2x	B_m_1_1	Bm11	Bm
8:a2	c_-2x	C_m_1_1	Cm11	Cm
8:a3	i_-2x	I_m_1_1	Im11	Im
9:b1	c_-2yc	C_1_c_1	C1c1	Cc
9:b2	a_-2yab	A_1_n_1	A1n1	An
9:b3	i_-2ya	I_1_a_1	I1a1	Ia
9:-b1	a_-2ya	A_1_a_1	A1a1	Aa
9:-b2	c_-2yac	C_1_n_1	C1n1	Cn
9:-b3	i_-2yc	I_1_c_1	I1c1	Ic
9:c1	a_-2a	A_1_1_a	A11a	Aa
9:c2	b_-2ab	B_1_1_n	B11n	Bn
9:c3	i_-2b	I_1_1_b	I11b	Ib
9:-c1	b_-2b	B_1_1_b	B11b	Bb
9:-c2	a_-2ab	A_1_1_n	A11n	An
9:-c3	i_-2a	I_1_1_a	I11a	Ia
9:a1	b_-2xb	B_b_1_1	Bb11	Bb
9:a2	c_-2xac	C_n_1_1	Cn11	Cn
9:a3	i_-2xc	I_c_1_1	Ic11	Ic
9:-a1	c_-2xc	C_c_1_1	Cc11	Cc
9:-a2	b_-2xab	B_n_1_1	Bn11	Bn
9:-a3	i_-2xb	I_b_1_1	Ib11	Ib
10:b	-p_2y	P_1_2/m_1	P12/m1	P2/m
10:c	-p_2	P_1_1_2/m	P112/m	P2/m
10:a	-p_2x	P_2/m_1_1	P2/m11	P2/m
11:b	-p_2yb	P_1_21/m_1	P121/m1	P21/m
11:c	-p_2c	P_1_1_21/m	P1121/m	P21/m
11:a	-p_2xa	P_21/m_1_1	P21/m11	P21/m
12:b1	-c_2y	C_1_2/m_1	C12/m1	C2/m
12:b2	-a_2y	A_1_2/m_1	A12/m1	A2/m
12:b3	-i_2y	I_1_2/m_1	I12/m1	I2/m
12:c1	-a_2	A_1_1_2/m	A112/m	A2/m
12:c2	-b_2	B_1_1_2/m	B112/m	B2/m
12:c3	-i_2	I_1_1_2/m	I112/m	I2/m
12:a1	-b_2x	B_2/m_1_1	B2/m11	B2/m
12:a2	-c_2x	C_2/m_1_1	C2/m11	C2/m
12:a3	-i_2x	I_2/m_1_1	I2/m11	I2/m
13:b1	-p_2yc	P_1_2/c_1	P12/c1	P2/c
13:b2	-p_2yac	P_1_2/n_1	P12/n1	P2/n
13:b3	-p_2ya	P_1_2/a_1	P12/a1	P2/a
13:c1	-p_2a	P_1_1_2/a	P112/a	P2/a
13:c2	-p_2ab	P_1_1_2/n	P112/n	P2/n
13:c3	-p_2b	P_1_1_2/b	P112/b	P2/b
13:a1	-p_2xb	P_2/b_1_1	P2/b11	P2/b
13:a2	-p_2xbc	P_2/n_1_1	P2/n11	P2/n
13:a3	-p_2xc	P_2/c_1_1	P2/c11	P2/c
14:b1	-p_2ybc	P_1_21/c_1	P121/c1	P21/c
14:b2	-p_2yn	P_1_21/n_1	P121/n1	P21/n
14:b3	-p_2yab	P_1_21/a_1	P121/a1	P21/a
14:c1	-p_2ac	P_1_1_21/a	P1121/a	P21/a
14:c2	-p_2n	P_1_1_21/n	P1121/n	P21/n
14:c3	-p_2bc	P_1_1_21/b	P1121/b	P21/b
14:a1	-p_2xab	P_21/b_1_1	P21/b11	P21/b
14:a2	-p_2xn	P_21/n_1_1	P21/n11	P21/n
14:a3	-p_2xac	P_21/c_1_1	P21/c11	P21/c
15:b1	-c_2yc	C_1_2/c_1	C12/c1	C2/c
15:b2	-a_2yab	A_1_2/n_1	A12/n1	A2/n
15:b3	-i_2ya	I_1_2/a_1	I12/a1	I2/a
15:-b1	-a_2ya	A_1_2/a_1	A12/a1	A2/a
15:-b2	-c_2yac	C_1_2/n_1	C12/n1	C2/n
15:-b3	-i_2yc	I_1_2/c_1	I12/c1	I2/c
15:c1	-a_2a	A_1_1_2/a	A112/a	A2/a
15:c2	-b_2ab	B_1_1_2/n	B112/n	B2/n
15:c3	-i_2b	I_1_1_2/b	I112/b	I2/b
15:-c1	-b_2b	B_1_1_2/b	B112/b	B2/b
15:-c2	-a_2ab	A_1_1_2/n	A112/n	A2/n
15:-c3	-i_2a	I_1_1_2/a	I112/a	I2/a
15:a1	-b_2xb	B_2/b_1_1	B2/b11	B2/b
15:a2	-c_2xac	C_2/n_1_1	C2/n11	C2/n
15:a3	-i_2xc	I_2/c_1_1	I2/c11	I2/c
15:-a1	-c_2xc	C_2/c_1_1	C2/c11	C2/c
15:-a2	-b_2xab	B_2/n_1_1	B2/n11	B2/n
15:-a3	-i_2xb	I_2/b_1_1	I2/b11	I2/b
16	p_2_2	P_2_2_2	P222	P222
17	p_2c_2	P_2_2_21	P2221	P2221
17:cab	p_2a_2a	P_21_2_2	P2122	P2122
17:bca	p_2_2b	P_2_21_2	P2212	P2212
18	p_2_2ab	P_21_21_2	P21212	P21212
18:cab	p_2bc_2	P_2_21_21	P22121	P22121
18:bca	p_2ac_2ac	P_21_2_21	P21221	P21221
19	p_2ac_2ab	P_21_21_21	P212121	P212121
20	c_2c_2	C_2_2_21	C2221	C2221
20:cab	a_2a_2a	A_21_2_2	A2122	A2122
20:bca	b_2_2b	B_2_21_2	B2212	B2212
21	c_2_2	C_2_2_2	C222	C222
21:cab	a_2_2	A_2_2_2	A222	A222
21:bca	b_2_2	B_2_2_2	B222	B222
22	f_2_2	F_2_2_2	F222	F222
23	i_2_2	I_2_2_2	I222	I222
24	i_2b_2c	I_21_21_21	I212121	I212121
25	p_2_-2	P_m_m_2	Pmm2	Pmm2
25:cab	p_-2_2	P_2_m_m	P2mm	P2mm
25:bca	p_-2_-2	P_m_2_m	Pm2m	Pm2m
26	p_2c_-2	P_m_c_21	Pmc21	Pmc21
26:ba-c	p_2c_-2c	P_c_m_21	Pcm21	Pcm21
26:cab	p_-2a_2a	P_21_m_a	P21ma	P21ma
26:-cba	p_-2_2a	P_21_a_m	P21am	P21am
26:bca	p_-2_-2b	P_b_21_m	Pb21m	Pb21m
26:a-cb	p_-2b_-2	P_m_21_b	Pm21b	Pm21b
27	p_2_-2c	P_c_c_2	Pcc2	Pcc2
27:cab	p_-2a_2	P_2_a_a	P2aa	P2aa
27:bca	p_-2b_-2b	P_b_2_b	Pb2b	Pb2b
28	p_2_-2a	P_m_a_2	Pma2	Pma2
28:ba-c	p_2_-2b	P_b_m_2	Pbm2	Pbm2
28:cab	p_-2b_2	P_2_m_b	P2mb	P2mb
28:-cba	p_-2c_2	P_2_c_m	P2cm	P2cm
28:bca	p_-2c_-2c	P_c_2_m	Pc2m	Pc2m
28:a-cb	p_-2a_-2a	P_m_2_a	Pm2a	Pm2a
29	p_2c_-2ac	P_c_a_21	Pca21	Pca21
29:ba-c	p_2c_-2b	P_b_c_21	Pbc21	Pbc21
29:cab	p_-2b_2a	P_21_a_b	P21ab	P21ab
29:-cba	p_-2ac_2a	P_21_c_a	P21ca	P21ca
29:bca	p_-2bc_-2c	P_c_21_b	Pc21b	Pc21b
29:a-cb	p_-2a_-2ab	P_b_21_a	Pb21a	Pb21a
30	p_2_-2bc	P_n_c_2	Pnc2	Pnc2
30:ba-c	p_2_-2ac	P_c_n_2	Pcn2	Pcn2
30:cab	p_-2ac_2	P_2_n_a	P2na	P2na
30:-cba	p_-2ab_2	P_2_a_n	P2an	P2an
30:bca	p_-2ab_-2ab	P_b_2_n	Pb2n	Pb2n
30:a-cb	p_-2bc_-2bc	P_n_2_b	Pn2b	Pn2b
31	p_2ac_-2	P_m_n_21	Pmn21	Pmn21
31:ba-c	p_2bc_-2bc	P_n_m_21	Pnm21	Pnm21
31:cab	p_-2ab_2ab	P_21_m_n	P21mn	P21mn
31:-cba	p_-2_2ac	P_21_n_m	P21nm	P21nm
31:bca	p_-2_-2bc	P_n_21_m	Pn21m	Pn21m
31:a-cb	p_-2ab_-2	P_m_21_n	Pm21n	Pm21n
32	p_2_-2ab	P_b_a_2	Pba2	Pba2
32:cab	p_-2bc_2	P_2_c_b	P2cb	P2cb
32:bca	p_-2ac_-2ac	P_c_2_a	Pc2a	Pc2a
33	p_2c_-2n	P_n_a_21	Pna21	Pna21
33:ba-c	p_2c_-2ab	P_b_n_21	Pbn21	Pbn21
33:cab	p_-2bc_2a	P_21_n_b	P21nb	P21nb
33:-cba	p_-2n_2a	P_21_c_n	P21cn	P21cn
33:bca	p_-2n_-2ac	P_c_21_n	Pc21n	Pc21n
33:a-cb	p_-2ac_-2n	P_n_21_a	Pn21a	Pn21a
34	p_2_-2n	P_n_n_2	Pnn2	Pnn2
34:cab	p_-2n_2	P_2_n_n	P2nn	P2nn
34:bca	p_-2n_-2n	P_n_2_n	Pn2n	Pn2n
35	c_2_-2	C_m_m_2	Cmm2	Cmm2
35:cab	a_-2_2	A_2_m_m	A2mm	A2mm
35:bca	b_-2_-2	B_m_2_m	Bm2m	Bm2m
36	c_2c_-2	C_m_c_21	Cmc21	Cmc21
36:ba-c	c_2c_-2c	C_c_m_21	Ccm21	Ccm21
36:cab	a_-2a_2a	A_21_m_a	A21ma	A21ma
36:-cba	a_-2_2a	A_21_a_m	A21am	A21am
36:bca	b_-2_-2b	B_b_21_m	Bb21m	Bb21m
36:a-cb	b_-2b_-2	B_m_21_b	Bm21b	Bm21b
37	c_2_-2c	C_c_c_2	Ccc2	Ccc2
37:cab	a_-2a_2	A_2_a_a	A2aa	A2aa
37:bca	b_-2b_-2b	B_b_2_b	Bb2b	Bb2b
38	a_2_-2	A_m_m_2	Amm2	Amm2
38:ba-c	b_2_-2	B_m_m_2	Bmm2	Bmm2
38:cab	b_-2_2	B_2_m_m	B2mm	B2mm
38:-cba	c_-2_2	C_2_m_m	C2mm	C2mm
38:bca	c_-2_-2	C_m_2_m	Cm2m	Cm2m
38:a-cb	a_-2_-2	A_m_2_m	Am2m	Am2m
39	a_2_-2b	A_e_m_2	Aem2	Aem2
39:ba-c	b_2_-2a	B_m_a_2	Bma2	Bma2
39:cab	b_-2a_2	B_2_c_m	B2cm	B2cm
39:-cba	c_-2a_2	C_2_m_b	C2mb	C2mb
39:bca	c_-2a_-2a	C_m_2_a	Cm2a	Cm2a
39:a-cb	a_-2b_-2b	A_c_2_m	Ac2m	Ac2m
40	a_2_-2a	A_m_a_2	Ama2	Ama2
40:ba-c	b_2_-2b	B_b_m_2	Bbm2	Bbm2
40:cab	b_-2b_2	B_2_m_b	B2mb	B2mb
40:-cba	c_-2c_2	C_2_c_m	C2cm	C2cm
40:bca	c_-2c_-2c	C_c_2_m	Cc2m	Cc2m
40:a-cb	a_-2a_-2a	A_m_2_a	Am2a	Am2a
41	a_2_-2ab	A_e_a_2	Aea2	Aea2
41:ba-c	b_2_-2ab	B_b_a_2	Bba2	Bba2
41:cab	b_-2ab_2	B_2_c_b	B2cb	B2cb
41:-cba	c_-2ac_2	C_2_c_b	C2cb	C2cb
41:bca	c_-2ac_-2ac	C_c_2_a	Cc2a	Cc2a
41:a-cb	a_-2ab_-2ab	A_c_2_a	Ac2a	Ac2a
42	f_2_-2	F_m_m_2	Fmm2	Fmm2
42:cab	f_-2_2	F_2_m_m	F2mm	F2mm
42:bca	f_-2_-2	F_m_2_m	Fm2m	Fm2m
43	f_2_-2d	F_d_d_2	Fdd2	Fdd2
43:cab	f_-2d_2	F_2_d_d	F2dd	F2dd
43:bca	f_-2d_-2d	F_d_2_d	Fd2d	Fd2d
44	i_2_-2	I_m_m_2	Imm2	Imm2
44:cab	i_-2_2	I_2_m_m	I2mm	I2mm
44:bca	i_-2_-2	I_m_2_m	Im2m	Im2m
45	i_2_-2c	I_b_a_2	Iba2	Iba2
45:cab	i_-2a_2	I_2_c_b	I2cb	I2cb
45:bca	i_-2b_-2b	I_c_2_a	Ic2a	Ic2a
46	i_2_-2a	I_m_a_2	Ima2	Ima2
46:ba-c	i_2_-2b	I_b_m_2	Ibm2	Ibm2
46:cab	i_-2b_2	I_2_m_b	I2mb	I2mb
46:-cba	i_-2c_2	I_2_c_m	I2cm	I2cm
46:bca	i_-2c_-2c	I_c_2_m	Ic2m	Ic2m
46:a-cb	i_-2a_-2a	I_m_2_a	Im2a	Im2a
47	-p_2_2	P_m_m_m	Pmmm	Pmmm
48:1	p_2_2_-1n	P_n_n_n:1	Pnnn:1	Pnnn
48:2	-p_2ab_2bc	P_n_n_n:2	Pnnn:2	Pnnn
49	-p_2_2c	P_c_c_m	Pccm	Pccm
49:cab	-p_2a_2	P_m_a_a	Pmaa	Pmaa
49:bca	-p_2b_2b	P_b_m_b	Pbmb	Pbmb
50:1	p_2_2_-1ab	P_b_a_n:1	Pban:1	Pban
50:2	-p_2ab_2b	P_b_a_n:2	Pban:2	Pban
50:1cab	p_2_2_-1bc	P_n_c_b:1	Pncb:1	Pncb
50:2cab	-p_2b_2bc	P_n_c_b:2	Pncb:2	Pncb
50:1bca	p_2_2_-1ac	P_c_n_a:1	Pcna:1	Pcna
50:2bca	-p_2a_2c	P_c_n_a:2	Pcna:2	Pcna
51	-p_2a_2a	P_m_m_a	Pmma	Pmma
51:ba-c	-p_2b_2	P_m_m_b	Pmmb	Pmmb
51:cab	-p_2_2b	P_b_m_m	Pbmm	Pbmm
51:-cba	-p_2c_2c	P_c_m_m	Pcmm	Pcmm
51:bca	-p_2c_2	P_m_c_m	Pmcm	Pmcm
51:a-cb	-p_2_2a	P_m_a_m	Pmam	Pmam
52	-p_2a_2bc	P_n_n_a	Pnna	Pnna
52:ba-c	-p_2b_2n	P_n_n_b	Pnnb	Pnnb
52:cab	-p_2n_2b	P_b_n_n	Pbnn	Pbnn
52:-cba	-p_2ab_2c	P_c_n_n	Pcnn	Pcnn
52:bca	-p_2ab_2n	P_n_c_n	Pncn	Pncn
52:a-cb	-p_2n_2bc	P_n_a_n	Pnan	Pnan
53	-p_2ac_2	P_m_n_a	Pmna	Pmna
53:ba-c	-p_2bc_2bc	P_n_m_b	Pnmb	Pnmb
53:cab	-p_2ab_2ab	P_b_m_n	Pbmn	Pbmn
53:-cba	-p_2_2ac	P_c_n_m	Pcnm	Pcnm
53:bca	-p_2_2bc	P_n_c_m	Pncm	Pncm
53:a-cb	-p_2ab_2	P_m_a_n	Pman	Pman
54	-p_2a_2ac	P_c_c_a	Pcca	Pcca
54:ba-c	-p_2b_2c	P_c_c_b	Pccb	Pccb
54:cab	-p_2a_2b	P_b_a_a	Pbaa	Pbaa
54:-cba	-p_2ac_2c	P_c_a_a	Pcaa	Pcaa
54:bca	-p_2bc_2b	P_b_c_b	Pbcb	Pbcb
54:a-cb	-p_2b_2ab	P_b_a_b	Pbab	Pbab
55	-p_2_2ab	P_b_a_m	Pbam	Pbam
55:cab	-p_2bc_2	P_m_c_b	Pmcb	Pmcb
55:bca	-p_2ac_2ac	P_c_m_a	Pcma	Pcma
56	-p_2ab_2ac	P_c_c_n	Pccn	Pccn
56:cab	-p_2ac_2bc	P_n_a_a	Pnaa	Pnaa
56:bca	-p_2bc_2ab	P_b_n_b	Pbnb	Pbnb
57	-p_2c_2b	P_b_c_m	Pbcm	Pbcm
57:ba-c	-p_2c_2ac	P_c_a_m	Pcam	Pcam
57:cab	-p_2ac_2a	P_m_c_a	Pmca	Pmca
57:-cba	-p_2b_2a	P_m_a_b	Pmab	Pmab
57:bca	-p_2a_2ab	P_b_m_a	Pbma	Pbma
57:a-cb	-p_2bc_2c	P_c_m_b	Pcmb	Pcmb
58	-p_2_2n	P_n_n_m	Pnnm	Pnnm
58:cab	-p_2n_2	P_m_n_n	Pmnn	Pmnn
58:bca	-p_2n_2n	P_n_m_n	Pnmn	Pnmn
59:1	p_2_2ab_-1ab	P_m_m_n:1	Pmmn:1	Pmmn
59:2	-p_2ab_2a	P_m_m_n:2	Pmmn:2	Pmmn
59:1cab	p_2bc_2_-1bc	P_n_m_m:1	Pnmm:1	Pnmm
59:2cab	-p_2c_2bc	P_n_m_m:2	Pnmm:2	Pnmm
59:1bca	p_2ac_2ac_-1ac	P_m_n_m:1	Pmnm:1	Pmnm
59:2bca	-p_2c_2a	P_m_n_m:2	Pmnm:2	Pmnm
60	-p_2n_2ab	P_b_c_n	Pbcn	Pbcn
60:ba-c	-p_2n_2c	P_c_a_n	Pcan	Pcan
60:cab	-p_2a_2n	P_n_c_a	Pnca	Pnca
60:-cba	-p_2bc_2n	P_n_a_b	Pnab	Pnab
60:bca	-p_2ac_2b	P_b_n_a	Pbna	Pbna
60:a-cb	-p_2b_2ac	P_c_n_b	Pcnb	Pcnb
61	-p_2ac_2ab	P_b_c_a	Pbca	Pbca
61:ba-c	-p_2bc_2ac	P_c_a_b	Pcab	Pcab
62	-p_2ac_2n	P_n_m_a	Pnma	Pnma
62:ba-c	-p_2bc_2a	P_m_n_b	Pmnb	Pmnb
62:cab	-p_2c_2ab	P_b_n_m	Pbnm	Pbnm
62:-cba	-p_2n_2ac	P_c_m_n	Pcmn	Pcmn
62:bca	-p_2n_2a	P_m_c_n	Pmcn	Pmcn
62:a-cb	-p_2c_2n	P_n_a_m	Pnam	Pnam
63	-c_2c_2	C_m_c_m	Cmcm	Cmcm
63:ba-c	-c_2c_2c	C_c_m_m	Ccmm	Ccmm
63:cab	-a_2a_2a	A_m_m_a	Amma	Amma
63:-cba	-a_2_2a	A_m_a_m	Amam	Amam
63:bca	-b_2_2b	B_b_m_m	Bbmm	Bbmm
63:a-cb	-b_2b_2	B_m_m_b	Bmmb	Bmmb
64	-c_2ac_2	C_m_c_e	Cmce	Cmce
64:ba-c	-c_2ac_2ac	C_c_m_b	Ccmb	Ccmb
64:cab	-a_2ab_2ab	A_b_m_a	Abma	Abma
64:-cba	-a_2_2ab	A_c_a_m	Acam	Acam
64:bca	-b_2_2ab	B_b_c_m	Bbcm	Bbcm
64:a-cb	-b_2ab_2	B_m_a_b	Bmab	Bmab
65	-c_2_2	C_m_m_m	Cmmm	Cmmm
65:cab	-a_2_2	A_m_m_m	Ammm	Ammm
65:bca	-b_2_2	B_m_m_m	Bmmm	Bmmm
66	-c_2_2c	C_c_c_m	Cccm	Cccm
66:cab	-a_2a_2	A_m_a_a	Amaa	Amaa
66:bca	-b_2b_2b	B_b_m_b	Bbmb	Bbmb
67	-c_2a_2	C_m_m_e	Cmme	Cmme
67:ba-c	-c_2a_2a	C_m_m_b	Cmmb	Cmmb
67:cab	-a_2b_2b	A_b_m_m	Abmm	Abmm
67:-cba	-a_2_2b	A_c_m_m	Acmm	Acmm
67:bca	-b_2_2a	B_m_c_m	Bmcm	Bmcm
67:a-cb	-b_2a_2	B_m_a_m	Bmam	Bmam
68:1	c_2_2_-1ac	C_c_c_e:1	Ccce:1	Ccce
68:2	-c_2a_2ac	C_c_c_e:2	Ccce:2	Ccce
68:1ba-	c_2_2_-1ac	C_c_c_b:1	Cccb:1	Cccb
68:2ba-	-c_2a_2c	C_c_c_b:2	Cccb:2	Cccb
68:1cab	a_2_2_-1ab	A_b_a_a:1	Abaa:1	Abaa
68:2cab	-a_2a_2b	A_b_a_a:2	Abaa:2	Abaa
68:1-cb	a_2_2_-1ab	A_c_a_a:1	Acaa:1	Acaa
68:2-cb	-a_2ab_2b	A_c_a_a:2	Acaa:2	Acaa
68:1bca	b_2_2_-1ab	B_b_c_b:1	Bbcb:1	Bbcb
68:2bca	-b_2ab_2b	B_b_c_b:2	Bbcb:2	Bbcb
68:1a-c	b_2_2_-1ab	B_b_a_b:1	Bbab:1	Bbab
68:2a-c	-b_2b_2ab	B_b_a_b:2	Bbab:2	Bbab
69	-f_2_2	F_m_m_m	Fmmm	Fmmm
70:1	f_2_2_-1d	F_d_d_d:1	Fddd:1	Fddd
70:2	-f_2uv_2vw	F_d_d_d:2	Fddd:2	Fddd
71	-i_2_2	I_m_m_m	Immm	Immm
72	-i_2_2c	I_b_a_m	Ibam	Ibam
72:cab	-i_2a_2	I_m_c_b	Imcb	Imcb
72:bca	-i_2b_2b	I_c_m_a	Icma	Icma
73	-i_2b_2c	I_b_c_a	Ibca	Ibca
73:ba-c	-i_2a_2b	I_c_a_b	Icab	Icab
74	-i_2b_2	I_m_m_a	Imma	Imma
74:ba-c	-i_2a_2a	I_m_m_b	Immb	Immb
74:cab	-i_2c_2c	I_b_m_m	Ibmm	Ibmm
74:-cba	-i_2_2b	I_c_m_m	Icmm	Icmm
74:bca	-i_2_2a	I_m_c_m	Imcm	Imcm
74:a-cb	-i_2c_2	I_m_a_m	Imam	Imam
75	p_4	P_4	P4	P4
76	p_4w	P_41	P41	P41
77	p_4c	P_42	P42	P42
78	p_4cw	P_43	P43	P43
79	i_4	I_4	I4	I4
80	i_4bw	I_41	I41	I41
81	p_-4	P_-4	P-4	P-4
82	i_-4	I_-4	I-4	I-4
83	-p_4	P_4/m	P4/m	P4/m
84	-p_4c	P_42/m	P42/m	P42/m
85:1	p_4ab_-1ab	P_4/n:1	P4/n:1	P4/n
85:2	-p_4a	P_4/n:2	P4/n:2	P4/n
86:1	p_4n_-1n	P_42/n:1	P42/n:1	P42/n
86:2	-p_4bc	P_42/n:2	P42/n:2	P42/n
87	-i_4	I_4/m	I4/m	I4/m
88:1	i_4bw_-1bw	I_41/a:1	I41/a:1	I41/a
88:2	-i_4ad	I_41/a:2	I41/a:2	I41/a
89	p_4_2	P_4_2_2	P422	P422
90	p_4ab_2ab	P_4_21_2	P4212	P4212
91	p_4w_2c	P_41_2_2	P4122	P4122
92	p_4abw_2nw	P_41_21_2	P41212	P41212
93	p_4c_2	P_42_2_2	P4222	P4222
94	p_4n_2n	P_42_21_2	P42212	P42212
95	p_4cw_2c	P_43_2_2	P4322	P4322
96	p_4nw_2abw	P_43_21_2	P43212	P43212
97	i_4_2	I_4_2_2	I422	I422
98	i_4bw_2bw	I_41_2_2	I4122	I4122
99	p_4_-2	P_4_m_m	P4mm	P4mm
100	p_4_-2ab	P_4_b_m	P4bm	P4bm
101	p_4c_-2c	P_42_c_m	P42cm	P42cm
102	p_4n_-2n	P_42_n_m	P42nm	P42nm
103	p_4_-2c	P_4_c_c	P4cc	P4cc
104	p_4_-2n	P_4_n_c	P4nc	P4nc
105	p_4c_-2	P_42_m_c	P42mc	P42mc
106	p_4c_-2ab	P_42_b_c	P42bc	P42bc
107	i_4_-2	I_4_m_m	I4mm	I4mm
108	i_4_-2c	I_4_c_m	I4cm	I4cm
109	i_4bw_-2	I_41_m_d	I41md	I41md
110	i_4bw_-2c	I_41_c_d	I41cd	I41cd
111	p_-4_2	P_-4_2_m	P-42m	P-42m
112	p_-4_2c	P_-4_2_c	P-42c	P-42c
113	p_-4_2ab	P_-4_21_m	P-421m	P-421m
114	p_-4_2n	P_-4_21_c	P-421c	P-421c
115	p_-4_-2	P_-4_m_2	P-4m2	P-4m2
116	p_-4_-2c	P_-4_c_2	P-4c2	P-4c2
117	p_-4_-2ab	P_-4_b_2	P-4b2	P-4b2
118	p_-4_-2n	P_-4_n_2	P-4n2	P-4n2
119	i_-4_-2	I_-4_m_2	I-4m2	I-4m2
120	i_-4_-2c	I_-4_c_2	I-4c2	I-4c2
121	i_-4_2	I_-4_2_m	I-42m	I-42m
122	i_-4_2bw	I_-4_2_d	I-42d	I-42d
123	-p_4_2	P_4/m_m_m	P4/mmm	P4/mmm
124	-p_4_2c	P_4/m_c_c	P4/mcc	P4/mcc
125:1	p_4_2_-1ab	P_4/n_b_m:1	P4/nbm:1	P4/nbm
125:2	-p_4a_2b	P_4/n_b_m:2	P4/nbm:2	P4/nbm
126:1	p_4_2_-1n	P_4/n_n_c:1	P4/nnc:1	P4/nnc
126:2	-p_4a_2bc	P_4/n_n_c:2	P4/nnc:2	P4/nnc
127	-p_4_2ab	P_4/m_b_m	P4/mbm	P4/mbm
128	-p_4_2n	P_4/m_n_c	P4/mnc	P4/mnc
129:1	p_4ab_2ab_-1ab	P_4/n_m_m:1	P4/nmm:1	P4/nmm
129:2	-p_4a_2a	P_4/n_m_m:2	P4/nmm:2	P4/nmm
130:1	p_4ab_2n_-1ab	P_4/n_c_c:1	P4/ncc:1	P4/ncc
130:2	-p_4a_2ac	P_4/n_c_c:2	P4/ncc:2	P4/ncc
131	-p_4c_2	P_42/m_m_c	P42/mmc	P42/mmc
132	-p_4c_2c	P_42/m_c_m	P42/mcm	P42/mcm
133:1	p_4n_2c_-1n	P_42/n_b_c:1	P42/nbc:1	P42/nbc
133:2	-p_4ac_2b	P_42/n_b_c:2	P42/nbc:2	P42/nbc
134:1	p_4n_2_-1n	P_42/n_n_m:1	P42/nnm:1	P42/nnm
134:2	-p_4ac_2bc	P_42/n_n_m:2	P42/nnm:2	P42/nnm
135	-p_4c_2ab	P_42/m_b_c	P42/mbc	P42/mbc
136	-p_4n_2n	P_42/m_n_m	P42/mnm	P42/mnm
137:1	p_4n_2n_-1n	P_42/n_m_c:1	P42/nmc:1	P42/nmc
137:2	-p_4ac_2a	P_42/n_m_c:2	P42/nmc:2	P42/nmc
138:1	p_4n_2ab_-1n	P_42/n_c_m:1	P42/ncm:1	P42/ncm
138:2	-p_4ac_2ac	P_42/n_c_m:2	P42/ncm:2	P42/ncm
139	-i_4_2	I_4/m_m_m	I4/mmm	I4/mmm
140	-i_4_2c	I_4/m_c_m	I4/mcm	I4/mcm
141:1	i_4bw_2bw_-1bw	I_41/a_m_d:1	I41/amd:1	I41/amd
141:2	-i_4bd_2	I_41/a_m_d:2	I41/amd:2	I41/amd
142:1	i_4bw_2aw_-1bw	I_41/a_c_d:1	I41/acd:1	I41/acd
142:2	-i_4bd_2c	I_41/a_c_d:2	I41/acd:2	I41/acd
143	p_3	P_3	P3	P3
144	p_31	P_31	P31	P31
145	p_32	P_32	P32	P32
146:h	r_3	R_3:h	R3:h	R3
146:r	p_3*	R_3:r	R3:r	R3
147	-p_3	P_-3	P-3	P-3
148:h	-r_3	R_-3:h	R-3:h	R-3
148:r	-p_3*	R_-3:r	R-3:r	R-3
149	p_3_2	P_3_1_2	P312	P312
150	p_3_2”	P_3_2_1	P321	P321
151	p_31_2_(0_0_4)	P_31_1_2	P3112	P3112
152	p_31_2”	P_31_2_1	P3121	P3121
153	p_32_2_(0_0_2)	P_32_1_2	P3212	P3212
154	p_32_2”	P_32_2_1	P3221	P3221
155:h	r_3_2”	R_3_2:h	R32:h	R32
155:r	p_3*_2	R_3_2:r	R32:r	R32
156	p_3_-2”	P_3_m_1	P3m1	P3m1
157	p_3_-2	P_3_1_m	P31m	P31m
158	p_3_-2”c	P_3_c_1	P3c1	P3c1
159	p_3_-2c	P_3_1_c	P31c	P31c
160:h	r_3_-2”	R_3_m:h	R3m:h	R3m
160:r	p_3*_-2	R_3_m:r	R3m:r	R3m
161:h	r_3_-2”c	R_3_c:h	R3c:h	R3c
161:r	p_3*_-2n	R_3_c:r	R3c:r	R3c
162	-p_3_2	P_-3_1_m	P-31m	P-31m
163	-p_3_2c	P_-3_1_c	P-31c	P-31c
164	-p_3_2”	P_-3_m_1	P-3m1	P-3m1
165	-p_3_2”c	P_-3_c_1	P-3c1	P-3c1
166:h	-r_3_2”	R_-3_m:h	R-3m:h	R-3m
166:r	-p_3*_2	R_-3_m:r	R-3m:r	R-3m
167:h	-r_3_2”c	R_-3_c:h	R-3c:h	R-3c
167:r	-p_3*_2n	R_-3_c:r	R-3c:r	R-3c
168	p_6	P_6	P6	P6
169	p_61	P_61	P61	P61
170	p_65	P_65	P65	P65
171	p_62	P_62	P62	P62
172	p_64	P_64	P64	P64
173	p_6c	P_63	P63	P63
174	p_-6	P_-6	P-6	P-6
175	-p_6	P_6/m	P6/m	P6/m
176	-p_6c	P_63/m	P63/m	P63/m
177	p_6_2	P_6_2_2	P622	P622
178	p_61_2_(0_0_5)	P_61_2_2	P6122	P6122
179	p_65_2_(0_0_1)	P_65_2_2	P6522	P6522
180	p_62_2_(0_0_4)	P_62_2_2	P6222	P6222
181	p_64_2_(0_0_2)	P_64_2_2	P6422	P6422
182	p_6c_2c	P_63_2_2	P6322	P6322
183	p_6_-2	P_6_m_m	P6mm	P6mm
184	p_6_-2c	P_6_c_c	P6cc	P6cc
185	p_6c_-2	P_63_c_m	P63cm	P63cm
186	p_6c_-2c	P_63_m_c	P63mc	P63mc
187	p_-6_2	P_-6_m_2	P-6m2	P-6m2
188	p_-6c_2	P_-6_c_2	P-6c2	P-6c2
189	p_-6_-2	P_-6_2_m	P-62m	P-62m
190	p_-6c_-2c	P_-6_2_c	P-62c	P-62c
191	-p_6_2	P_6/m_m_m	P6/mmm	P6/mmm
192	-p_6_2c	P_6/m_c_c	P6/mcc	P6/mcc
193	-p_6c_2	P_63/m_c_m	P63/mcm	P63/mcm
194	-p_6c_2c	P_63/m_m_c	P63/mmc	P63/mmc
195	p_2_2_3	P_2_3	P23	P23
196	f_2_2_3	F_2_3	F23	F23
197	i_2_2_3	I_2_3	I23	I23
198	p_2ac_2ab_3	P_21_3	P213	P213
199	i_2b_2c_3	I_21_3	I213	I213
200	-p_2_2_3	P_m_-3	Pm-3	Pm-3
201:1	p_2_2_3_-1n	P_n_-3:1	Pn-3:1	Pn-3
201:2	-p_2ab_2bc_3	P_n_-3:2	Pn-3:2	Pn-3
202	-f_2_2_3	F_m_-3	Fm-3	Fm-3
203:1	f_2_2_3_-1d	F_d_-3:1	Fd-3:1	Fd-3
203:2	-f_2uv_2vw_3	F_d_-3:2	Fd-3:2	Fd-3
204	-i_2_2_3	I_m_-3	Im-3	Im-3
205	-p_2ac_2ab_3	P_a_-3	Pa-3	Pa-3
206	-i_2b_2c_3	I_a_-3	Ia-3	Ia-3
207	p_4_2_3	P_4_3_2	P432	P432
208	p_4n_2_3	P_42_3_2	P4232	P4232
209	f_4_2_3	F_4_3_2	F432	F432
210	f_4d_2_3	F_41_3_2	F4132	F4132
211	i_4_2_3	I_4_3_2	I432	I432
212	p_4acd_2ab_3	P_43_3_2	P4332	P4332
213	p_4bd_2ab_3	P_41_3_2	P4132	P4132
214	i_4bd_2c_3	I_41_3_2	I4132	I4132
215	p_-4_2_3	P_-4_3_m	P-43m	P-43m
216	f_-4_2_3	F_-4_3_m	F-43m	F-43m
217	i_-4_2_3	I_-4_3_m	I-43m	I-43m
218	p_-4n_2_3	P_-4_3_n	P-43n	P-43n
219	f_-4a_2_3	F_-4_3_c	F-43c	F-43c
220	i_-4bd_2c_3	I_-4_3_d	I-43d	I-43d
221	-p_4_2_3	P_m_-3_m	Pm-3m	Pm-3m
222:1	p_4_2_3_-1n	P_n_-3_n:1	Pn-3n:1	Pn-3n
222:2	-p_4a_2bc_3	P_n_-3_n:2	Pn-3n:2	Pn-3n
223	-p_4n_2_3	P_m_-3_n	Pm-3n	Pm-3n
224:1	p_4n_2_3_-1n	P_n_-3_m:1	Pn-3m:1	Pn-3m
224:2	-p_4bc_2bc_3	P_n_-3_m:2	Pn-3m:2	Pn-3m
225	-f_4_2_3	F_m_-3_m	Fm-3m	Fm-3m
226	-f_4a_2_3	F_m_-3_c	Fm-3c	Fm-3c
227:1	f_4d_2_3_-1d	F_d_-3_m:1	Fd-3m:1	Fd-3m
227:2	-f_4vw_2vw_3	F_d_-3_m:2	Fd-3m:2	Fd-3m
228:1	f_4d_2_3_-1ad	F_d_-3_c:1	Fd-3c:1	Fd-3c
228:2	-f_4ud_2vw_3	F_d_-3_c:2	Fd-3c:2	Fd-3c
229	-i_4_2_3	I_m_-3_m	Im-3m	Im-3m
230	-i_4bd_2c_3	I_a_-3_d	Ia-3d	Ia-3d