SpaceGroupFactory.cpp

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// Mantid Repository : https://github.com/mantidproject/mantid
//
// Copyright © 2018 ISIS Rutherford Appleton Laboratory UKRI,
//   NScD Oak Ridge National Laboratory, European Spallation Source,
//   Institut Laue - Langevin & CSNS, Institute of High Energy Physics, CAS
// SPDX - License - Identifier: GPL - 3.0 +
#include "MantidGeometry/Crystal/SpaceGroupFactory.h"
#include "MantidGeometry/Crystal/CenteringGroup.h"
#include "MantidGeometry/Crystal/GroupTransformation.h"
#include "MantidGeometry/Crystal/ProductOfCyclicGroups.h"
#include "MantidGeometry/Crystal/SymmetryOperationSymbolParser.h"
#include "MantidKernel/Exception.h"
#include "MantidKernel/LibraryManager.h"
 
#include <algorithm>
#include <boost/algorithm/string.hpp>
#include <locale>
#include <memory>
#include <utility>
 
namespace Mantid::Geometry {
 
bool isValidGeneratorString(const std::string &generatorString) {
  std::vector<std::string> generatorStrings;
  boost::split(generatorStrings, generatorString, boost::is_any_of(";"));
 
  return std::all_of(generatorStrings.cbegin(), generatorStrings.cend(), [&](const auto &identifier) {
    try {
      SymmetryOperationSymbolParser::parseIdentifier(identifier);
    } catch (const Kernel::Exception::ParseError &) {
      return false;
    }
    return true;
  });
}
 
std::vector<std::string> operator*(const SymmetryOperation &symOp, const std::vector<std::string> &strings) {
  if (strings.size() < 3) {
    throw std::invalid_argument("There must be at least three strings in the "
                                "vector for SymmetryOperation * "
                                "std::vector<std::string>.");
  }
 
  std::vector<int> transformedIndices = symOp.matrix() * std::vector<int>{0, 1, 2};
 
  std::vector<std::string> transformedStrings;
  std::transform(transformedIndices.cbegin(), transformedIndices.cend(), std::back_inserter(transformedStrings),
                 [=](int index) { return strings[abs(index)]; });
 
  if (transformedStrings.size() < strings.size()) {
    transformedStrings.insert(transformedStrings.end(), strings.cbegin() + transformedStrings.size(), strings.cend());
  }
 
  return transformedStrings;
}
 
AbstractSpaceGroupGenerator::AbstractSpaceGroupGenerator(size_t number, std::string hmSymbol,
                                                         std::string generatorInformation)
    : m_number(number), m_hmSymbol(std::move(hmSymbol)), m_generatorString(std::move(generatorInformation)),
      m_prototype() {}
 
SpaceGroup_const_sptr AbstractSpaceGroupGenerator::getPrototype() {
  if (!hasValidPrototype()) {
    m_prototype = generatePrototype();
  }
 
  return m_prototype;
}
 
SpaceGroup_const_sptr AbstractSpaceGroupGenerator::generatePrototype() {
  Group_const_sptr generatingGroup = generateGroup();
 
  if (!generatingGroup) {
    throw std::runtime_error("Could not create group from supplied symmetry operations.");
  }
 
  return std::make_shared<const SpaceGroup>(m_number, m_hmSymbol, *generatingGroup);
}
 
AlgorithmicSpaceGroupGenerator::AlgorithmicSpaceGroupGenerator(size_t number, const std::string &hmSymbol,
                                                               const std::string &generatorInformation)
    : AbstractSpaceGroupGenerator(number, hmSymbol, generatorInformation) {
  if (!isValidGeneratorString(generatorInformation)) {
    throw std::runtime_error("Generator string could not be parsed: " + generatorInformation);
  }
}
 
Group_const_sptr AlgorithmicSpaceGroupGenerator::generateGroup() const {
  Group_const_sptr baseGroup = GroupFactory::create<ProductOfCyclicGroups>(getGeneratorString());
  Group_const_sptr centeringGroup = GroupFactory::create<CenteringGroup>(getCenteringSymbol());
 
  return baseGroup * centeringGroup;
}
 
std::string AlgorithmicSpaceGroupGenerator::getCenteringSymbol() const { return getHMSymbol().substr(0, 1); }
 
TransformationSpaceGroupGenerator::TransformationSpaceGroupGenerator(size_t number, const std::string &hmSymbol,
                                                                     const std::string &generatorInformation)
    : AbstractSpaceGroupGenerator(number, hmSymbol, generatorInformation) {
  setBaseAndTransformation(generatorInformation);
}
 
Group_const_sptr TransformationSpaceGroupGenerator::generateGroup() const {
  SpaceGroup_const_sptr baseGroup = getBaseSpaceGroup();
 
  GroupTransformation transformation(m_transformation);
 
  return std::make_shared<Group>(transformation(*baseGroup));
}
 
SpaceGroup_const_sptr TransformationSpaceGroupGenerator::getBaseSpaceGroup() const {
  return SpaceGroupFactory::Instance().createSpaceGroup(m_baseGroupHMSymbol);
}
 
void TransformationSpaceGroupGenerator::setBaseAndTransformation(const std::string &generatorInformation) {
  std::vector<std::string> parts;
  boost::split(parts, generatorInformation, boost::is_any_of("|"));
 
  if (parts.size() != 2) {
    throw std::invalid_argument("Not a valid string for generation of "
                                "transformed space groups. Correct format is "
                                "'HM symbol | transformation'.");
  }
 
  m_baseGroupHMSymbol = boost::trim_copy(parts.front());
  m_transformation = parts.back();
}
 
TabulatedSpaceGroupGenerator::TabulatedSpaceGroupGenerator(size_t number, const std::string &hmSymbol,
                                                           const std::string &generatorInformation)
    : AbstractSpaceGroupGenerator(number, hmSymbol, generatorInformation) {
  if (!isValidGeneratorString(generatorInformation)) {
    throw std::runtime_error("Generator string could not be parsed: " + generatorInformation);
  }
}
 
Group_const_sptr TabulatedSpaceGroupGenerator::generateGroup() const {
  return GroupFactory::create<Group>(getGeneratorString());
}
 
SpaceGroup_const_sptr SpaceGroupFactoryImpl::createSpaceGroup(const std::string &hmSymbol) {
  if (!isSubscribed(hmSymbol)) {
    throw std::invalid_argument("Space group with symbol '" + hmSymbol + "' is not registered.");
  }
 
  return constructFromPrototype(getPrototype(hmSymbol));
}
 
bool SpaceGroupFactoryImpl::isSubscribed(const std::string &hmSymbol) const {
  return m_generatorMap.find(hmSymbol) != m_generatorMap.end();
}
 
bool SpaceGroupFactoryImpl::isSubscribed(size_t number) const { return m_numberMap.find(number) != m_numberMap.end(); }
 
std::vector<std::string> SpaceGroupFactoryImpl::subscribedSpaceGroupSymbols() const {
  std::vector<std::string> symbols;
  symbols.reserve(m_generatorMap.size());
  std::transform(m_generatorMap.cbegin(), m_generatorMap.cend(), std::back_inserter(symbols),
                 [](const auto &generator) { return generator.first; });
  return symbols;
}
 
std::vector<std::string> SpaceGroupFactoryImpl::subscribedSpaceGroupSymbols(size_t number) const {
  std::vector<std::string> symbols;
 
  auto keyPair = m_numberMap.equal_range(number);
 
  for (auto it = keyPair.first; it != keyPair.second; ++it) {
    symbols.emplace_back(it->second);
  }
 
  return symbols;
}
 
std::vector<size_t> SpaceGroupFactoryImpl::subscribedSpaceGroupNumbers() const {
  std::vector<size_t> numbers;
  numbers.reserve(m_numberMap.size());
 
  for (auto it = m_numberMap.begin(); it != m_numberMap.end(); it = m_numberMap.upper_bound(it->first)) {
    numbers.emplace_back(it->first);
  }
 
  return numbers;
}
 
std::vector<std::string> SpaceGroupFactoryImpl::subscribedSpaceGroupSymbols(const PointGroup_sptr &pointGroup) {
  if (m_pointGroupMap.empty()) {
    fillPointGroupMap();
  }
 
  std::string pointGroupSymbol = pointGroup->getSymbol();
 
  std::vector<std::string> symbols;
  auto keyPair = m_pointGroupMap.equal_range(pointGroupSymbol);
 
  for (auto it = keyPair.first; it != keyPair.second; ++it) {
    symbols.emplace_back(it->second);
  }
 
  return symbols;
}
 
void SpaceGroupFactoryImpl::unsubscribeSpaceGroup(const std::string &hmSymbol) {
  if (!isSubscribed(hmSymbol)) {
    throw std::invalid_argument("Cannot unsubscribe space group that is not registered.");
  }
 
  auto eraseGenerator = m_generatorMap.find(hmSymbol);
  AbstractSpaceGroupGenerator_sptr generator = eraseGenerator->second;
 
  auto eraseNumber = m_numberMap.find(generator->getNumber());
 
  m_numberMap.erase(eraseNumber);
  m_generatorMap.erase(eraseGenerator);
}
 
void SpaceGroupFactoryImpl::subscribeGeneratedSpaceGroup(size_t number, const std::string &hmSymbol,
                                                         const std::string &generators) {
  subscribeUsingGenerator<AlgorithmicSpaceGroupGenerator>(number, hmSymbol, generators);
}
 
void SpaceGroupFactoryImpl::subscribeTabulatedSpaceGroup(size_t number, const std::string &hmSymbol,
                                                         const std::string &symmetryOperations) {
  subscribeUsingGenerator<TabulatedSpaceGroupGenerator>(number, hmSymbol, symmetryOperations);
}
 
std::string SpaceGroupFactoryImpl::getTransformedSymbolOrthorhombic(const std::string &hmSymbol,
                                                                    const std::string &transformation) const {
  // Split the symbol up in its components
  std::vector<std::string> symbolComponents;
  boost::split(symbolComponents, hmSymbol, boost::is_any_of(" "));
 
  // Get the centering, that needs to be transformed as well.
  std::string centeringLowerCase = boost::to_lower_copy(symbolComponents.front());
 
  // Remove the centering, use only the a, b, c-part of the symbol
  symbolComponents.erase(symbolComponents.begin());
 
  // Use inverse symmetry operation to permute symbol
  SymmetryOperation op = SymmetryOperation(transformation).inverse();
 
  // Symbols that need to be replaced in the permutations
  std::vector<std::string> axes{"a", "b", "c"};
  std::vector<std::string> permutedAxes = op * axes;
 
  // Permutation of the symbol, re-append centering symbol, lower case.
  std::vector<std::string> permuted = op * symbolComponents;
  permuted.insert(permuted.begin(), centeringLowerCase);
 
  // Transform those components for which it is necessary (a, b, c)
  std::transform(permuted.begin(), permuted.end(), permuted.begin(), [&](const std::string &symbol) {
    auto needsTransformation = std::find(permutedAxes.cbegin(), permutedAxes.cend(), symbol);
 
    if (needsTransformation == permutedAxes.cend()) {
      return symbol;
    }
 
    return axes[std::distance(permutedAxes.cbegin(), needsTransformation)];
  });
 
  // Transform centering string back to upper case.
  boost::to_upper(*permuted.begin());
 
  return boost::join(permuted, " ");
}
 
SpaceGroup_const_sptr SpaceGroupFactoryImpl::constructFromPrototype(const SpaceGroup_const_sptr &prototype) const {
  return std::make_shared<const SpaceGroup>(*prototype);
}
 
void SpaceGroupFactoryImpl::fillPointGroupMap() {
  m_pointGroupMap.clear();
 
  for (auto &generator : m_generatorMap) {
    SpaceGroup_const_sptr spaceGroup = getPrototype(generator.first);
 
    m_pointGroupMap.emplace(spaceGroup->getPointGroup()->getSymbol(), generator.first);
  }
}
 
SpaceGroup_const_sptr SpaceGroupFactoryImpl::getPrototype(const std::string &hmSymbol) {
  AbstractSpaceGroupGenerator_sptr generator = m_generatorMap.find(hmSymbol)->second;
 
  if (!generator) {
    throw std::runtime_error("No generator for symbol '" + hmSymbol + "'");
  }
 
  return generator->getPrototype();
}
 
void SpaceGroupFactoryImpl::subscribe(const AbstractSpaceGroupGenerator_sptr &generator) {
  if (!generator) {
    throw std::runtime_error("Cannot register null-generator.");
  }
 
  m_numberMap.emplace(generator->getNumber(), generator->getHMSymbol());
  m_generatorMap.emplace(generator->getHMSymbol(), generator);
 
  // Clear the point group map
  m_pointGroupMap.clear();
}
 
SpaceGroupFactoryImpl::SpaceGroupFactoryImpl() : m_numberMap(), m_generatorMap(), m_pointGroupMap() {
  Kernel::LibraryManager::Instance();
}
 
/* Space groups according to International Tables for Crystallography,
 * using the generators specified there.
 */
// Triclinic
DECLARE_TABULATED_SPACE_GROUP(1, "P 1", "x,y,z")
DECLARE_GENERATED_SPACE_GROUP(2, "P -1", "-x,-y,-z")
 
/* Monoclinic space groups.
 *
 * Unique axes b and c are given, as well as 3 cell choices where
 * applicable. Since there are only so few monoclinic space group
 * types, all the transformations are given explicitly. It would
 * actually be shorter to provide all group definition explicitly
 * with their generators, but this way the relationship between
 * the groups is more pronounced.
 */
DECLARE_GENERATED_SPACE_GROUP(3, "P 1 2 1", "-x,y,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(3, "P 1 1 2", "P 1 2 1 | y,z,x")
 
DECLARE_GENERATED_SPACE_GROUP(4, "P 1 21 1", "-x,y+1/2,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(4, "P 1 1 21", "P 1 21 1 | y,z,x")
 
DECLARE_GENERATED_SPACE_GROUP(5, "C 1 2 1", "-x,y,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(5, "A 1 2 1", "C 1 2 1 | -x+z,y,-x")
DECLARE_TRANSFORMED_SPACE_GROUP(5, "I 1 2 1", "C 1 2 1 | -z,y,x-z")
DECLARE_TRANSFORMED_SPACE_GROUP(5, "A 1 1 2", "C 1 2 1 | y,z,x")
DECLARE_TRANSFORMED_SPACE_GROUP(5, "B 1 1 2", "A 1 1 2 | -y,x-y,z")
DECLARE_TRANSFORMED_SPACE_GROUP(5, "I 1 1 2", "A 1 1 2 | -x+y,-x,z")
 
DECLARE_GENERATED_SPACE_GROUP(6, "P 1 m 1", "x,-y,z")
DECLARE_TRANSFORMED_SPACE_GROUP(6, "P 1 1 m", "P 1 m 1 | y,z,x")
 
DECLARE_GENERATED_SPACE_GROUP(7, "P 1 c 1", "x,-y,z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(7, "P 1 n 1", "P 1 c 1 | -x+z,y,-x")
DECLARE_TRANSFORMED_SPACE_GROUP(7, "P 1 a 1", "P 1 c 1 | -z,y,x-z")
DECLARE_TRANSFORMED_SPACE_GROUP(7, "P 1 1 a", "P 1 c 1 | y,z,x")
DECLARE_TRANSFORMED_SPACE_GROUP(7, "P 1 1 n", "P 1 1 a | -y,x-y,z")
DECLARE_TRANSFORMED_SPACE_GROUP(7, "P 1 1 b", "P 1 1 a | -x+y,-x,z")
 
DECLARE_GENERATED_SPACE_GROUP(8, "C 1 m 1", "x,-y,z")
DECLARE_TRANSFORMED_SPACE_GROUP(8, "A 1 m 1", "C 1 m 1 | -x+z,y,-x")
DECLARE_TRANSFORMED_SPACE_GROUP(8, "I 1 m 1", "C 1 m 1 | -z,y,x-z")
DECLARE_TRANSFORMED_SPACE_GROUP(8, "A 1 1 m", "C 1 m 1 | y,z,x")
DECLARE_TRANSFORMED_SPACE_GROUP(8, "B 1 1 m", "A 1 1 m | -y,x-y,z")
DECLARE_TRANSFORMED_SPACE_GROUP(8, "I 1 1 m", "A 1 1 m | -x+y,-x,z")
 
DECLARE_GENERATED_SPACE_GROUP(9, "C 1 c 1", "x,-y,z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(9, "A 1 n 1", "C 1 c 1 | -x+z,y,-x")
DECLARE_TRANSFORMED_SPACE_GROUP(9, "I 1 a 1", "C 1 c 1 | -z,y,x-z")
DECLARE_TRANSFORMED_SPACE_GROUP(9, "A 1 1 a", "C 1 c 1 | y,z,x")
DECLARE_TRANSFORMED_SPACE_GROUP(9, "B 1 1 n", "A 1 1 a | -y,x-y,z")
DECLARE_TRANSFORMED_SPACE_GROUP(9, "I 1 1 b", "A 1 1 a | -x+y,-x,z")
 
DECLARE_GENERATED_SPACE_GROUP(10, "P 1 2/m 1", "-x,y,-z; -x,-y,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(10, "P 1 1 2/m", "P 1 2/m 1 | y,z,x")
 
DECLARE_GENERATED_SPACE_GROUP(11, "P 1 21/m 1", "-x,y+1/2,-z; -x,-y,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(11, "P 1 1 21/m", "P 1 21/m 1 | y,z,x")
 
DECLARE_GENERATED_SPACE_GROUP(12, "C 1 2/m 1", "-x,y,-z; -x,-y,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(12, "A 1 2/m 1", "C 1 2/m 1 | -x+z,y,-x")
DECLARE_TRANSFORMED_SPACE_GROUP(12, "I 1 2/m 1", "C 1 2/m 1 | -z,y,x-z")
DECLARE_TRANSFORMED_SPACE_GROUP(12, "A 1 1 2/m", "C 1 2/m 1 | y,z,x")
DECLARE_TRANSFORMED_SPACE_GROUP(12, "B 1 1 2/m", "A 1 1 2/m | -y,x-y,z")
DECLARE_TRANSFORMED_SPACE_GROUP(12, "I 1 1 2/m", "A 1 1 2/m | -x+y,-x,z")
 
DECLARE_GENERATED_SPACE_GROUP(13, "P 1 2/c 1", "-x,y,-z+1/2; -x,-y,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(13, "P 1 2/n 1", "P 1 2/c 1 | -x+z,y,-x")
DECLARE_TRANSFORMED_SPACE_GROUP(13, "P 1 2/a 1", "P 1 2/c 1 | -z,y,x-z")
DECLARE_TRANSFORMED_SPACE_GROUP(13, "P 1 1 2/a", "P 1 2/c 1 | y,z,x")
DECLARE_TRANSFORMED_SPACE_GROUP(13, "P 1 1 2/n", "P 1 1 2/a | -y,x-y,z")
DECLARE_TRANSFORMED_SPACE_GROUP(13, "P 1 1 2/b", "P 1 1 2/a | -x+y,-x,z")
 
DECLARE_GENERATED_SPACE_GROUP(14, "P 1 21/c 1", "-x,y+1/2,-z+1/2; -x,-y,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(14, "P 1 21/n 1", "P 1 21/c 1 | -x+z,y,-x")
DECLARE_TRANSFORMED_SPACE_GROUP(14, "P 1 21/a 1", "P 1 21/c 1 | -z,y,x-z")
DECLARE_TRANSFORMED_SPACE_GROUP(14, "P 1 1 21/a", "P 1 21/c 1 | y,z,x")
DECLARE_TRANSFORMED_SPACE_GROUP(14, "P 1 1 21/n", "P 1 1 21/a | -y,x-y,z")
DECLARE_TRANSFORMED_SPACE_GROUP(14, "P 1 1 21/b", "P 1 1 21/a | -x+y,-x,z")
 
DECLARE_GENERATED_SPACE_GROUP(15, "C 1 2/c 1", "-x,y,-z+1/2; -x,-y,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(15, "A 1 2/n 1", "C 1 2/c 1 | -x+z,y,-x")
DECLARE_TRANSFORMED_SPACE_GROUP(15, "I 1 2/a 1", "C 1 2/c 1 | -z,y,x-z")
DECLARE_TRANSFORMED_SPACE_GROUP(15, "A 1 1 2/a", "C 1 2/c 1 | y,z,x")
DECLARE_TRANSFORMED_SPACE_GROUP(15, "B 1 1 2/n", "A 1 1 2/a | -y,x-y,z")
DECLARE_TRANSFORMED_SPACE_GROUP(15, "I 1 1 2/b", "A 1 1 2/a | -x+y,-x,z")
 
/* Orthorhombic space groups have a special subscribe-method in the
 * factory, because each group has potentially 6 different settings.
 * In addition some groups have more than one origin choice.
 */
DECLARE_ORTHORHOMBIC_SPACE_GROUP(16, "P 2 2 2", "-x,y,-z; x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(17, "P 2 2 21", "-x,-y,z+1/2; -x,y,-z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(18, "P 21 21 2", "-x,-y,z; -x+1/2,y+1/2,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(19, "P 21 21 21", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(20, "C 2 2 21", "-x,-y,z+1/2; -x,y,-z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(21, "C 2 2 2", "-x,-y,z; -x,y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(22, "F 2 2 2", "-x,-y,z; -x,y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(23, "I 2 2 2", "-x,-y,z; -x,y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(24, "I 21 21 21", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(25, "P m m 2", "-x,-y,z; x,-y,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(26, "P m c 21", "-x,-y,z+1/2; x,-y,z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(27, "P c c 2", "-x,-y,z; x,-y,z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(28, "P m a 2", "-x,-y,z; x+1/2,-y,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(29, "P c a 21", "-x,-y,z+1/2; x+1/2,-y,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(30, "P n c 2", "-x,-y,z; x,-y+1/2,z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(31, "P m n 21", "-x+1/2,-y,z+1/2; x+1/2,-y,z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(32, "P b a 2", "-x,-y,z; x+1/2,-y+1/2,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(33, "P n a 21", "-x,-y,z+1/2; x+1/2,-y+1/2,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(34, "P n n 2", "-x,-y,z; x+1/2,-y+1/2,z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(35, "C m m 2", "-x,-y,z; x,-y,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(36, "C m c 21", "-x,-y,z+1/2; x,-y,z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(37, "C c c 2", "-x,-y,z; x,-y,z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(38, "A m m 2", "-x,-y,z; x,-y,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(39, "A e m 2", "-x,-y,z; x,-y,z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(40, "A m a 2", "-x,-y,z; x+1/2,-y,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(41, "A e a 2", "-x,-y,z; x+1/2,-y+1/2,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(42, "F m m 2", "-x,-y,z; x,-y,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(43, "F d d 2", "-x,-y,z; x+1/4,-y+1/4,z+1/4")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(44, "I m m 2", "-x,-y,z; x,-y,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(45, "I b a 2", "-x,-y,z; x+1/2,-y+1/2,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(46, "I m a 2", "-x,-y,z; x+1/2,-y,z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(47, "P m m m", "-x,-y,z; -x,y,-z; -x,-y,-z")
 
DECLARE_ORTHORHOMBIC_SPACE_GROUP(48, "P n n n", "-x,-y,z; -x,y,-z; -x+1/2,-y+1/2,-z+1/2")
DECLARE_TRANSFORMED_ORTHORHOMBIC_SPACE_GROUP(48, "P n n n :2", "P n n n | x-1/4,y-1/4,z-1/4")
 
DECLARE_ORTHORHOMBIC_SPACE_GROUP(49, "P c c m", "-x,-y,z; -x,y,-z+1/2; -x,-y,-z")
 
DECLARE_ORTHORHOMBIC_SPACE_GROUP(50, "P b a n", "-x,-y,z; -x,y,-z; -x+1/2,-y+1/2,-z")
DECLARE_TRANSFORMED_ORTHORHOMBIC_SPACE_GROUP(50, "P b a n :2", "P b a n | x-1/4,y-1/4,z")
 
DECLARE_ORTHORHOMBIC_SPACE_GROUP(51, "P m m a", "-x+1/2,-y,z; -x,y,-z; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(52, "P n n a", "-x+1/2,-y,z; -x+1/2,y+1/2,-z+1/2; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(53, "P m n a", "-x+1/2,-y,z+1/2; -x+1/2,y,-z+1/2; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(54, "P c c a", "-x+1/2,-y,z; -x,y,-z+1/2; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(55, "P b a m", "-x,-y,z; -x+1/2,y+1/2,-z; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(56, "P c c n", "-x+1/2,-y+1/2,z; -x,y+1/2,-z+1/2; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(57, "P b c m", "-x,-y,z+1/2; -x,y+1/2,-z+1/2; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(58, "P n n m", "-x,-y,z; -x+1/2,y+1/2,-z+1/2; -x,-y,-z")
 
DECLARE_ORTHORHOMBIC_SPACE_GROUP(59, "P m m n", "-x,-y,z; -x+1/2,y+1/2,-z; -x+1/2,-y+1/2,-z")
DECLARE_TRANSFORMED_ORTHORHOMBIC_SPACE_GROUP(59, "P m m n :2", "P m m n | x-1/4,y-1/4,z")
 
DECLARE_ORTHORHOMBIC_SPACE_GROUP(60, "P b c n", "-x+1/2,-y+1/2,z+1/2; -x,y,-z+1/2; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(61, "P b c a", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(62, "P n m a", "-x+1/2,-y,z+1/2; -x,y+1/2,-z; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(63, "C m c m", "-x,-y,z+1/2; -x,y,-z+1/2; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(64, "C m c e", "-x,-y+1/2,z+1/2; -x,y+1/2,-z+1/2; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(65, "C m m m", "-x,-y,z; -x,y,-z; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(66, "C c c m", "-x,-y,z; -x,y,-z+1/2; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(67, "C m m e", "-x,-y+1/2,z; -x,y+1/2,-z; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(68, "C c c e", "-x+1/2,-y+1/2,z; -x,y,-z; -x,-y+1/2,-z+1/2")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(69, "F m m m", "-x,-y,z; -x,y,-z; -x,-y,-z")
 
DECLARE_ORTHORHOMBIC_SPACE_GROUP(70, "F d d d", "-x,-y,z; -x,y,-z; -x+1/4,-y+1/4,-z+1/4")
DECLARE_TRANSFORMED_ORTHORHOMBIC_SPACE_GROUP(70, "F d d d :2", "F d d d | x+1/8,y+1/8,z+1/8")
 
DECLARE_ORTHORHOMBIC_SPACE_GROUP(71, "I m m m", "-x,-y,z; -x,y,-z; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(72, "I b a m", "-x,-y,z; -x+1/2,y+1/2,-z; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(73, "I b c a", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2; -x,-y,-z")
DECLARE_ORTHORHOMBIC_SPACE_GROUP(74, "I m m a", "-x,-y+1/2,z; -x,y+1/2,-z; -x,-y,-z")
 
// Tetragonal
DECLARE_GENERATED_SPACE_GROUP(75, "P 4", "-x,-y,z; -y,x,z")
DECLARE_GENERATED_SPACE_GROUP(76, "P 41", "-x,-y,z+1/2; -y,x,z+1/4")
DECLARE_GENERATED_SPACE_GROUP(77, "P 42", "-x,-y,z; -y,x,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(78, "P 43", "-x,-y,z+1/2; -y,x,z+3/4")
DECLARE_GENERATED_SPACE_GROUP(79, "I 4", "-x,-y,z; -y,x,z")
DECLARE_GENERATED_SPACE_GROUP(80, "I 41", "-x+1/2,-y+1/2,z+1/2; -y,x+1/2,z+1/4")
DECLARE_GENERATED_SPACE_GROUP(81, "P -4", "-x,-y,z; y,-x,-z")
DECLARE_GENERATED_SPACE_GROUP(82, "I -4", "-x,-y,z; y,-x,-z")
DECLARE_GENERATED_SPACE_GROUP(83, "P 4/m", "-x,-y,z; -y,x,z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(84, "P 42/m", "-x,-y,z; -y,x,z+1/2; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(85, "P 4/n", "-x,-y,z; -y+1/2,x+1/2,z; -x+1/2,-y+1/2,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(85, "P 4/n :2", "P 4/n | x+1/4,y-1/4,z")
 
DECLARE_GENERATED_SPACE_GROUP(86, "P 42/n", "-x,-y,z; -y+1/2,x+1/2,z+1/2; -x+1/2,-y+1/2,-z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(86, "P 42/n :2", "P 42/n | x+1/4,y+1/4,z+1/4")
 
DECLARE_GENERATED_SPACE_GROUP(87, "I 4/m", "-x,-y,z; -y,x,z; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(88, "I 41/a", "-x+1/2,-y+1/2,z+1/2; -y,x+1/2,z+1/4; -x,-y+1/2,-z+1/4")
DECLARE_TRANSFORMED_SPACE_GROUP(88, "I 41/a :2", "I 41/a | x,y+1/4,z+1/8")
 
DECLARE_GENERATED_SPACE_GROUP(89, "P 4 2 2", "-x,-y,z; -y,x,z; -x,y,-z")
DECLARE_GENERATED_SPACE_GROUP(90, "P 4 21 2", "-x,-y,z; -y+1/2,x+1/2,z; -x+1/2,y+1/2,-z")
DECLARE_GENERATED_SPACE_GROUP(91, "P 41 2 2", "-x,-y,z+1/2; -y,x,z+1/4; -x,y,-z")
DECLARE_GENERATED_SPACE_GROUP(92, "P 41 21 2", "-x,-y,z+1/2; -y+1/2,x+1/2,z+1/4; -x+1/2,y+1/2,-z+1/4")
DECLARE_GENERATED_SPACE_GROUP(93, "P 42 2 2", "-x,-y,z; -y,x,z+1/2; -x,y,-z")
DECLARE_GENERATED_SPACE_GROUP(94, "P 42 21 2", "-x,-y,z; -y+1/2,x+1/2,z+1/2; -x+1/2,y+1/2,-z+1/2")
DECLARE_GENERATED_SPACE_GROUP(95, "P 43 2 2", "-x,-y,z+1/2; -y,x,z+3/4; -x,y,-z")
DECLARE_GENERATED_SPACE_GROUP(96, "P 43 21 2", "-x,-y,z+1/2; -y+1/2,x+1/2,z+3/4; -x+1/2,y+1/2,-z+3/4")
DECLARE_GENERATED_SPACE_GROUP(97, "I 4 2 2", "-x,-y,z; -y,x,z; -x,y,-z")
DECLARE_GENERATED_SPACE_GROUP(98, "I 41 2 2", "-x+1/2,-y+1/2,z+1/2; -y,x+1/2,z+1/4; -x+1/2,y,-z+3/4")
DECLARE_GENERATED_SPACE_GROUP(99, "P 4 m m", "-x,-y,z; -y,x,z; x,-y,z")
DECLARE_GENERATED_SPACE_GROUP(100, "P 4 b m", "-x,-y,z; -y,x,z; x+1/2,-y+1/2,z")
DECLARE_GENERATED_SPACE_GROUP(101, "P 42 c m", "-x,-y,z; -y,x,z+1/2; x,-y,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(102, "P 42 n m", "-x,-y,z; -y+1/2,x+1/2,z+1/2; x+1/2,-y+1/2,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(103, "P 4 c c", "-x,-y,z; -y,x,z; x,-y,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(104, "P 4 n c", "-x,-y,z; -y,x,z; x+1/2,-y+1/2,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(105, "P 42 m c", "-x,-y,z; -y,x,z+1/2; x,-y,z")
DECLARE_GENERATED_SPACE_GROUP(106, "P 42 b c", "-x,-y,z; -y,x,z+1/2; x+1/2,-y+1/2,z")
DECLARE_GENERATED_SPACE_GROUP(107, "I 4 m m", "-x,-y,z; -y,x,z; x,-y,z")
DECLARE_GENERATED_SPACE_GROUP(108, "I 4 c m", "-x,-y,z; -y,x,z; x,-y,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(109, "I 41 m d", "-x+1/2,-y+1/2,z+1/2; -y,x+1/2,z+1/4; x,-y,z")
DECLARE_GENERATED_SPACE_GROUP(110, "I 41 c d", "-x+1/2,-y+1/2,z+1/2; -y,x+1/2,z+1/4; x,-y,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(111, "P -4 2 m", "-x,-y,z; y,-x,-z; -x,y,-z")
DECLARE_GENERATED_SPACE_GROUP(112, "P -4 2 c", "-x,-y,z; y,-x,-z; -x,y,-z+1/2")
DECLARE_GENERATED_SPACE_GROUP(113, "P -4 21 m", "-x,-y,z; y,-x,-z; -x+1/2,y+1/2,-z")
DECLARE_GENERATED_SPACE_GROUP(114, "P -4 21 c", "-x,-y,z; y,-x,-z; -x+1/2,y+1/2,-z+1/2")
DECLARE_GENERATED_SPACE_GROUP(115, "P -4 m 2", "-x,-y,z; y,-x,-z; x,-y,z")
DECLARE_GENERATED_SPACE_GROUP(116, "P -4 c 2", "-x,-y,z; y,-x,-z; x,-y,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(117, "P -4 b 2", "-x,-y,z; y,-x,-z; x+1/2,-y+1/2,z")
DECLARE_GENERATED_SPACE_GROUP(118, "P -4 n 2", "-x,-y,z; y,-x,-z; x+1/2,-y+1/2,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(119, "I -4 m 2", "-x,-y,z; y,-x,-z; x,-y,z")
DECLARE_GENERATED_SPACE_GROUP(120, "I -4 c 2", "-x,-y,z; y,-x,-z; x,-y,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(121, "I -4 2 m", "-x,-y,z; y,-x,-z; -x,y,-z")
DECLARE_GENERATED_SPACE_GROUP(122, "I -4 2 d", "-x,-y,z; y,-x,-z; -x+1/2,y,-z+3/4")
DECLARE_GENERATED_SPACE_GROUP(123, "P 4/m m m", "-x,-y,z; -y,x,z; -x,y,-z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(124, "P 4/m c c", "-x,-y,z; -y,x,z; -x,y,-z+1/2; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(125, "P 4/n b m", "-x,-y,z; -y,x,z; -x,y,-z; -x+1/2,-y+1/2,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(125, "P 4/n b m :2", "P 4/n b m | x+1/4,y+1/4,z")
 
DECLARE_GENERATED_SPACE_GROUP(126, "P 4/n n c", "-x,-y,z; -y,x,z; -x,y,-z; -x+1/2,-y+1/2,-z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(126, "P 4/n n c :2", "P 4/n n c | x+1/4,y+1/4,z+1/4")
 
DECLARE_GENERATED_SPACE_GROUP(127, "P 4/m b m", "-x,-y,z; -y,x,z; -x+1/2,y+1/2,-z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(128, "P 4/m n c", "-x,-y,z; -y,x,z; -x+1/2,y+1/2,-z+1/2; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(129, "P 4/n m m", "-x,-y,z; -y+1/2,x+1/2,z; -x+1/2,y+1/2,-z; -x+1/2,-y+1/2,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(129, "P 4/n m m :2", "P 4/n m m | x+1/4,y-1/4,z")
 
DECLARE_GENERATED_SPACE_GROUP(130, "P 4/n c c", "-x,-y,z; -y+1/2,x+1/2,z; -x+1/2,y+1/2,-z+1/2; -x+1/2,-y+1/2,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(130, "P 4/n c c :2", "P 4/n c c | x+1/4,y-1/4,z")
 
DECLARE_GENERATED_SPACE_GROUP(131, "P 42/m m c", "-x,-y,z; -y,x,z+1/2; -x,y,-z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(132, "P 42/m c m", "-x,-y,z; -y,x,z+1/2; -x,y,-z+1/2; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(133, "P 42/n b c", "-x,-y,z; -y+1/2,x+1/2,z+1/2; -x,y,-z+1/2; -x+1/2,-y+1/2,-z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(133, "P 42/n b c :2", "P 42/n b c | x+1/4,y-1/4,z+1/4")
 
DECLARE_GENERATED_SPACE_GROUP(134, "P 42/n n m", "-x,-y,z; -y+1/2,x+1/2,z+1/2; -x,y,-z; -x+1/2,-y+1/2,-z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(134, "P 42/n n m :2", "P 42/n n m | x+1/4,y-1/4,z+1/4")
 
DECLARE_GENERATED_SPACE_GROUP(135, "P 42/m b c", "-x,-y,z; -y,x,z+1/2; -x+1/2,y+1/2,-z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(136, "P 42/m n m", "-x,-y,z; -y+1/2,x+1/2,z+1/2; -x+1/2,y+1/2,-z+1/2; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(137, "P 42/n m c",
                              "-x,-y,z; -y+1/2,x+1/2,z+1/2; -x+1/2,y+1/2,-z+1/2; -x+1/2,-y+1/2,-z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(137, "P 42/n m c :2", "P 42/n m c | x+1/4,y-1/4,z+1/4")
 
DECLARE_GENERATED_SPACE_GROUP(138, "P 42/n c m", "-x,-y,z; -y+1/2,x+1/2,z+1/2; -x+1/2,y+1/2,-z; -x+1/2,-y+1/2,-z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(138, "P 42/n c m :2", "P 42/n c m | x+1/4,y-1/4,z+1/4")
 
DECLARE_GENERATED_SPACE_GROUP(139, "I 4/m m m", "-x,-y,z; -y,x,z; -x,y,-z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(140, "I 4/m c m", "-x,-y,z; -y,x,z; -x,y,-z+1/2; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(141, "I 41/a m d",
                              "-x+1/2,-y+1/2,z+1/2; -y,x+1/2,z+1/4; -x+1/2,y,-z+3/4; -x,-y+1/2,-z+1/4")
DECLARE_TRANSFORMED_SPACE_GROUP(141, "I 41/a m d :2", "I 41/a m d | x,y-1/4,z+1/8")
 
DECLARE_GENERATED_SPACE_GROUP(142, "I 41/a c d",
                              "-x+1/2,-y+1/2,z+1/2; -y,x+1/2,z+1/4; -x+1/2,y,-z+1/4; -x,-y+1/2,-z+1/4")
DECLARE_TRANSFORMED_SPACE_GROUP(142, "I 41/a c d :2", "I 41/a c d | x,y-1/4,z+1/8")
 
// Trigonal
DECLARE_GENERATED_SPACE_GROUP(143, "P 3", "-y,x-y,z")
DECLARE_GENERATED_SPACE_GROUP(144, "P 31", "-y,x-y,z+1/3")
DECLARE_GENERATED_SPACE_GROUP(145, "P 32", "-y,x-y,z+2/3")
 
DECLARE_GENERATED_SPACE_GROUP(146, "R 3", "-y,x-y,z")
DECLARE_TRANSFORMED_SPACE_GROUP(146, "R 3 :r", "R 3 | 2/3x-1/3y-1/3z, 1/3x+1/3y-2/3z, 1/3x+1/3y+1/3z")
 
DECLARE_GENERATED_SPACE_GROUP(147, "P -3", "-y,x-y,z; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(148, "R -3", "-y,x-y,z; -x,-y,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(148, "R -3 :r", "R -3 | 2/3x-1/3y-1/3z, 1/3x+1/3y-2/3z, 1/3x+1/3y+1/3z")
 
DECLARE_GENERATED_SPACE_GROUP(149, "P 3 1 2", "-y,x-y,z; -y,-x,-z")
DECLARE_GENERATED_SPACE_GROUP(150, "P 3 2 1", "-y,x-y,z; y,x,-z")
DECLARE_GENERATED_SPACE_GROUP(151, "P 31 1 2", "-y,x-y,z+1/3; -y,-x,-z+2/3")
DECLARE_GENERATED_SPACE_GROUP(152, "P 31 2 1", "-y,x-y,z+1/3; y,x,-z")
DECLARE_GENERATED_SPACE_GROUP(153, "P 32 1 2", "-y,x-y,z+2/3; -y,-x,-z+1/3")
DECLARE_GENERATED_SPACE_GROUP(154, "P 32 2 1", "-y,x-y,z+2/3; y,x,-z")
 
DECLARE_GENERATED_SPACE_GROUP(155, "R 32", "-y,x-y,z; y,x,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(155, "R 32 :r", "R 32 | 2/3x-1/3y-1/3z, 1/3x+1/3y-2/3z, 1/3x+1/3y+1/3z")
 
DECLARE_GENERATED_SPACE_GROUP(156, "P 3 m 1", "-y,x-y,z; -y,-x,z")
DECLARE_GENERATED_SPACE_GROUP(157, "P 3 1 m", "-y,x-y,z; y,x,z")
DECLARE_GENERATED_SPACE_GROUP(158, "P 3 c 1", "-y,x-y,z; -y,-x,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(159, "P 3 1 c", "-y,x-y,z; y,x,z+1/2")
 
DECLARE_GENERATED_SPACE_GROUP(160, "R 3 m", "-y,x-y,z; -y,-x,z")
DECLARE_TRANSFORMED_SPACE_GROUP(160, "R 3 m :r", "R 3 m | 2/3x-1/3y-1/3z, 1/3x+1/3y-2/3z, 1/3x+1/3y+1/3z")
 
DECLARE_GENERATED_SPACE_GROUP(161, "R 3 c", "-y,x-y,z; -y,-x,z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(161, "R 3 c :r", "R 3 c | 2/3x-1/3y-1/3z, 1/3x+1/3y-2/3z, 1/3x+1/3y+1/3z")
 
DECLARE_GENERATED_SPACE_GROUP(162, "P -3 1 m", "-y,x-y,z; -y,-x,-z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(163, "P -3 1 c", "-y,x-y,z; -y,-x,-z+1/2; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(164, "P -3 m 1", "-y,x-y,z; y,x,-z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(165, "P -3 c 1", "-y,x-y,z; y,x,-z+1/2; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(166, "R -3 m", "-y,x-y,z; y,x,-z; -x,-y,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(166, "R -3 m :r", "R -3 m | 2/3x-1/3y-1/3z, 1/3x+1/3y-2/3z, 1/3x+1/3y+1/3z")
 
DECLARE_GENERATED_SPACE_GROUP(167, "R -3 c", "-y,x-y,z; y,x,-z+1/2; -x,-y,-z")
DECLARE_TRANSFORMED_SPACE_GROUP(167, "R -3 c :r", "R -3 c | 2/3x-1/3y-1/3z, 1/3x+1/3y-2/3z, 1/3x+1/3y+1/3z")
 
// Hexagonal
DECLARE_GENERATED_SPACE_GROUP(168, "P 6", "-y,x-y,z; -x,-y,z")
DECLARE_GENERATED_SPACE_GROUP(169, "P 61", "-y,x-y,z+1/3; -x,-y,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(170, "P 65", "-y,x-y,z+2/3; -x,-y,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(171, "P 62", "-y,x-y,z+2/3; -x,-y,z")
DECLARE_GENERATED_SPACE_GROUP(172, "P 64", "-y,x-y,z+1/3; -x,-y,z")
DECLARE_GENERATED_SPACE_GROUP(173, "P 63", "-y,x-y,z; -x,-y,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(174, "P -6", "-y,x-y,z; x,y,-z")
DECLARE_GENERATED_SPACE_GROUP(175, "P 6/m", "-y,x-y,z; -x,-y,z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(176, "P 63/m", "-y,x-y,z; -x,-y,z+1/2; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(177, "P 6 2 2", "-y,x-y,z; -x,-y,z; y,x,-z")
DECLARE_GENERATED_SPACE_GROUP(178, "P 61 2 2", "-y,x-y,z+1/3; -x,-y,z+1/2; y,x,-z+1/3")
DECLARE_GENERATED_SPACE_GROUP(179, "P 65 2 2", "-y,x-y,z+2/3; -x,-y,z+1/2; y,x,-z+2/3")
DECLARE_GENERATED_SPACE_GROUP(180, "P 62 2 2", "-y,x-y,z+2/3; -x,-y,z; y,x,-z+2/3")
DECLARE_GENERATED_SPACE_GROUP(181, "P 64 2 2", "-y,x-y,z+1/3; -x,-y,z; y,x,-z+1/3")
DECLARE_GENERATED_SPACE_GROUP(182, "P 63 2 2", "-y,x-y,z; -x,-y,z+1/2; y,x,-z")
DECLARE_GENERATED_SPACE_GROUP(183, "P 6 m m", "-y,x-y,z; -x,-y,z; -y,-x,z")
DECLARE_GENERATED_SPACE_GROUP(184, "P 6 c c", "-y,x-y,z; -x,-y,z; -y,-x,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(185, "P 63 c m", "-y,x-y,z; -x,-y,z+1/2; -y,-x,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(186, "P 63 m c", "-y,x-y,z; -x,-y,z+1/2; -y,-x,z")
DECLARE_GENERATED_SPACE_GROUP(187, "P -6 m 2", "-y,x-y,z; x,y,-z; -y,-x,z")
DECLARE_GENERATED_SPACE_GROUP(188, "P -6 c 2", "-y,x-y,z; x,y,-z+1/2; -y,-x,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(189, "P -6 2 m", "-y,x-y,z; x,y,-z; y,x,-z")
DECLARE_GENERATED_SPACE_GROUP(190, "P -6 2 c", "-y,x-y,z; x,y,-z+1/2; y,x,-z")
DECLARE_GENERATED_SPACE_GROUP(191, "P 6/m m m", "-y,x-y,z; x,y,-z; y,x,-z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(192, "P 6/m c c", "-y,x-y,z; -x,-y,z; y,x,-z+1/2; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(193, "P 63/m c m", "-y,x-y,z; -x,-y,z+1/2; y,x,-z+1/2; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(194, "P 63/m m c", "-y,x-y,z; -x,-y,z+1/2; y,x,-z; -x,-y,-z")
 
// Cubic
DECLARE_GENERATED_SPACE_GROUP(195, "P 2 3", "-x,-y,z; -x,y,-z; z,x,y")
DECLARE_GENERATED_SPACE_GROUP(196, "F 2 3", "-x,-y,z; -x,y,-z; z,x,y")
DECLARE_GENERATED_SPACE_GROUP(197, "I 2 3", "-x,-y,z; -x,y,-z; z,x,y")
DECLARE_GENERATED_SPACE_GROUP(198, "P 21 3", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2; z,x,y")
DECLARE_GENERATED_SPACE_GROUP(199, "I 21 3", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2; z,x,y")
DECLARE_GENERATED_SPACE_GROUP(200, "P m -3", "-x,-y,z; -x,y,-z; z,x,y; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(201, "P n -3", "-x,-y,z; -x,y,-z; z,x,y; -x+1/2,-y+1/2,-z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(201, "P n -3 :2", "P n -3 | x+1/4,y+1/4,z+1/4")
 
DECLARE_GENERATED_SPACE_GROUP(202, "F m -3", "-x,-y,z; -x,y,-z; z,x,y; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(203, "F d -3", "-x,-y,z; -x,y,-z; z,x,y; -x+1/4,-y+1/4,-z+1/4")
DECLARE_TRANSFORMED_SPACE_GROUP(203, "F d -3 :2", "F d -3 | x+1/8,y+1/8,z+1/8")
 
DECLARE_GENERATED_SPACE_GROUP(204, "I m -3", "-x,-y,z; -x,y,-z; z,x,y; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(205, "P a -3", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2; z,x,y; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(206, "I a -3", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2; z,x,y; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(207, "P 4 3 2", "-x,-y,z; -x,y,-z; z,x,y; y,x,-z")
DECLARE_GENERATED_SPACE_GROUP(208, "P 42 3 2", "-x,-y,z; -x,y,-z; z,x,y; y+1/2,x+1/2,-z+1/2")
DECLARE_GENERATED_SPACE_GROUP(209, "F 4 3 2", "-x,-y,z; -x,y,-z; z,x,y; y,x,-z")
 
DECLARE_GENERATED_SPACE_GROUP(210, "F 41 3 2", "-x,-y+1/2,z+1/2; -x+1/2,y+1/2,-z; z,x,y; y+3/4,x+1/4,-z+3/4")
DECLARE_GENERATED_SPACE_GROUP(211, "I 4 3 2", "-x,-y,z; -x,y,-z; z,x,y; y,x,-z")
DECLARE_GENERATED_SPACE_GROUP(212, "P 43 3 2", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2; z,x,y; y+1/4,x+3/4,-z+3/4")
DECLARE_GENERATED_SPACE_GROUP(213, "P 41 3 2", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2; z,x,y; y+3/4,x+1/4,-z+1/4")
DECLARE_GENERATED_SPACE_GROUP(214, "I 41 3 2", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2; z,x,y; y+3/4,x+1/4,-z+1/4")
DECLARE_GENERATED_SPACE_GROUP(215, "P -4 3 m", "-x,-y,z; -x,y,-z; z,x,y; y,x,z")
DECLARE_GENERATED_SPACE_GROUP(216, "F -4 3 m", "-x,-y,z; -x,y,-z; z,x,y; y,x,z")
DECLARE_GENERATED_SPACE_GROUP(217, "I -4 3 m", "-x,-y,z; -x,y,-z; z,x,y; y,x,z")
DECLARE_GENERATED_SPACE_GROUP(218, "P -4 3 n", "-x,-y,z; -x,y,-z; z,x,y; y+1/2,x+1/2,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(219, "F -4 3 c", "-x,-y,z; -x,y,-z; z,x,y; y+1/2,x+1/2,z+1/2")
DECLARE_GENERATED_SPACE_GROUP(220, "I -4 3 d", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2; z,x,y; y+1/4,x+1/4,z+1/4")
DECLARE_GENERATED_SPACE_GROUP(221, "P m -3 m", "-x,-y,z; -x,y,-z; z,x,y; y,x,-z; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(222, "P n -3 n", "-x,-y,z; -x,y,-z; z,x,y; y,x,-z; -x+1/2,-y+1/2,-z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(222, "P n -3 n :2", "P n -3 n | x+1/4,y+1/4,z+1/4")
 
DECLARE_GENERATED_SPACE_GROUP(223, "P m -3 n", "-x,-y,z; -x,y,-z; z,x,y; y+1/2,x+1/2,-z+1/2; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(224, "P n -3 m", "-x,-y,z; -x,y,-z; z,x,y; y+1/2,x+1/2,-z+1/2; -x+1/2,-y+1/2,-z+1/2")
DECLARE_TRANSFORMED_SPACE_GROUP(224, "P n -3 m :2", "P n -3 m | x+1/4,y+1/4,z+1/4")
 
DECLARE_GENERATED_SPACE_GROUP(225, "F m -3 m", "-x,-y,z; -x,y,-z; z,x,y; y,x,-z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(226, "F m -3 c", "-x,-y,z; -x,y,-z; z,x,y; y+1/2,x+1/2,-z+1/2; -x,-y,-z")
 
DECLARE_GENERATED_SPACE_GROUP(227, "F d -3 m",
                              "-x,-y+1/2,z+1/2; -x+1/2,y+1/2,-z; z,x,y; "
                              "y+3/4,x+1/4,-z+3/4; -x+1/4,-y+1/4,-z+1/4")
DECLARE_TRANSFORMED_SPACE_GROUP(227, "F d -3 m :2", "F d -3 m | x+1/8,y+1/8,z+1/8")
 
DECLARE_GENERATED_SPACE_GROUP(228, "F d -3 c",
                              "-x,-y+1/2,z+1/2; -x+1/2,y+1/2,-z; z,x,y; "
                              "y+3/4,x+1/4,-z+3/4; -x+3/4,-y+3/4,-z+3/4")
DECLARE_TRANSFORMED_SPACE_GROUP(228, "F d -3 c :2", "F d -3 c | x+3/8,y+3/8,z+3/8")
 
DECLARE_GENERATED_SPACE_GROUP(229, "I m -3 m", "-x,-y,z; -x,y,-z; z,x,y; y,x,-z; -x,-y,-z")
DECLARE_GENERATED_SPACE_GROUP(230, "I a -3 d", "-x+1/2,-y,z+1/2; -x,y+1/2,-z+1/2; z,x,y; y+3/4,x+1/4,-z+1/4; -x,-y,-z")
 
} // namespace Mantid::Geometry