diff --git a/cmake/dca_config.cmake b/cmake/dca_config.cmake
index 976f24253..1d660cead 100644
--- a/cmake/dca_config.cmake
+++ b/cmake/dca_config.cmake
@@ -100,15 +100,16 @@ endif()
 # TODO: Add more point groups and lattices.
 
 # Point group
-set(DCA_POINT_GROUP "D4" CACHE STRING "Point group symmetry, options are: C6 | D4 | no_symmetry<2> | no_symmetry<3>.")
-set_property(CACHE DCA_POINT_GROUP PROPERTY STRINGS C6 D4 no_symmetry<2> no_symmetry<3>)
+set(DCA_POINT_GROUP "D4" CACHE STRING "Point group symmetry, options are: D6 | D4 | no_symmetry<2> | no_symmetry<3>.")
+set_property(CACHE DCA_POINT_GROUP PROPERTY STRINGS D6 D4 no_symmetry<2> no_symmetry<3>)
 
-if (DCA_POINT_GROUP STREQUAL "C6")
+if (DCA_POINT_GROUP STREQUAL "D6")
   set(DCA_POINT_GROUP_INCLUDE
-    "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_hexagonal.hpp")
+    "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp")
 
 elseif (DCA_POINT_GROUP STREQUAL "D4")
-  set(DCA_POINT_GROUP_INCLUDE "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp")
+  set(DCA_POINT_GROUP_INCLUDE
+    "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp")
 
 elseif (DCA_POINT_GROUP STREQUAL "no_symmetry<2>")
   set(DCA_POINT_GROUP_INCLUDE "dca/phys/domains/cluster/symmetries/point_groups/no_symmetry.hpp")
@@ -116,7 +117,7 @@ elseif (DCA_POINT_GROUP STREQUAL "no_symmetry<3>")
   set(DCA_POINT_GROUP_INCLUDE "dca/phys/domains/cluster/symmetries/point_groups/no_symmetry.hpp")
 
 else()
-  message(FATAL_ERROR "Please set DCA_POINT_GROUP to a valid option: C6 | D4 | no_symmetry<2> | no_symmetry<3>.")
+  message(FATAL_ERROR "Please set DCA_POINT_GROUP to a valid option: D6 | D4 | no_symmetry<2> | no_symmetry<3>.")
 endif()
 
 # Lattice type
diff --git a/include/dca/phys/dca_step/symmetrization/derive_point_group.hpp b/include/dca/phys/dca_step/symmetrization/derive_point_group.hpp
new file mode 100644
index 000000000..01649dc59
--- /dev/null
+++ b/include/dca/phys/dca_step/symmetrization/derive_point_group.hpp
@@ -0,0 +1,202 @@
+// Copyright (C) 2018 UT-Battelle, LLC
+// All rights reserved.
+//
+// See LICENSE for terms of usage.
+// See CITATION.md for citation guidelines, if DCA++ is used for scientific publications.
+//
+// Author: Tyler Sax (tylersax@gmail.com)
+//
+// Derives a 2D model's point group -- the built-in 2D holohedry pool filtered by the lattice geometry
+// and by H0 invariance -- and compares it against the group realized from the model's declared
+// DCA_point_group. Production symmetrization continues to use the DECLARED group:
+// deriveAndComparePointGroup restores the declared symmetry state before returning and only
+// reports divergences. The reports build a per-model map of models whose declaration disagrees with
+// the symmetry encoded in their own Hamiltonian, which drives later declaration-free upgrades.
+
+#ifndef DCA_PHYS_DCA_STEP_SYMMETRIZATION_DERIVE_POINT_GROUP_HPP
+#define DCA_PHYS_DCA_STEP_SYMMETRIZATION_DERIVE_POINT_GROUP_HPP
+
+#include <cmath>
+#include <complex>
+#include <sstream>
+#include <stdexcept>
+#include <string>
+#include <type_traits>
+#include <vector>
+
+#include "dca/function/domains.hpp"
+#include "dca/function/function.hpp"
+#include "dca/phys/dca_step/symmetrization/solve_orbital_op_signs.hpp"
+#include "dca/phys/domains/cluster/cluster_symmetry.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
+#include "dca/phys/domains/cluster/symmetrization_algorithms/search_maximal_symmetry_group.hpp"
+#include "dca/phys/domains/cluster/symmetrization_algorithms/set_symmetry_matrices.hpp"
+#include "dca/phys/domains/quantum/electron_band_domain.hpp"
+#include "dca/phys/domains/quantum/electron_spin_domain.hpp"
+#include "dca/phys/domains/quantum/point_group_symmetry_element.hpp"
+
+namespace dca {
+namespace phys {
+// dca::phys::
+
+namespace detail {
+// dca::phys::detail::
+
+// Geometric equality of two symmetry elements (same linear part .O and fractional translation .t),
+// at the tolerance the symmetry search uses to deduplicate operations.
+inline bool isSameSymmetryElement(const domains::point_group_symmetry_element& a,
+                                  const domains::point_group_symmetry_element& b) {
+  constexpr double tol = 1e-6;
+  if (a.DIMENSION != b.DIMENSION)
+    return false;
+  double diff = 0.;
+  for (int i = 0; i < a.DIMENSION * a.DIMENSION; ++i)
+    diff = std::max(diff, std::abs(a.O[i] - b.O[i]));
+  for (int i = 0; i < a.DIMENSION; ++i)
+    diff = std::max(diff, std::abs(a.t[i] - b.t[i]));
+  return diff < tol;
+}
+
+inline bool containsSymmetryElement(const std::vector<domains::point_group_symmetry_element>& set,
+                                    const domains::point_group_symmetry_element& op) {
+  for (const auto& element : set)
+    if (isSameSymmetryElement(element, op))
+      return true;
+  return false;
+}
+
+// Detects the initializeH0 lattice interface: the legacy lattices cannot feed the
+// H0-invariance gate, so derivation skips them.
+template <typename Lattice, typename Parameters, typename H0Type, typename = void>
+struct LatticeHasInitializeH0 : std::false_type {};
+template <typename Lattice, typename Parameters, typename H0Type>
+struct LatticeHasInitializeH0<Lattice, Parameters, H0Type,
+                              std::void_t<decltype(Lattice::initializeH0(
+                                  std::declval<const Parameters&>(), std::declval<H0Type&>()))>>
+    : std::true_type {};
+
+// Makes PointGroup the live group of one cluster family, overwriting the previously realized group:
+// clear the operation lists (the search only appends), re-search under the same geometric filter,
+// resize the per-operation functions to the new count, and rebuild the symmetry tables. Assumes the
+// family was already populated once by cluster_domain_symmetry_initializer.
+template <typename KCluster, typename PointGroup>
+void installPointGroup() {
+  using CS_k = domains::cluster_symmetry<KCluster>;
+  using CS_r = domains::cluster_symmetry<typename CS_k::dual_type>;
+  using Family = typename CS_k::cluster_family_type;
+
+  CS_k::sym_unit_cell_dmn_t::parameter_type::get_elements().clear();
+  CS_k::sym_super_cell_dmn_t::parameter_type::get_elements().clear();
+  domains::search_maximal_symmetry_group<Family, PointGroup, domains::UNIT_CELL>::execute();
+  domains::search_maximal_symmetry_group<Family, PointGroup, domains::SUPER_CELL>::execute();
+  CS_k::get_symmetry_matrix().reset();
+  CS_r::get_symmetry_matrix().reset();
+  CS_k::get_mapped_point().reset();
+  CS_k::get_fold_phase().reset();
+  CS_k::get_orbital_op().reset();
+  domains::set_symmetry_matrices<Family>::execute();
+}
+
+}  // namespace detail
+// dca::phys::
+
+// Derives the point group of a 2D model for one cluster family and returns a human-readable report
+// of how it diverges from the declared group -- empty when they agree. Must be called right after
+// cluster_domain_symmetry_initializer<RClusterDmn, DCA_point_group>::execute() for the same family;
+// the declared symmetry state is restored before returning, so production behavior is unchanged.
+// 3D models (no 3D pool yet) and legacy lattices (no initializeH0) skip derivation entirely.
+template <typename RClusterDmn, typename Model, typename Parameters>
+std::string deriveAndComparePointGroup(const Parameters& parameters) {
+  using Lattice = typename Model::lattice_type;
+  using RCluster = typename RClusterDmn::parameter_type;
+  using KCluster = typename domains::cluster_symmetry<RCluster>::dual_type;
+  using BDmn = func::dmn_0<domains::electron_band_domain>;
+  using SDmn = func::dmn_0<domains::electron_spin_domain>;
+  using NuDmn = func::dmn_variadic<BDmn, SDmn>;
+  using H0Type =
+      func::function<std::complex<double>, func::dmn_variadic<NuDmn, NuDmn, func::dmn_0<KCluster>>>;
+
+  if constexpr (Lattice::DIMENSION != 2 ||
+                !detail::LatticeHasInitializeH0<Lattice, Parameters, H0Type>::value) {
+    (void)parameters;
+    return "";
+  }
+  else {
+    using KSymDmn = typename domains::cluster_symmetry<KCluster>::sym_super_cell_dmn_t;
+    using Element = domains::point_group_symmetry_element;
+
+    // The realized declared group, populated by the initializer that just ran. Only the first
+    // get_size() entries are the live group: on a re-initialization (some tests run update_domains
+    // twice per process) the search zeroes the count but appends to the never-cleared element
+    // vector, so the vector can be longer than the group. The reference tracks the singleton
+    // through the group installs below; the copy preserves the declared ops.
+    const std::vector<Element>& realized = KSymDmn::parameter_type::get_elements();
+    const int n_declared = KSymDmn::parameter_type::get_size();
+    const std::vector<Element> declared(realized.begin(), realized.begin() + n_declared);
+
+    // Derive: candidate pool -> same geometric filter -> H0-invariance gate.
+    detail::installPointGroup<KCluster, domains::holohedry_pool_2D>();
+
+    H0Type H0;
+    Model::initializeH0(parameters, H0);
+
+    std::vector<Element> derived;
+    for (const int s : verifiedSymmetryOps<KCluster>(H0))
+      derived.push_back(realized[s]);
+
+    // Restore the declared state, and verify the restoration: every consumer of the symmetry
+    // domains must see exactly the state the declared initializer produced.
+    detail::installPointGroup<KCluster, typename Lattice::DCA_point_group>();
+
+    if (KSymDmn::parameter_type::get_size() != n_declared)
+      throw std::logic_error(
+          "deriveAndComparePointGroup: failed to restore the declared symmetry state of " +
+          KCluster::get_name() + ".");
+    for (std::size_t i = 0; i < declared.size(); ++i)
+      if (!detail::isSameSymmetryElement(realized[i], declared[i]))
+        throw std::logic_error(
+            "deriveAndComparePointGroup: failed to restore the declared symmetry state of " +
+            KCluster::get_name() + ".");
+
+    // Compare as sets and report the divergence, if any, naming the ops that differ.
+    std::vector<std::string> underdeclared;
+    for (const auto& op : derived)
+      if (!detail::containsSymmetryElement(declared, op))
+        underdeclared.push_back(op.describe());
+
+    std::vector<std::string> unverified;
+    for (const auto& op : declared)
+      if (!detail::containsSymmetryElement(derived, op))
+        unverified.push_back(op.describe());
+
+    if (underdeclared.empty() && unverified.empty())
+      return "";
+
+    auto join = [](const std::vector<std::string>& ops) {
+      std::string s;
+      for (std::size_t i = 0; i < ops.size(); ++i)
+        s += (i ? ", " : "") + ops[i];
+      return s;
+    };
+
+    std::ostringstream report;
+    report << "derived-symmetry check [" << KCluster::get_name() << "]:\n"
+           << "\tdeclared group: " << declared.size() << " op(s); derived group: " << derived.size()
+           << " op(s).\n";
+    if (!underdeclared.empty())
+      report << "\tunder-declared: " << underdeclared.size()
+             << " derived H0-verified symmetry op(s) beyond the declared group "
+             << "(symmetry averaging left unexploited): " << join(underdeclared) << ".\n";
+    if (!unverified.empty())
+      report << "\tWARNING: " << unverified.size()
+             << " declared op(s) not reproduced by the derivation -- the op fails the "
+             << "H0-invariance check (or, the lattice is not in the pool's standard orientation): "
+             << join(unverified) << ".\n";
+    return report.str();
+  }
+}
+
+}  // namespace phys
+}  // namespace dca
+
+#endif  // DCA_PHYS_DCA_STEP_SYMMETRIZATION_DERIVE_POINT_GROUP_HPP
diff --git a/include/dca/phys/dca_step/symmetrization/solve_orbital_op_signs.hpp b/include/dca/phys/dca_step/symmetrization/solve_orbital_op_signs.hpp
index ae2d5768f..80e602c55 100644
--- a/include/dca/phys/dca_step/symmetrization/solve_orbital_op_signs.hpp
+++ b/include/dca/phys/dca_step/symmetrization/solve_orbital_op_signs.hpp
@@ -29,9 +29,11 @@
 //
 // The analytic models shipped with DCA++ have at most n_b = 3 orbitals, so once the gauge
 // sigma(0) = +1 is fixed there are at most 2^(n_b-1) <= 4 candidate sign vectors. We simply
-// enumerate them and keep the one that satisfies every coupling constraint. Three conditions make
+// enumerate them and keep the one that satisfies every coupling constraint. Four conditions make
 // an operation fail:
 //
+//   * a missing band image in the symmetry table (the -1 value recorded by set_symmetry_matrices
+//     when it finds no admissible image) means the op has no candidate permutation at all;
 //   * a magnitude mismatch (an entry vanishing on only one side) means the permutation is not a
 //     symmetry of H0;
 //   * a non-real or non-unit ratio means the op needs a non-signed-permutation U_S (genuine orbital
@@ -78,8 +80,18 @@ void solveSignsForOp(int s, int nb, int nk, const SymFunc& sym, const FoldFunc&
   // half of U_S (which orbital maps to which); it is k-independent for a point-group op, so read it
   // at k = 0. Only the signs that decorate it are unknown.
   std::vector<int> image(nb);
-  for (int b = 0; b < nb; ++b)
+  for (int b = 0; b < nb; ++b) {
     image[b] = sym(0, b, s).second;
+    // set_symmetry_matrices records a silent -1 sentinel when its position/flavor matching finds
+    // no admissible image (its own throw is commented out), so the table cannot be assumed
+    // complete here. Reject the op before the sentinel is used to index H0.
+    if (image[b] < 0 || image[b] >= nb)
+      throw std::out_of_range(
+          "solveOrbitalOpSignsFromH0: no band image in the symmetry table for band " +
+          std::to_string(b) + " under op " + std::to_string(s) +
+          " -- the geometric position/flavor matching found no admissible image, so the op has no "
+          "candidate permutation.");
+  }
 
   // Gather one sign-product constraint per nonzero coupling, over all cluster momenta. Each nonzero
   // coupling forces sigma(b0) sigma(b1) to the +/-1 fold-corrected ratio of the two H0 entries. The
diff --git a/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_hexagonal.hpp b/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_hexagonal.hpp
deleted file mode 100644
index 70086253d..000000000
--- a/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_hexagonal.hpp
+++ /dev/null
@@ -1,81 +0,0 @@
-// Copyright (C) 2018 ETH Zurich
-// Copyright (C) 2018 UT-Battelle, LLC
-// All rights reserved.
-//
-// See LICENSE for terms of usage.
-// See CITATION.md for citation guidelines, if DCA++ is used for scientific publications.
-//
-// Author: Peter Staar (taa@zurich.ibm.com)
-//
-// 2D hexagonal.
-
-#ifndef DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_HEXAGONAL_HPP
-#define DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_HEXAGONAL_HPP
-
-#include "dca/phys/domains/cluster/symmetries/symmetry_operations/2d/Cn_2d.hpp"
-#include "dca/phys/domains/cluster/symmetries/symmetry_operations/2d/Sn_2d.hpp"
-#include "dca/util/type_list.hpp"
-
-namespace dca {
-namespace phys {
-namespace domains {
-// dca::phys::domains::
-
-// Group actions
-typedef Cn_2D<1, 3> Cn_2D_1_3_type;
-typedef Cn_2D<2, 3> Cn_2D_2_3_type;
-
-typedef Sn_2D<0, 3> Sn_2D_0_3_type;
-typedef Sn_2D<1, 3> Sn_2D_1_3_type;
-typedef Sn_2D<2, 3> Sn_2D_2_3_type;
-
-typedef Cn_2D<1, 6> Cn_2D_1_6_type;
-typedef Cn_2D<2, 6> Cn_2D_2_6_type;
-typedef Cn_2D<3, 6> Cn_2D_3_6_type;
-typedef Cn_2D<4, 6> Cn_2D_4_6_type;
-typedef Cn_2D<5, 6> Cn_2D_5_6_type;
-
-typedef Sn_2D<0, 6> Sn_2D_0_6_type;
-typedef Sn_2D<1, 6> Sn_2D_1_6_type;
-typedef Sn_2D<2, 6> Sn_2D_2_6_type;
-typedef Sn_2D<3, 6> Sn_2D_3_6_type;
-typedef Sn_2D<4, 6> Sn_2D_4_6_type;
-typedef Sn_2D<5, 6> Sn_2D_5_6_type;
-
-// Point group: set of group actions
-struct C3 {
-  typedef dca::util::Typelist<Cn_2D_1_3_type, Cn_2D_2_3_type> point_group_type_list;
-};
-
-struct S3 {
-  typedef dca::util::Typelist<Sn_2D_0_3_type, Sn_2D_1_3_type, Sn_2D_2_3_type> point_group_type_list;
-};
-
-struct D3 {
-  typedef dca::util::Typelist<Cn_2D_1_3_type, Cn_2D_2_3_type, Sn_2D_0_3_type, Sn_2D_1_3_type, Sn_2D_2_3_type>
-      point_group_type_list;
-};
-
-struct C6 {
-  typedef dca::util::Typelist<Cn_2D_1_6_type, Cn_2D_2_6_type, Cn_2D_3_6_type, Cn_2D_4_6_type, Cn_2D_5_6_type>
-      point_group_type_list;
-};
-
-struct S6 {
-  typedef dca::util::Typelist<Sn_2D_0_6_type, Sn_2D_1_6_type, Sn_2D_2_6_type, Sn_2D_3_6_type,
-                              Sn_2D_4_6_type, Sn_2D_5_6_type>
-      point_group_type_list;
-};
-
-struct D6 {
-  typedef dca::util::Typelist<Cn_2D_1_6_type, Cn_2D_2_6_type, Cn_2D_3_6_type, Cn_2D_4_6_type,
-                              Cn_2D_5_6_type, Sn_2D_0_6_type, Sn_2D_1_6_type, Sn_2D_2_6_type,
-                              Sn_2D_3_6_type, Sn_2D_4_6_type, Sn_2D_5_6_type>
-      point_group_type_list;
-};
-
-}  // domains
-}  // phys
-}  // dca
-
-#endif  // DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_HEXAGONAL_HPP
diff --git a/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_oblique.hpp b/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_oblique.hpp
deleted file mode 100644
index 3d52cc7ad..000000000
--- a/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_oblique.hpp
+++ /dev/null
@@ -1,33 +0,0 @@
-// Copyright (C) 2018 ETH Zurich
-// Copyright (C) 2018 UT-Battelle, LLC
-// All rights reserved.
-//
-// See LICENSE for terms of usage.
-// See CITATION.md for citation guidelines, if DCA++ is used for scientific publications.
-//
-// Author: Peter Staar (taa@zurich.ibm.com)
-//
-// 2D oblique.
-
-#ifndef DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_OBLIQUE_HPP
-#define DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_OBLIQUE_HPP
-
-#include "dca/phys/domains/cluster/symmetries/symmetry_operations/2d/Cn_2d.hpp"
-#include "dca/util/type_list.hpp"
-
-namespace dca {
-namespace phys {
-namespace domains {
-// dca::phys::domains::
-
-// Group actions
-typedef Cn_2D<1, 2> Cn_2D_1_2_type;
-
-// Point group: set of group actions
-typedef dca::util::Typelist<Cn_2D_1_2_type> C2;
-
-}  // domains
-}  // phys
-}  // dca
-
-#endif  // DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_OBLIQUE_HPP
diff --git a/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_rectangular.hpp b/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_rectangular.hpp
deleted file mode 100644
index 05f7e3135..000000000
--- a/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_rectangular.hpp
+++ /dev/null
@@ -1,38 +0,0 @@
-// Copyright (C) 2018 ETH Zurich
-// Copyright (C) 2018 UT-Battelle, LLC
-// All rights reserved.
-//
-// See LICENSE for terms of usage.
-// See CITATION.md for citation guidelines, if DCA++ is used for scientific publications.
-//
-// Author: Peter Staar (taa@zurich.ibm.com)
-//
-// 2D rectangular.
-
-#ifndef DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_RECTANGULAR_HPP
-#define DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_RECTANGULAR_HPP
-
-#include "dca/phys/domains/cluster/symmetries/symmetry_operations/2d/Sn_2d.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_oblique.hpp"  // for C2
-#include "dca/util/type_list.hpp"
-
-namespace dca {
-namespace phys {
-namespace domains {
-// dca::phys::domains::
-
-// Group actions
-typedef Sn_2D<0, 4> Sn_2D_0_4_type;
-typedef Sn_2D<1, 4> Sn_2D_1_4_type;
-
-// Point group: set of group actions
-typedef dca::util::Typelist<Sn_2D_0_4_type> Sx_2d;
-typedef dca::util::Typelist<Sn_2D_1_4_type> Sy_2d;
-
-typedef dca::util::Append<C2, dca::util::Append<Sx_2d, Sy_2d>::type>::type D2;
-
-}  // domains
-}  // phys
-}  // dca
-
-#endif  // DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_RECTANGULAR_HPP
diff --git a/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp b/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp
deleted file mode 100644
index f0738f3a6..000000000
--- a/include/dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp
+++ /dev/null
@@ -1,74 +0,0 @@
-// Copyright (C) 2018 ETH Zurich
-// Copyright (C) 2018 UT-Battelle, LLC
-// All rights reserved.
-//
-// See LICENSE for terms of usage.
-// See CITATION.md for citation guidelines, if DCA++ is used for scientific publications.
-//
-// Author: Peter Staar (taa@zurich.ibm.com)
-//
-// 2D square.
-
-#ifndef DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_SQUARE_HPP
-#define DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_SQUARE_HPP
-
-#include "dca/phys/domains/cluster/symmetries/symmetry_operations/2d/Cn_2d.hpp"
-#include "dca/phys/domains/cluster/symmetries/symmetry_operations/2d/Sn_2d.hpp"
-#include "dca/phys/domains/cluster/symmetries/symmetry_operations/identity_group_operation.hpp"
-#include "dca/util/type_list.hpp"
-
-namespace dca {
-namespace phys {
-namespace domains {
-// dca::phys::domains::
-
-// Group actions
-typedef Cn_2D<1, 4> Cn_2D_1_4_type;
-typedef Cn_2D<2, 4> Cn_2D_2_4_type;
-typedef Cn_2D<3, 4> Cn_2D_3_4_type;
-typedef Cn_2D<4, 4> Cn_2D_4_4_type;
-
-typedef Sn_2D<0, 8> Sn_2D_0_8_type;
-typedef Sn_2D<1, 8> Sn_2D_1_8_type;
-typedef Sn_2D<2, 8> Sn_2D_2_8_type;
-typedef Sn_2D<3, 8> Sn_2D_3_8_type;
-typedef Sn_2D<4, 8> Sn_2D_4_8_type;
-typedef Sn_2D<5, 8> Sn_2D_5_8_type;
-typedef Sn_2D<6, 8> Sn_2D_6_8_type;
-typedef Sn_2D<7, 8> Sn_2D_7_8_type;
-typedef Sn_2D<8, 8> Sn_2D_8_8_type;
-
-// Point group: set of group actions
-struct C4 {
-  typedef dca::util::Typelist<Cn_2D_1_4_type, Cn_2D_2_4_type, Cn_2D_3_4_type, Cn_2D_4_4_type>
-      point_group_type_list;
-};
-
-struct S4 {
-  typedef dca::util::Typelist<Sn_2D_0_8_type, Sn_2D_2_8_type, Sn_2D_4_8_type, Sn_2D_6_8_type>
-      point_group_type_list;
-};
-
-struct S4_plus {
-  typedef dca::util::Typelist<identity_group_operation<2>, Cn_2D_2_4_type, Sn_2D_0_8_type, Sn_2D_2_8_type>
-      point_group_type_list;
-};
-
-struct S8 {
-  typedef dca::util::Typelist<Sn_2D_0_8_type, Sn_2D_1_8_type, Sn_2D_2_8_type, Sn_2D_3_8_type,
-                              Sn_2D_4_8_type, Sn_2D_5_8_type, Sn_2D_6_8_type, Sn_2D_7_8_type,
-                              identity_group_operation<2>>
-      point_group_type_list;
-};
-
-struct D4 {
-  using point_group_type_list =
-      dca::util::Typelist<Sn_2D<0, 8>, Sn_2D<1, 8>, Sn_2D<2, 8>, Sn_2D<3, 8>, Cn_2D<1, 4>,
-                          Cn_2D<2, 4>, Cn_2D<3, 4>, identity_group_operation<2>>;
-};
-
-}  // namespace domains
-}  // namespace phys
-}  // namespace dca
-
-#endif  // DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_2D_SQUARE_HPP
diff --git a/include/dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp b/include/dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp
new file mode 100644
index 000000000..c7eec398f
--- /dev/null
+++ b/include/dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp
@@ -0,0 +1,60 @@
+// Copyright (C) 2018 UT-Battelle, LLC
+// All rights reserved.
+//
+// See LICENSE for terms of usage.
+// See CITATION.md for citation guidelines, if DCA++ is used for scientific publications.
+//
+// Author: Tyler Sax
+//
+// The two 2D crystallographic holohedries and the candidate pool they generate.
+//
+// D4 (square holohedry, order 8) and D6 (hexagonal holohedry, order 12) are the two maximal point
+// groups in 2D. By the crystallographic restriction, every 2D crystallographic point group is a
+// subgroup of one of them, so they are the only primitives needed: the lower-symmetry groups
+// (rectangular D2, oblique C2, trigonal C3/D3, ...) are not declared per model but derived at
+// runtime from the pool by the geometry filter (search_symmetry_group: is_lattice_compatible +
+// is_unit_cell_compatible) and the H0-invariance gate.
+
+#ifndef DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_HOLOHEDRIES_2D_HPP
+#define DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_HOLOHEDRIES_2D_HPP
+
+#include "dca/phys/domains/cluster/symmetries/symmetry_operations/2d/Cn_2d.hpp"
+#include "dca/phys/domains/cluster/symmetries/symmetry_operations/2d/Sn_2d.hpp"
+#include "dca/phys/domains/cluster/symmetries/symmetry_operations/identity_group_operation.hpp"
+#include "dca/util/type_list.hpp"
+
+namespace dca {
+namespace phys {
+namespace domains {
+// dca::phys::domains::
+
+// Square holohedry (order 8): the three C4 rotations, four mirrors (reflection lines at n*pi/4,
+// n={0...3}), and the identity.
+struct D4 {
+  using point_group_type_list =
+      dca::util::Typelist<Sn_2D<0, 8>, Sn_2D<1, 8>, Sn_2D<2, 8>, Sn_2D<3, 8>, Cn_2D<1, 4>,
+                          Cn_2D<2, 4>, Cn_2D<3, 4>, identity_group_operation<2>>;
+};
+
+// Hexagonal holohedry (order 12): the five C6 rotations, six mirrors (reflection lines at n*pi/6,
+// n = 0..5), and the identity.
+struct D6 {
+  using point_group_type_list =
+      dca::util::Typelist<Cn_2D<1, 6>, Cn_2D<2, 6>, Cn_2D<3, 6>, Cn_2D<4, 6>, Cn_2D<5, 6>,
+                          Sn_2D<0, 12>, Sn_2D<1, 12>, Sn_2D<2, 12>, Sn_2D<3, 12>, Sn_2D<4, 12>,
+                          Sn_2D<5, 12>, identity_group_operation<2>>;
+};
+
+// Universal 2D candidate pool = square holohedry D4  U  hexagonal holohedry D6. Overlapping ops
+// (identity, the C2 rotation) appear twice; search_symmetry_group::is_duplicate collapses them at
+// populate time, so the union need not be hand-deduplicated.
+struct holohedry_pool_2D {
+  using point_group_type_list =
+      dca::util::Append<D4::point_group_type_list, D6::point_group_type_list>::type;
+};
+
+}  // namespace domains
+}  // namespace phys
+}  // namespace dca
+
+#endif  // DCA_PHYS_DOMAINS_CLUSTER_SYMMETRIES_POINT_GROUPS_2D_HOLOHEDRIES_2D_HPP
diff --git a/include/dca/phys/domains/quantum/point_group_symmetry_element.hpp b/include/dca/phys/domains/quantum/point_group_symmetry_element.hpp
index c61569ab8..9a808aae2 100644
--- a/include/dca/phys/domains/quantum/point_group_symmetry_element.hpp
+++ b/include/dca/phys/domains/quantum/point_group_symmetry_element.hpp
@@ -13,6 +13,7 @@
 #define DCA_PHYS_DOMAINS_QUANTUM_POINT_GROUP_SYMMETRY_ELEMENT_HPP
 
 #include <complex>
+#include <string>
 #include <vector>
 
 namespace dca {
@@ -38,6 +39,9 @@ class point_group_symmetry_element {
 
   void transform(const double* t0, double* t1) const;
 
+  // A short human-readable label for the operation.
+  std::string describe() const;
+
   template <class stream_type>
   void to_JSON(stream_type& ss) const;
 
diff --git a/include/dca/phys/models/analytic_hamiltonians/fe_as_lattice.hpp b/include/dca/phys/models/analytic_hamiltonians/fe_as_lattice.hpp
index 802beae01..6a31b41e1 100644
--- a/include/dca/phys/models/analytic_hamiltonians/fe_as_lattice.hpp
+++ b/include/dca/phys/models/analytic_hamiltonians/fe_as_lattice.hpp
@@ -17,7 +17,7 @@
 
 #include "dca/function/domains.hpp"
 #include "dca/function/function.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/bilayer_lattice.hpp"
 #include "dca/phys/models/analytic_hamiltonians/cluster_shape_type.hpp"
 #include "dca/phys/models/traits.hpp"
diff --git a/include/dca/phys/models/analytic_hamiltonians/threeband_hubbard.hpp b/include/dca/phys/models/analytic_hamiltonians/threeband_hubbard.hpp
index e4a5ab094..34c6220e3 100644
--- a/include/dca/phys/models/analytic_hamiltonians/threeband_hubbard.hpp
+++ b/include/dca/phys/models/analytic_hamiltonians/threeband_hubbard.hpp
@@ -21,7 +21,7 @@
 
 #include "dca/function/domains.hpp"
 #include "dca/function/function.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/cluster/symmetries/point_groups/no_symmetry.hpp"
 #include "dca/phys/models/analytic_hamiltonians/cluster_shape_type.hpp"
 #include "dca/util/type_list.hpp"
diff --git a/include/dca/phys/models/analytic_hamiltonians/twoband_Cu.hpp b/include/dca/phys/models/analytic_hamiltonians/twoband_Cu.hpp
index c8b2e65a0..56dff502e 100644
--- a/include/dca/phys/models/analytic_hamiltonians/twoband_Cu.hpp
+++ b/include/dca/phys/models/analytic_hamiltonians/twoband_Cu.hpp
@@ -21,7 +21,7 @@
 
 #include "dca/function/domains.hpp"
 #include "dca/function/function.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/cluster/symmetries/point_groups/no_symmetry.hpp"
 #include "dca/phys/models/analytic_hamiltonians/cluster_shape_type.hpp"
 #include "dca/util/type_list.hpp"
diff --git a/include/dca/phys/parameters/parameters.hpp b/include/dca/phys/parameters/parameters.hpp
index b9bf52383..b1f66a280 100644
--- a/include/dca/phys/parameters/parameters.hpp
+++ b/include/dca/phys/parameters/parameters.hpp
@@ -40,6 +40,7 @@
 #include "dca/phys/parameters/model_parameters.hpp"
 #include "dca/phys/parameters/output_parameters.hpp"
 #include "dca/phys/parameters/physics_parameters.hpp"
+#include "dca/phys/dca_step/symmetrization/derive_point_group.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
 #include "dca/phys/domains/cluster/cluster_domain_family.hpp"
 #include "dca/phys/domains/cluster/cluster_domain_initializer.hpp"
@@ -138,6 +139,16 @@ class Parameters : public AnalysisParameters,
   void update_model();
   void update_domains();
 
+  // Derives the point group of one cluster family from the holohedry pool + H0 and prints how it
+  // diverges from the declared group (rank 0; silent on agreement). The declared symmetry state is
+  // restored by the check, so production symmetrization is unchanged.
+  template <typename RDmn>
+  void checkDerivedSymmetry() const {
+    const std::string report = deriveAndComparePointGroup<RDmn, Model>(*this);
+    if (!report.empty() && concurrency_.id() == concurrency_.first())
+      std::cout << report;
+  }
+
   int get_buffer_size(const concurrency_type& concurrency) const;
   void pack(const concurrency_type& concurrency, char* buffer, int buffer_size, int& position) const;
   void unpack(const concurrency_type& concurrency, char* buffer, int buffer_size, int& position);
@@ -285,8 +296,8 @@ void Parameters<Concurrency, Threading, Profiler, Model, RandomNumberGenerator,
     read_obj.open_file(filename);
     this->readWrite(read_obj);
     // Detect model-section input mistakes while the parsed tree is still live (before close_file).
-    // Only the JSON reader tracks which sections were read; HDF5 as input is realistically 
-    // machine-generated, so the access check is not implemented for that path 
+    // Only the JSON reader tracks which sections were read; HDF5 as input is realistically
+    // machine-generated, so the access check is not implemented for that path
     if constexpr (std::is_same_v<Reader, dca::io::JSONReader>) {
       checkModelSections(read_obj.topLevelGroupAccess(), filename);
     }
@@ -335,6 +346,10 @@ void Parameters<Concurrency, Threading, Profiler, Model, RandomNumberGenerator,
                                                             DomainsParameters::get_cluster());
   domains::cluster_domain_symmetry_initializer<
       RClusterDmn, typename Model::lattice_type::DCA_point_group>::execute();
+  // The check runs immediately after the initializer: the derivation temporarily overwrites
+  // this family's symmetry singletons with the pool-derived group and then restores exactly
+  // the state the line above just produced.
+  checkDerivedSymmetry<RClusterDmn>();
 
   if (concurrency_.id() == concurrency_.first())
     KClusterDmn::parameter_type::print(std::cout);
@@ -344,6 +359,7 @@ void Parameters<Concurrency, Threading, Profiler, Model, RandomNumberGenerator,
                                                            DomainsParameters::get_sp_host());
   domains::cluster_domain_symmetry_initializer<
       RSpHostDmn, typename Model::lattice_type::DCA_point_group>::execute();
+  checkDerivedSymmetry<RSpHostDmn>();
 
   domains::MomentumExchangeDomain::initialize(*this);
 
@@ -355,6 +371,7 @@ void Parameters<Concurrency, Threading, Profiler, Model, RandomNumberGenerator,
                                                           AnalysisParameters::get_q_host());
   domains::cluster_domain_symmetry_initializer<
       RQHostDmn, typename Model::lattice_type::DCA_point_group>::execute();
+  checkDerivedSymmetry<RQHostDmn>();
 
   if (concurrency_.id() == concurrency_.first())
     KQHostDmn::parameter_type::print(std::cout);
@@ -364,6 +381,7 @@ void Parameters<Concurrency, Threading, Profiler, Model, RandomNumberGenerator,
                                                           AnalysisParameters::get_q_host_fine());
   domains::cluster_domain_symmetry_initializer<
       RQFineDmn, typename Model::lattice_type::DCA_point_group>::execute();
+  checkDerivedSymmetry<RQFineDmn>();
 
   if (concurrency_.id() == concurrency_.first())
     KQFineDmn::parameter_type::print(std::cout);
@@ -381,6 +399,7 @@ void Parameters<Concurrency, Threading, Profiler, Model, RandomNumberGenerator,
   }
   domains::cluster_domain_symmetry_initializer<
       RTpHostDmn, typename Model::lattice_type::DCA_point_group>::execute();
+  checkDerivedSymmetry<RTpHostDmn>();
 
   if (concurrency_.id() == concurrency_.first())
     KTpHostDmn::parameter_type::print(std::cout);
diff --git a/src/phys/domains/quantum/point_group_symmetry_element.cpp b/src/phys/domains/quantum/point_group_symmetry_element.cpp
index 9c40ea952..82f1e824b 100644
--- a/src/phys/domains/quantum/point_group_symmetry_element.cpp
+++ b/src/phys/domains/quantum/point_group_symmetry_element.cpp
@@ -11,7 +11,10 @@
 
 #include "dca/phys/domains/quantum/point_group_symmetry_element.hpp"
 
+#include <cmath>
 #include <cstring>
+#include <numeric>
+#include <sstream>
 
 namespace dca {
 namespace phys {
@@ -80,6 +83,70 @@ void point_group_symmetry_element::transform(const double* t0, double* t1) const
     t1[i] += t[i];
 }
 
+namespace {
+// Writes the angle (num / den) * pi in lowest terms, e.g. 0, pi, pi/2, 2pi/3.
+std::string formatPiFraction(int num, int den) {
+  if (num == 0)
+    return "0";
+
+  const int g = std::gcd(num, den);
+  num /= g;
+  den /= g;
+
+  std::ostringstream angle;
+  if (num > 1)
+    angle << num;
+  angle << "pi";
+  if (den > 1)
+    angle << "/" << den;
+
+  return angle.str();
+}
+}  // namespace
+
+std::string point_group_symmetry_element::describe() const {
+  std::ostringstream label;
+
+  if (DIMENSION == 2) {
+    // O is column major, so its first column (O[0], O[1]) is (cos, sin) of the rotation angle for a
+    // proper rotation (det +1) and (cos, sin) of twice the mirror-line angle for a reflection (det
+    // -1). Each operation gets a Schoenflies name (Cn^k for rotation, and sigma for reflection) and
+    // a geometric description.
+    //
+    // Rotations are decoded in twelfths of a full turn, and mirror lines are decoded in twelfths of
+    // pi. Both are reported in radians.
+    constexpr int kTurn = 12;
+    const double det = O[0] * O[3] - O[2] * O[1];
+    if (det > 0.) {
+      const long raw = std::lround(std::atan2(O[1], O[0]) * kTurn / (2. * M_PI));
+      const int t = static_cast<int>((raw % kTurn + kTurn) % kTurn);
+      if (t == 0) {
+        label << "E (identity)";
+      }
+      else {
+        const int g = std::gcd(t, kTurn);
+        label << "C" << kTurn / g;
+        if (t / g > 1)
+          label << "^" << t / g;
+        // A rotation by t twelfths of a turn is an angle of 2 pi t / 12 = t pi / 6.
+        label << " (rotation " << formatPiFraction(t, kTurn / 2) << " rad)";
+      }
+    }
+    else {
+      // The mirror-line angle is half the angle carried by the first column, so halving turns
+      // twelfths of a turn into twelfths of pi and the mirror line is m * pi / 12.
+      const long raw = std::lround(std::atan2(O[1], O[0]) * kTurn / (2. * M_PI));
+      const int m = static_cast<int>((raw % kTurn + kTurn) % kTurn);
+      label << "sigma (mirror @ " << formatPiFraction(m, kTurn) << " rad)";
+    }
+  }
+  else {
+    label << DIMENSION << "D symmetry element";
+  }
+
+  return label.str();
+}
+
 }  // namespace domains
 }  // namespace phys
 }  // namespace dca
diff --git a/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_interband/bilayer_lattice_Nc1_interband.cpp b/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_interband/bilayer_lattice_Nc1_interband.cpp
index c068ac3a1..7ea7e098a 100644
--- a/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_interband/bilayer_lattice_Nc1_interband.cpp
+++ b/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_interband/bilayer_lattice_Nc1_interband.cpp
@@ -29,7 +29,7 @@
 #include "dca/config/profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctaux/ctaux_cluster_solver.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/quantum/electron_band_domain.hpp"
 #include "dca/phys/domains/quantum/electron_spin_domain.hpp"
 #include "dca/phys/domains/time_and_frequency/frequency_domain.hpp"
diff --git a/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_intra_plus_interband/bilayer_lattice_Nc1_intra_plus_interband.cpp b/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_intra_plus_interband/bilayer_lattice_Nc1_intra_plus_interband.cpp
index 7b3427730..500c8db46 100644
--- a/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_intra_plus_interband/bilayer_lattice_Nc1_intra_plus_interband.cpp
+++ b/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_intra_plus_interband/bilayer_lattice_Nc1_intra_plus_interband.cpp
@@ -40,7 +40,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/config/profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctaux/ctaux_cluster_solver.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/quantum/electron_band_domain.hpp"
 #include "dca/phys/domains/quantum/electron_spin_domain.hpp"
 #include "dca/phys/domains/time_and_frequency/frequency_domain.hpp"
diff --git a/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_intraband/bilayer_lattice_Nc1_intraband.cpp b/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_intraband/bilayer_lattice_Nc1_intraband.cpp
index 16f8d9bef..68a2bc03f 100644
--- a/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_intraband/bilayer_lattice_Nc1_intraband.cpp
+++ b/test/integration/cluster_solver/ctaux/bilayer_lattice/Nc1_intraband/bilayer_lattice_Nc1_intraband.cpp
@@ -38,7 +38,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/config/profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctaux/ctaux_cluster_solver.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/quantum/electron_band_domain.hpp"
 #include "dca/phys/domains/quantum/electron_spin_domain.hpp"
 #include "dca/phys/domains/time_and_frequency/frequency_domain.hpp"
diff --git a/test/integration/cluster_solver/ctaux/square_lattice/Nc4_nn/square_lattice_Nc4_nn.cpp b/test/integration/cluster_solver/ctaux/square_lattice/Nc4_nn/square_lattice_Nc4_nn.cpp
index 4c22ccb6b..b13c0a084 100644
--- a/test/integration/cluster_solver/ctaux/square_lattice/Nc4_nn/square_lattice_Nc4_nn.cpp
+++ b/test/integration/cluster_solver/ctaux/square_lattice/Nc4_nn/square_lattice_Nc4_nn.cpp
@@ -39,7 +39,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/config/profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctaux/ctaux_cluster_solver.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/quantum/electron_band_domain.hpp"
 #include "dca/phys/domains/quantum/electron_spin_domain.hpp"
 #include "dca/phys/domains/time_and_frequency/frequency_domain.hpp"
diff --git a/test/integration/cluster_solver/ctaux/square_lattice/Nc4_onSite_plus_nn/square_lattice_Nc4_onSite_plus_nn.cpp b/test/integration/cluster_solver/ctaux/square_lattice/Nc4_onSite_plus_nn/square_lattice_Nc4_onSite_plus_nn.cpp
index bf972c72e..60d0307a5 100644
--- a/test/integration/cluster_solver/ctaux/square_lattice/Nc4_onSite_plus_nn/square_lattice_Nc4_onSite_plus_nn.cpp
+++ b/test/integration/cluster_solver/ctaux/square_lattice/Nc4_onSite_plus_nn/square_lattice_Nc4_onSite_plus_nn.cpp
@@ -39,7 +39,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/config/profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctaux/ctaux_cluster_solver.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/quantum/electron_band_domain.hpp"
 #include "dca/phys/domains/quantum/electron_spin_domain.hpp"
 #include "dca/phys/domains/time_and_frequency/frequency_domain.hpp"
diff --git a/test/integration/cluster_solver/ctint/ctint_double_update_comparison_test.cpp b/test/integration/cluster_solver/ctint/ctint_double_update_comparison_test.cpp
index 33104e949..1af69d9fc 100644
--- a/test/integration/cluster_solver/ctint/ctint_double_update_comparison_test.cpp
+++ b/test/integration/cluster_solver/ctint/ctint_double_update_comparison_test.cpp
@@ -38,7 +38,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/phys/dca_step/cluster_solver/ctint/walker/ctint_walker_cpu_submatrix.hpp"
 #include "test/unit/phys/dca_step/cluster_solver/ctint/walker/walker_wrapper_submatrix.hpp"
 #include "test/unit/phys/dca_step/cluster_solver/ctint/walker/walker_wrapper.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/fe_as_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_threading/no_threading.hpp"
diff --git a/test/integration/cluster_solver/ctint/ctint_fe_as_test.cpp b/test/integration/cluster_solver/ctint/ctint_fe_as_test.cpp
index d2722521c..dea7874a9 100644
--- a/test/integration/cluster_solver/ctint/ctint_fe_as_test.cpp
+++ b/test/integration/cluster_solver/ctint/ctint_fe_as_test.cpp
@@ -33,7 +33,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/phys/dca_data/dca_data.hpp"
 #include "dca/config/profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctint/ctint_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/fe_as_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_threading/no_threading.hpp"
diff --git a/test/integration/cluster_solver/ctint/ctint_hund_lattice_test.cpp b/test/integration/cluster_solver/ctint/ctint_hund_lattice_test.cpp
index 4b3ac5b4c..d0af55bb1 100644
--- a/test/integration/cluster_solver/ctint/ctint_hund_lattice_test.cpp
+++ b/test/integration/cluster_solver/ctint/ctint_hund_lattice_test.cpp
@@ -32,7 +32,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/phys/dca_data/dca_data.hpp"
 #include "dca/config/profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctint/ctint_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/hund_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_threading/no_threading.hpp"
diff --git a/test/integration/cluster_solver/ctint/ctint_square_lattice_test.cpp b/test/integration/cluster_solver/ctint/ctint_square_lattice_test.cpp
index d58c0cfb8..63d867952 100644
--- a/test/integration/cluster_solver/ctint/ctint_square_lattice_test.cpp
+++ b/test/integration/cluster_solver/ctint/ctint_square_lattice_test.cpp
@@ -32,7 +32,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/phys/dca_data/dca_data.hpp"
 #include "dca/config/profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctint/ctint_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_threading/no_threading.hpp"
diff --git a/test/integration/cluster_solver/ctint/ctint_square_lattice_test_gpu.cpp b/test/integration/cluster_solver/ctint/ctint_square_lattice_test_gpu.cpp
index 79464fcf9..248827a7c 100644
--- a/test/integration/cluster_solver/ctint/ctint_square_lattice_test_gpu.cpp
+++ b/test/integration/cluster_solver/ctint/ctint_square_lattice_test_gpu.cpp
@@ -35,7 +35,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/phys/dca_data/dca_data.hpp"
 #include "dca/config/profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctint/ctint_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_threading/no_threading.hpp"
diff --git a/test/integration/cluster_solver/ctint/ctint_square_lattice_tp_test.cpp b/test/integration/cluster_solver/ctint/ctint_square_lattice_tp_test.cpp
index 17ee70a33..62cf87ca5 100644
--- a/test/integration/cluster_solver/ctint/ctint_square_lattice_tp_test.cpp
+++ b/test/integration/cluster_solver/ctint/ctint_square_lattice_tp_test.cpp
@@ -36,7 +36,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/config/profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctint/ctint_cluster_solver.hpp"
 #include "dca/phys/dca_step/cluster_solver/stdthread_qmci/stdthread_qmci_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
diff --git a/test/integration/cluster_solver/shared_tools/accumulation/tp/test_setup.hpp b/test/integration/cluster_solver/shared_tools/accumulation/tp/test_setup.hpp
index 8f94854b1..b58f6f406 100644
--- a/test/integration/cluster_solver/shared_tools/accumulation/tp/test_setup.hpp
+++ b/test/integration/cluster_solver/shared_tools/accumulation/tp/test_setup.hpp
@@ -19,7 +19,7 @@
 #include "dca/io/json/json_reader.hpp"
 #include "dca/phys/parameters/parameters.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctint/structs/interaction_vertices.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/quantum/electron_band_domain.hpp"
 #include "dca/phys/domains/quantum/electron_spin_domain.hpp"
 #include "dca/phys/domains/time_and_frequency/frequency_domain.hpp"
diff --git a/test/integration/cluster_solver/stdthread_qmci/gpu/ctint_double_update_comparison_test.cpp b/test/integration/cluster_solver/stdthread_qmci/gpu/ctint_double_update_comparison_test.cpp
index 445bebfa7..4c50f4a96 100644
--- a/test/integration/cluster_solver/stdthread_qmci/gpu/ctint_double_update_comparison_test.cpp
+++ b/test/integration/cluster_solver/stdthread_qmci/gpu/ctint_double_update_comparison_test.cpp
@@ -26,7 +26,7 @@
 #include "dca/phys/dca_data/dca_data.hpp"
 #include "dca/parallel//no_concurrency/no_concurrency.hpp"
 #include "test/unit/phys/dca_step/cluster_solver/ctint/walker/walker_wrapper_submatrix.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/fe_as_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_threading/no_threading.hpp"
diff --git a/test/integration/cluster_solver/stdthread_qmci/gpu/stdthread_ctaux_gpu_tp_test.cpp b/test/integration/cluster_solver/stdthread_qmci/gpu/stdthread_ctaux_gpu_tp_test.cpp
index 9d78666f3..cfcf665bf 100644
--- a/test/integration/cluster_solver/stdthread_qmci/gpu/stdthread_ctaux_gpu_tp_test.cpp
+++ b/test/integration/cluster_solver/stdthread_qmci/gpu/stdthread_ctaux_gpu_tp_test.cpp
@@ -40,7 +40,7 @@ using McOptions = MockMcOptions<Scalar>;
 
 #include "dca/phys/dca_step/cluster_solver/ctaux/ctaux_cluster_solver.hpp"
 #include "dca/phys/dca_step/cluster_solver/stdthread_qmci/stdthread_qmci_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
diff --git a/test/integration/cluster_solver/stdthread_qmci/gpu/stdthread_ctint_gpu_tp_test.cpp b/test/integration/cluster_solver/stdthread_qmci/gpu/stdthread_ctint_gpu_tp_test.cpp
index 004a48c49..827b01e29 100644
--- a/test/integration/cluster_solver/stdthread_qmci/gpu/stdthread_ctint_gpu_tp_test.cpp
+++ b/test/integration/cluster_solver/stdthread_qmci/gpu/stdthread_ctint_gpu_tp_test.cpp
@@ -35,7 +35,7 @@ using McOptions = MockMcOptions<Scalar>;
 
 #include "dca/phys/dca_step/cluster_solver/ctint/ctint_cluster_solver.hpp"
 #include "dca/phys/dca_step/cluster_solver/stdthread_qmci/stdthread_qmci_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
diff --git a/test/integration/cluster_solver/stdthread_qmci/stdthread_ctaux_tp_test.cpp b/test/integration/cluster_solver/stdthread_qmci/stdthread_ctaux_tp_test.cpp
index b9516c765..09f712210 100644
--- a/test/integration/cluster_solver/stdthread_qmci/stdthread_ctaux_tp_test.cpp
+++ b/test/integration/cluster_solver/stdthread_qmci/stdthread_ctaux_tp_test.cpp
@@ -41,7 +41,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/phys/dca_step/cluster_solver/ctaux/ctaux_cluster_solver.hpp"
 #include "dca/phys/dca_step/cluster_solver/stdthread_qmci/stdthread_qmci_cluster_solver.hpp"
 #include "dca/phys/domains/cluster/momentum_exchange_domain.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/time_and_frequency/frequency_exchange_domain.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
diff --git a/test/integration/cluster_solver/stdthread_qmci/stdthread_ctint_cluster_solver_io_test.cpp b/test/integration/cluster_solver/stdthread_qmci/stdthread_ctint_cluster_solver_io_test.cpp
index ffe5db075..0e2ab0944 100644
--- a/test/integration/cluster_solver/stdthread_qmci/stdthread_ctint_cluster_solver_io_test.cpp
+++ b/test/integration/cluster_solver/stdthread_qmci/stdthread_ctint_cluster_solver_io_test.cpp
@@ -37,7 +37,7 @@ using McOptions = MockMcOptions<Scalar>;
 
 #include "dca/phys/dca_step/cluster_solver/ctint/ctint_cluster_solver.hpp"
 #include "dca/phys/dca_step/cluster_solver/stdthread_qmci/stdthread_qmci_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/mpi_concurrency/mpi_concurrency.hpp"
diff --git a/test/integration/cluster_solver/stdthread_qmci/stdthread_ctint_cluster_solver_test.hpp b/test/integration/cluster_solver/stdthread_qmci/stdthread_ctint_cluster_solver_test.hpp
index dd724a41e..872eb3e9c 100644
--- a/test/integration/cluster_solver/stdthread_qmci/stdthread_ctint_cluster_solver_test.hpp
+++ b/test/integration/cluster_solver/stdthread_qmci/stdthread_ctint_cluster_solver_test.hpp
@@ -28,7 +28,7 @@
 #include "dca/config/profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctint/ctint_cluster_solver.hpp"
 #include "dca/phys/dca_step/cluster_solver/stdthread_qmci/stdthread_qmci_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
diff --git a/test/integration/exact_diagonalization_advanced/ed_cluster_solver_test_helper.hpp b/test/integration/exact_diagonalization_advanced/ed_cluster_solver_test_helper.hpp
index 662a4f030..d503a6c1d 100644
--- a/test/integration/exact_diagonalization_advanced/ed_cluster_solver_test_helper.hpp
+++ b/test/integration/exact_diagonalization_advanced/ed_cluster_solver_test_helper.hpp
@@ -21,7 +21,7 @@
 #include "dca/phys/dca_data/dca_data.hpp"
 #include "dca/phys/dca_data/dca_data_real_freq.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/quantum/electron_band_domain.hpp"
 #include "dca/phys/domains/quantum/electron_spin_domain.hpp"
 #include "dca/phys/domains/time_and_frequency/frequency_domain.hpp"
diff --git a/test/integration/exact_diagonalization_advanced/hamiltonian_test.cpp b/test/integration/exact_diagonalization_advanced/hamiltonian_test.cpp
index 469a7eb9c..3aa3a0dfc 100644
--- a/test/integration/exact_diagonalization_advanced/hamiltonian_test.cpp
+++ b/test/integration/exact_diagonalization_advanced/hamiltonian_test.cpp
@@ -23,7 +23,7 @@
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
 #include "dca/phys/dca_step/cluster_solver/exact_diagonalization_advanced/fock_space.hpp"
 #include "dca/phys/dca_step/cluster_solver/exact_diagonalization_advanced/options.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 
diff --git a/test/integration/phys/lattice_mapping/lattice_mapping_sp_test.cpp b/test/integration/phys/lattice_mapping/lattice_mapping_sp_test.cpp
index f397210dd..3e5c75fd6 100644
--- a/test/integration/phys/lattice_mapping/lattice_mapping_sp_test.cpp
+++ b/test/integration/phys/lattice_mapping/lattice_mapping_sp_test.cpp
@@ -39,7 +39,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/profiling/null_profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/cluster_solver_id.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/quantum/electron_band_domain.hpp"
 #include "dca/phys/domains/quantum/electron_spin_domain.hpp"
 #include "dca/phys/domains/time_and_frequency/frequency_domain.hpp"
diff --git a/test/integration/phys/symmetrization/CMakeLists.txt b/test/integration/phys/symmetrization/CMakeLists.txt
index fd3ba98d8..7653c0dea 100644
--- a/test/integration/phys/symmetrization/CMakeLists.txt
+++ b/test/integration/phys/symmetrization/CMakeLists.txt
@@ -22,6 +22,17 @@ dca_add_gtest(symmetrize_characterization_threeband_D4_test
         TEST_DEFINES ThreebandD4
         )
 
+# Regression case for solveSignsForOp's -1 guard: Kagome produces the -1 which the solver
+# must reject instead of indexing H0 out of bounds.
+dca_add_gtest(symmetrize_characterization_kagome_test
+        FAST
+        GTEST_MPI_MAIN
+        INCLUDE_DIRS ${DCA_INCLUDES};${PROJECT_SOURCE_DIR};${FFTW_INCLUDE_DIR}
+        LIBS  ${DCA_LIBS}
+        CUSTOM_SOURCE symmetrize_characterization_test.cpp
+        TEST_DEFINES KagomeNoSym
+        )
+
 dca_add_gtest(symmetrize_characterization_singleband_chain_test
         FAST
         GTEST_MPI_MAIN
diff --git a/test/integration/phys/symmetrization/kagome_input.json b/test/integration/phys/symmetrization/kagome_input.json
new file mode 100644
index 000000000..0df6aae0d
--- /dev/null
+++ b/test/integration/phys/symmetrization/kagome_input.json
@@ -0,0 +1,31 @@
+{
+  "physics": {
+    "beta"                      :  1.,
+    "chemical-potential"        : 0.
+  },
+
+  "Kagome-Hubbard-model":
+  {
+    "t"       : 1.,
+    "U"       : 2.
+  },
+
+  "domains": {
+    "real-space-grids": {
+      "cluster": [[2, 0], [0, 2]],
+      "sp-host": [[8, 0], [0, 8]]
+    },
+
+    "imaginary-time": {
+      "sp-time-intervals": 128
+    },
+
+    "imaginary-frequency": {
+      "sp-fermionic-frequencies": 64
+    }
+  },
+
+  "DCA": {
+    "interacting-orbitals": [0, 1, 2]
+  }
+}
diff --git a/test/integration/phys/symmetrization/symmetrize_characterization_test.cpp b/test/integration/phys/symmetrization/symmetrize_characterization_test.cpp
index 558f10bf1..495ab4729 100644
--- a/test/integration/phys/symmetrization/symmetrize_characterization_test.cpp
+++ b/test/integration/phys/symmetrization/symmetrize_characterization_test.cpp
@@ -40,6 +40,7 @@
 
 #include "dca/platform/dca_gpu.h"
 #include "dca/phys/dca_step/symmetrization/symmetrize.hpp"
+#include "dca/phys/dca_step/symmetrization/derive_point_group.hpp"
 #include "dca/phys/dca_step/symmetrization/solve_orbital_op_signs.hpp"
 
 #include <algorithm>
@@ -58,10 +59,11 @@
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
 #include "dca/phys/dca_algorithms/compute_free_greens_function.hpp"
 #include "dca/phys/domains/cluster/cluster_symmetry.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/cluster/symmetries/point_groups/no_symmetry.hpp"
 #include "dca/phys/parameters/parameters.hpp"
 #include "dca/profiling/null_profiler.hpp"
+#include "dca/phys/models/analytic_hamiltonians/Kagome_hubbard.hpp"
 #include "dca/phys/models/analytic_hamiltonians/singleband_chain.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/analytic_hamiltonians/threeband_hubbard.hpp"
@@ -107,6 +109,7 @@ struct SquareD4 {
   static std::vector<int> expectedFailingKOps() { return {}; }
   static std::vector<int> expectedFailingROps() { return {}; }
   static std::vector<int> expectedUnverifiedKOps() { return {}; }
+  static constexpr int expected_num_derived_symmetries = 8;
 };
 
 // CASE 2 -- three-band Hubbard (Emery/CuO2) model on D4. This case fails the
@@ -126,6 +129,7 @@ struct ThreebandD4 {
   static std::vector<int> expectedFailingKOps() { return {}; }
   static std::vector<int> expectedFailingROps() { return {}; }
   static std::vector<int> expectedUnverifiedKOps() { return {}; }
+  static constexpr int expected_num_derived_symmetries = 8;
 };
 
 // CASE 3 -- 1D chain. Since this model has identity-only symmetry, this test exercises
@@ -141,6 +145,25 @@ struct SinglebandChain {
   static std::vector<int> expectedFailingKOps() { return {}; }
   static std::vector<int> expectedFailingROps() { return {}; }
   static std::vector<int> expectedUnverifiedKOps() { return {}; }
+  static constexpr int expected_num_derived_symmetries = 4;
+};
+
+// CASE 4 -- Kagome lattice declared with no_symmetry<2>, mirroring the production
+// instantiation in test/unit/phys/dca_step/cluster_solver/test_setup.hpp. This is the
+// regression case for solveSignsForOp's -1 guard: the hexagonal has the 12 D6 ops,
+// but the orbital position+flavor matching in set_symmetry_matrices filters them all out,
+// and records (-1, -1). The H0 sign solver must reject the ops instead of indexing H0 with
+// band -1 (an OOB).
+struct KagomeNoSym {
+  using Scalar = double;
+  using Lattice = dca::phys::models::KagomeHubbard<dca::phys::domains::no_symmetry<2>>;
+  static constexpr char Input[] = "kagome_input.json";
+  static constexpr int expected_num_symmetries = 1;
+  static std::vector<std::string> expectedFailingReps() { return {}; }
+  static std::vector<int> expectedFailingKOps() { return {}; }
+  static std::vector<int> expectedFailingROps() { return {}; }
+  static std::vector<int> expectedUnverifiedKOps() { return {}; }
+  static constexpr int expected_num_derived_symmetries = 2;
 };
 
 // Templated test fixture
@@ -514,6 +537,48 @@ TYPED_TEST(SymmetrizeCharacterizationTest, GroupShadowCrossCheck) {
          "sign-consistency check rejects changed vs the expectation.";
 }
 
+// TEST 5c: DerivedGroupReport -- the production derive-and-report step. update_domains now derives
+// each 2D model's point group from scratch (2D holohedry pool -> geometry filter -> H0 gate),
+// reports any divergence from the declared group, and keeps symmetrizing with the declared group.
+// This exercises the same routine directly and asserts the three things production relies on:
+// (1) the declared symmetry state is restored, so the derivation is invisible downstream
+// (2) the report is silent when derived == declared, and names the divergence otherwise
+// (3) the derived-group size matches the trait expectation
+TYPED_TEST(SymmetrizeCharacterizationTest, DerivedGroupReport) {
+  using Fixture = SymmetrizeCharacterizationTest<TypeParam>;
+  using RClusterDmn = typename Fixture::RClusterDmn;
+  using KCluster = typename Fixture::KCluster;
+  using Model = typename Fixture::Model;
+  using CS_k = dca::phys::domains::cluster_symmetry<KCluster>;
+
+  const int n_ops_before = Fixture::KSymDmn::dmn_size();
+  const auto symmetry_matrix_before = CS_k::get_symmetry_matrix();
+
+  const std::string report =
+      dca::phys::deriveAndComparePointGroup<RClusterDmn, Model>(this->parameters_);
+
+  // (1) restoration: production must be left in exactly the declared state.
+  ASSERT_EQ(Fixture::KSymDmn::dmn_size(), n_ops_before);
+  const auto& symmetry_matrix_after = CS_k::get_symmetry_matrix();
+  for (int i = 0; i < symmetry_matrix_before.size(); ++i)
+    ASSERT_EQ(symmetry_matrix_after(i), symmetry_matrix_before(i))
+        << "symmetry table changed at linear index " << i;
+
+  // (2) + (3): silent on match; on divergence the report names the derived-group size and the
+  // under-declaration.
+  if (TypeParam::expected_num_derived_symmetries == TypeParam::expected_num_symmetries) {
+    EXPECT_TRUE(report.empty()) << "unexpected divergence report:\n" << report;
+  }
+  else {
+    const std::string expected_size = "derived group: " +
+                                      std::to_string(TypeParam::expected_num_derived_symmetries) +
+                                      " op(s)";
+    EXPECT_NE(report.find(expected_size), std::string::npos) << report;
+    EXPECT_NE(report.find("under-declared"), std::string::npos) << report;
+    std::cout << "[derived group] " << TypeParam::Input << ":\n" << report;
+  }
+}
+
 // TEST 6: MappedPointShadow -- the mapped_point accessor. The band-independent ".first" of the
 // legacy symmetry matrix (the image of each momentum under each op) is promoted into its own
 // (cluster-point x op) accessor. Verify it reproduces .first for every band, which confirms both
@@ -606,11 +671,12 @@ TYPED_TEST(SymmetrizeCharacterizationTest, PerOpMapRealSpace) {
       << "Real-space per-op red/green map changed vs the expected failing set.";
 }
 
-// TEST 8 (a/b/c): rejection fixtures for the sign-consistency solver.
+// TEST 8 (a/b/c/d): rejection fixtures for the sign-consistency solver.
 //
-// solveOrbitalOpSignsFromH0 has three throw branches. No shipped model exercises any of the
-// three on a correct H0 -- the spine cases all pass -- so we perturb a copy of the (exactly
-// symmetric) H0 to break one operation and assert the solver throws for the intended reason.
+// solveOrbitalOpSignsFromH0 has four throw branches. No shipped model exercises any of the
+// four on a correct H0 and a complete symmetry table -- the spine cases all pass -- so we
+// perturb a copy of the (exactly symmetric) H0 (8a-8c), or the table itself (8d), to break one
+// operation and assert the solver throws for the intended reason.
 //
 // These need a multi-orbital model. A single band has no off-diagonal coupling to corrupt,
 // and on the spine's square cluster every nonzero single-band entry sits at a k that is a
@@ -640,6 +706,29 @@ TYPED_TEST(SymmetrizeCharacterizationTest, RejectsBrokenOffDiagonalCoupling) {
   const auto t = findStrongestOffDiagonal(this->H0_, nb, nk);
   ASSERT_GT(t.mag, kTolerance) << "no off-diagonal coupling to perturb.";
 
+  // The perturbation is only visible to an op that maps the corrupted entry to a *different*
+  // (unperturbed) H0 entry. Both the entry and its Hermitian partner are perturbed, so an op
+  // landing on either sees a consistent H0 and cannot throw. An identity-only group (e.g. a
+  // multi-band lattice declared with no_symmetry) therefore has nothing to detect: skip.
+  {
+    const auto& sym = dca::phys::domains::cluster_symmetry<KCluster>::get_symmetry_matrix();
+    const int n_ops = Fixture::KSymDmn::dmn_size();
+    bool movable = false;
+    for (int s = 0; s < n_ops && !movable; ++s) {
+      const int k_new = sym(t.k, 0, s).first;
+      const int b0_new = sym(0, t.b0, s).second;
+      const int b1_new = sym(0, t.b1, s).second;
+      const bool fixes = k_new == t.k && b0_new == t.b0 && b1_new == t.b1;
+      const bool hermitian_partner = k_new == t.k && b0_new == t.b1 && b1_new == t.b0;
+      movable = !fixes && !hermitian_partner;
+    }
+    if (!movable) {
+      std::cout << "[skip] " << TypeParam::Input
+                << ": no op moves the corrupted coupling to an unperturbed entry.\n";
+      return;
+    }
+  }
+
   // Vanish: the op connecting this entry to its (still nonzero) image sees it gone on one side only.
   {
     auto H0p = this->H0_;
@@ -765,4 +854,36 @@ TEST(SolveOrbitalOpSigns, BoundaryFoldDoesNotMaskPxParity) {
   EXPECT_DOUBLE_EQ(u_s(2, 2, 0), 1.);
 }
 
+// 8d -- missing band image: the geometric position/flavor matching in set_symmetry_matrices
+// records -1 when it finds no admissible image, so the solver cannot assume a complete table.
+// Unlike 8a-8c this perturbs the symmetry table, not H0: the whole-group populator must throw,
+// and the non-throwing classifier must omit the op instead of indexing H0 with -1 (an OOB). The
+// table is shared static state, so it gets restored.
+TYPED_TEST(SymmetrizeCharacterizationTest, RejectsMissingBandImage) {
+  using Fixture = SymmetrizeCharacterizationTest<TypeParam>;
+  using KCluster = typename Fixture::KCluster;
+
+  const std::vector<int> baseline = dca::phys::verifiedSymmetryOps<KCluster>(this->H0_);
+  ASSERT_FALSE(baseline.empty()) << "no verified op to corrupt (identity should always pass).";
+  const int s_target = baseline.front();
+
+  auto& sym = dca::phys::domains::cluster_symmetry<KCluster>::get_symmetry_matrix();
+  const int saved_image = sym(0, 0, s_target).second;
+  sym(0, 0, s_target).second = -1;
+
+  const std::string msg =
+      captureThrowMessage([&] { dca::phys::solveOrbitalOpSignsFromH0<KCluster>(this->H0_); });
+  const std::vector<int> verified = dca::phys::verifiedSymmetryOps<KCluster>(this->H0_);
+
+  sym(0, 0, s_target).second = saved_image;
+
+  ASSERT_FALSE(msg.empty()) << "sign solver did not reject a -1 sentinel band image.";
+  EXPECT_NE(msg.find("no band image"), std::string::npos) << "actual: " << msg;
+
+  std::vector<int> expected = baseline;
+  expected.erase(std::find(expected.begin(), expected.end(), s_target));
+  EXPECT_EQ(verified, expected)
+      << "classifier did not cleanly omit (only) the op with the sentinel image.";
+}
+
 }  // namespace
diff --git a/test/integration/statistical_tests/bilayer_lattice/bilayer_lattice_setup.hpp b/test/integration/statistical_tests/bilayer_lattice/bilayer_lattice_setup.hpp
index 2632fb73d..8d3c4c328 100644
--- a/test/integration/statistical_tests/bilayer_lattice/bilayer_lattice_setup.hpp
+++ b/test/integration/statistical_tests/bilayer_lattice/bilayer_lattice_setup.hpp
@@ -25,7 +25,7 @@
 #include "dca/phys/dca_step/cluster_solver/ctaux/ctaux_cluster_solver.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctint/ctint_cluster_solver.hpp"
 #include "dca/phys/dca_step/cluster_solver/stdthread_qmci/stdthread_qmci_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/math/random/random.hpp"
 #include "dca/math/statistical_testing/function_cut.hpp"
 #include "dca/math/statistical_testing/statistical_testing.hpp"
diff --git a/test/integration/statistical_tests/rashba_lattice/rashba_lattice_setup.hpp b/test/integration/statistical_tests/rashba_lattice/rashba_lattice_setup.hpp
index 05ff0e97d..9e414b258 100644
--- a/test/integration/statistical_tests/rashba_lattice/rashba_lattice_setup.hpp
+++ b/test/integration/statistical_tests/rashba_lattice/rashba_lattice_setup.hpp
@@ -29,7 +29,7 @@
 #include "dca/phys/dca_step/cluster_mapping/double_counting_correction.hpp"
 #include "dca/phys/dca_step/cluster_mapping/update_chemical_potential.hpp"
 #include "dca/phys/dca_step/cluster_solver/stdthread_qmci/stdthread_qmci_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/math/random/random.hpp"
 #include "dca/phys/dca_step/cluster_mapping/update_chemical_potential.hpp"
 #include "dca/math/statistical_testing/function_cut.hpp"
diff --git a/test/integration/statistical_tests/square_lattice/square_lattice_setup.hpp b/test/integration/statistical_tests/square_lattice/square_lattice_setup.hpp
index d3b84bde4..d485cd7c9 100644
--- a/test/integration/statistical_tests/square_lattice/square_lattice_setup.hpp
+++ b/test/integration/statistical_tests/square_lattice/square_lattice_setup.hpp
@@ -25,7 +25,7 @@
 #include "dca/phys/dca_step/cluster_solver/ctaux/ctaux_cluster_solver.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctint/ctint_cluster_solver.hpp"
 #include "dca/phys/dca_step/cluster_solver/stdthread_qmci/stdthread_qmci_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/math/random/random.hpp"
 #include "dca/math/statistical_testing/function_cut.hpp"
 #include "dca/math/statistical_testing/statistical_testing.hpp"
diff --git a/test/performance/phys/accumulation/sp_accumulation_performance_test.cpp b/test/performance/phys/accumulation/sp_accumulation_performance_test.cpp
index cc7be41f5..8328f887d 100644
--- a/test/performance/phys/accumulation/sp_accumulation_performance_test.cpp
+++ b/test/performance/phys/accumulation/sp_accumulation_performance_test.cpp
@@ -34,7 +34,7 @@ using Profiler = dca::profiling::CountingProfiler<dca::profiling::time_event<std
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
 #include "dca/parallel/no_threading/no_threading.hpp"
 #include "dca/phys/dca_data/dca_data.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/bilayer_lattice.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/parameters/parameters.hpp"
diff --git a/test/performance/phys/accumulation/tp_accumulation_performance_test.cpp b/test/performance/phys/accumulation/tp_accumulation_performance_test.cpp
index 7cf9b55ac..d5be6befd 100644
--- a/test/performance/phys/accumulation/tp_accumulation_performance_test.cpp
+++ b/test/performance/phys/accumulation/tp_accumulation_performance_test.cpp
@@ -34,7 +34,7 @@ using Profiler = dca::profiling::CountingProfiler<dca::profiling::time_event<std
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
 #include "dca/parallel/no_threading/no_threading.hpp"
 #include "dca/phys/dca_data/dca_data.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/bilayer_lattice.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/parameters/parameters.hpp"
diff --git a/test/performance/phys/ctaux/ctaux_walker_performance_test.cpp b/test/performance/phys/ctaux/ctaux_walker_performance_test.cpp
index 2462d0ad7..b7c34d919 100644
--- a/test/performance/phys/ctaux/ctaux_walker_performance_test.cpp
+++ b/test/performance/phys/ctaux/ctaux_walker_performance_test.cpp
@@ -28,7 +28,7 @@ using McOptions = MockMcOptions<double>;
 #include "dca/io/json/json_reader.hpp"
 #include "dca/math/random/std_random_wrapper.hpp"
 #include "dca/phys/dca_data/dca_data.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
diff --git a/test/performance/phys/ctint/ctint_walker_performance_test.cpp b/test/performance/phys/ctint/ctint_walker_performance_test.cpp
index e58793e58..d3db7fcd9 100644
--- a/test/performance/phys/ctint/ctint_walker_performance_test.cpp
+++ b/test/performance/phys/ctint/ctint_walker_performance_test.cpp
@@ -30,7 +30,7 @@ using McOptions = MockMcOptions<double>;
 }  // namespace dca
 
 #include "dca/phys/dca_step/cluster_solver/ctint/walker/ctint_walker_choice.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/bilayer_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
diff --git a/test/system-level/analysis/dca_post_interpolation_pp_up_down/analysis_dca_post_interpolation_pp_up_down_test.cpp b/test/system-level/analysis/dca_post_interpolation_pp_up_down/analysis_dca_post_interpolation_pp_up_down_test.cpp
index 641ba3111..1a9df8028 100644
--- a/test/system-level/analysis/dca_post_interpolation_pp_up_down/analysis_dca_post_interpolation_pp_up_down_test.cpp
+++ b/test/system-level/analysis/dca_post_interpolation_pp_up_down/analysis_dca_post_interpolation_pp_up_down_test.cpp
@@ -42,7 +42,7 @@ using McOptions = MockMcOptions<Scalar>;
 
 #include "dca/phys/dca_analysis/bse_solver/bse_solver.hpp"
 #include "dca/phys/dca_data/dca_data.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/phys/parameters/parameters.hpp"
diff --git a/test/system-level/analysis/dca_pp_up_down_full/analysis_dca_pp_up_down_full_test.cpp b/test/system-level/analysis/dca_pp_up_down_full/analysis_dca_pp_up_down_full_test.cpp
index 367483335..202c41f37 100644
--- a/test/system-level/analysis/dca_pp_up_down_full/analysis_dca_pp_up_down_full_test.cpp
+++ b/test/system-level/analysis/dca_pp_up_down_full/analysis_dca_pp_up_down_full_test.cpp
@@ -42,7 +42,7 @@ using McOptions = MockMcOptions<Scalar>;
 
 #include "dca/phys/dca_analysis/bse_solver/bse_solver.hpp"
 #include "dca/phys/dca_data/dca_data.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/phys/parameters/parameters.hpp"
diff --git a/test/system-level/analysis/dca_pp_up_down_sym/analysis_dca_pp_up_down_sym_test.cpp b/test/system-level/analysis/dca_pp_up_down_sym/analysis_dca_pp_up_down_sym_test.cpp
index 2c177a041..c54db984e 100644
--- a/test/system-level/analysis/dca_pp_up_down_sym/analysis_dca_pp_up_down_sym_test.cpp
+++ b/test/system-level/analysis/dca_pp_up_down_sym/analysis_dca_pp_up_down_sym_test.cpp
@@ -40,7 +40,7 @@ using McOptions = MockMcOptions<Scalar>;
 
 #include "dca/phys/dca_analysis/bse_solver/bse_solver.hpp"
 #include "dca/phys/dca_data/dca_data.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/phys/parameters/parameters.hpp"
diff --git a/test/system-level/analysis/dcaplus_pp_up_down_full/analysis_dcaplus_pp_up_down_full_test.cpp b/test/system-level/analysis/dcaplus_pp_up_down_full/analysis_dcaplus_pp_up_down_full_test.cpp
index b74cc6f33..054c2e063 100644
--- a/test/system-level/analysis/dcaplus_pp_up_down_full/analysis_dcaplus_pp_up_down_full_test.cpp
+++ b/test/system-level/analysis/dcaplus_pp_up_down_full/analysis_dcaplus_pp_up_down_full_test.cpp
@@ -42,7 +42,7 @@ using McOptions = MockMcOptions<Scalar>;
 
 #include "dca/phys/dca_analysis/bse_solver/bse_solver.hpp"
 #include "dca/phys/dca_data/dca_data.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/phys/parameters/parameters.hpp"
diff --git a/test/system-level/analysis/dcaplus_pp_up_down_sym/analysis_dcaplus_pp_up_down_sym_test.cpp b/test/system-level/analysis/dcaplus_pp_up_down_sym/analysis_dcaplus_pp_up_down_sym_test.cpp
index c6591b8cd..cd5c54d61 100644
--- a/test/system-level/analysis/dcaplus_pp_up_down_sym/analysis_dcaplus_pp_up_down_sym_test.cpp
+++ b/test/system-level/analysis/dcaplus_pp_up_down_sym/analysis_dcaplus_pp_up_down_sym_test.cpp
@@ -40,7 +40,7 @@ using McOptions = MockMcOptions<Scalar>;
 
 #include "dca/phys/dca_analysis/bse_solver/bse_solver.hpp"
 #include "dca/phys/dca_data/dca_data.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/phys/parameters/parameters.hpp"
diff --git a/test/system-level/dca/ctint_sp_DCA_mpi_thread_test.cpp b/test/system-level/dca/ctint_sp_DCA_mpi_thread_test.cpp
index e4d397e1b..7a34b0a21 100644
--- a/test/system-level/dca/ctint_sp_DCA_mpi_thread_test.cpp
+++ b/test/system-level/dca/ctint_sp_DCA_mpi_thread_test.cpp
@@ -36,7 +36,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/phys/dca_loop/dca_loop.hpp"
 #include "dca/phys/dca_data/dca_data.hpp"
 #include "dca/phys/dca_step/cluster_solver/stdthread_qmci/stdthread_qmci_cluster_solver.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/phys/parameters/parameters.hpp"
diff --git a/test/system-level/dca/dca_sp_DCA+_mpi_cuda_test.cpp b/test/system-level/dca/dca_sp_DCA+_mpi_cuda_test.cpp
index 5ef504b71..b05250392 100644
--- a/test/system-level/dca/dca_sp_DCA+_mpi_cuda_test.cpp
+++ b/test/system-level/dca/dca_sp_DCA+_mpi_cuda_test.cpp
@@ -40,7 +40,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/phys/dca_loop/dca_loop.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctaux/ctaux_cluster_solver.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/phys/parameters/parameters.hpp"
diff --git a/test/system-level/dca/dca_sp_DCA+_mpi_test.cpp b/test/system-level/dca/dca_sp_DCA+_mpi_test.cpp
index 1d15396d6..d18e42508 100644
--- a/test/system-level/dca/dca_sp_DCA+_mpi_test.cpp
+++ b/test/system-level/dca/dca_sp_DCA+_mpi_test.cpp
@@ -41,7 +41,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/phys/dca_loop/dca_loop.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctaux/ctaux_cluster_solver.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/phys/parameters/parameters.hpp"
diff --git a/test/system-level/dca/dca_sp_DCA+_thread_test.cpp b/test/system-level/dca/dca_sp_DCA+_thread_test.cpp
index 53dc07d1a..eef8e76e4 100644
--- a/test/system-level/dca/dca_sp_DCA+_thread_test.cpp
+++ b/test/system-level/dca/dca_sp_DCA+_thread_test.cpp
@@ -41,7 +41,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/phys/dca_step/cluster_solver/stdthread_qmci/stdthread_qmci_cluster_solver.hpp"
 
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/quantum/electron_band_domain.hpp"
 #include "dca/phys/domains/quantum/electron_spin_domain.hpp"
 #include "dca/phys/domains/time_and_frequency/frequency_domain.hpp"
diff --git a/test/unit/math/function_transform/function_transform_test.cpp b/test/unit/math/function_transform/function_transform_test.cpp
index 6a1f68dba..f9354754b 100644
--- a/test/unit/math/function_transform/function_transform_test.cpp
+++ b/test/unit/math/function_transform/function_transform_test.cpp
@@ -16,7 +16,7 @@
 
 #include "dca/io/json/json_reader.hpp"
 #include "dca/io/hdf5/hdf5_reader.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 
 #include "test/mock_mcconfig.hpp"
 namespace dca {
diff --git a/test/unit/phys/dca_algorithms/compute_band_structure_test.cpp b/test/unit/phys/dca_algorithms/compute_band_structure_test.cpp
index f1b236bb4..e98fd127d 100644
--- a/test/unit/phys/dca_algorithms/compute_band_structure_test.cpp
+++ b/test/unit/phys/dca_algorithms/compute_band_structure_test.cpp
@@ -22,7 +22,7 @@
 #include "dca/parallel/no_threading/no_threading.hpp"
 #include "dca/profiling/null_profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/cluster_solver_id.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/quantum/brillouin_zone_cut_domain.hpp"
 #include "dca/phys/domains/quantum/electron_band_domain.hpp"
 #include "dca/phys/domains/quantum/electron_spin_domain.hpp"
diff --git a/test/unit/phys/dca_algorithms/compute_free_greens_function_test.cpp b/test/unit/phys/dca_algorithms/compute_free_greens_function_test.cpp
index 03117e75e..c572e99b7 100644
--- a/test/unit/phys/dca_algorithms/compute_free_greens_function_test.cpp
+++ b/test/unit/phys/dca_algorithms/compute_free_greens_function_test.cpp
@@ -21,7 +21,7 @@
 #include "dca/function/domains.hpp"
 #include "dca/function/function.hpp"
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/bilayer_lattice.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
diff --git a/test/unit/phys/dca_algorithms/compute_greens_function_test.cpp b/test/unit/phys/dca_algorithms/compute_greens_function_test.cpp
index 9f485b0c7..92577e123 100644
--- a/test/unit/phys/dca_algorithms/compute_greens_function_test.cpp
+++ b/test/unit/phys/dca_algorithms/compute_greens_function_test.cpp
@@ -21,7 +21,7 @@
 #include "dca/function/function.hpp"
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
 #include "dca/phys/dca_algorithms/compute_free_greens_function.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/bilayer_lattice.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
diff --git a/test/unit/phys/dca_step/cluster_solver/ctint/ct_int_solver_methods_test.cpp b/test/unit/phys/dca_step/cluster_solver/ctint/ct_int_solver_methods_test.cpp
index ac8802612..6508d3fd0 100644
--- a/test/unit/phys/dca_step/cluster_solver/ctint/ct_int_solver_methods_test.cpp
+++ b/test/unit/phys/dca_step/cluster_solver/ctint/ct_int_solver_methods_test.cpp
@@ -14,7 +14,7 @@ using McOptions = MockMcOptions<Scalar>;
 #include "dca/phys/dca_step/cluster_solver/ctint/ctint_cluster_solver.hpp"
 #include "dca/testing/gtest_h_w_warning_blocking.h"
 #include "test/unit/phys/dca_step/cluster_solver/stub_rng.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
 #include "dca/parallel/no_threading/no_threading.hpp"
diff --git a/test/unit/phys/dca_step/cluster_solver/test_setup.hpp b/test/unit/phys/dca_step/cluster_solver/test_setup.hpp
index 2cf8821e3..0cbc408f9 100644
--- a/test/unit/phys/dca_step/cluster_solver/test_setup.hpp
+++ b/test/unit/phys/dca_step/cluster_solver/test_setup.hpp
@@ -21,7 +21,7 @@
 
 #include "dca/phys/parameters/parameters.hpp"
 #include "dca/phys/dca_step/cluster_solver/ctint/structs/interaction_vertices.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/quantum/electron_band_domain.hpp"
 #include "dca/phys/domains/quantum/electron_spin_domain.hpp"
 #include "dca/phys/domains/time_and_frequency/frequency_domain.hpp"
diff --git a/test/unit/phys/dca_step/lattice_mapping/deconvolution/deconvolution_routines_test.cpp b/test/unit/phys/dca_step/lattice_mapping/deconvolution/deconvolution_routines_test.cpp
index 1c59a3bed..524255a93 100644
--- a/test/unit/phys/dca_step/lattice_mapping/deconvolution/deconvolution_routines_test.cpp
+++ b/test/unit/phys/dca_step/lattice_mapping/deconvolution/deconvolution_routines_test.cpp
@@ -19,7 +19,7 @@
 #include "dca/parallel/no_concurrency/no_concurrency.hpp"
 #include "dca/profiling/null_profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/cluster_solver_id.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 
diff --git a/test/unit/phys/domains/cluster/CMakeLists.txt b/test/unit/phys/domains/cluster/CMakeLists.txt
index 821b36155..716421069 100644
--- a/test/unit/phys/domains/cluster/CMakeLists.txt
+++ b/test/unit/phys/domains/cluster/CMakeLists.txt
@@ -13,5 +13,7 @@ dca_add_gtest(cluster_domain_test
   GTEST_MAIN
   LIBS ${LAPACK_LIBRARIES} function enumerations quantum_domains cluster_domains dca_io FFTW::Double ${DCA_GPU_LIBS})
 
+add_subdirectory(symmetries)
+
 # deprecated (requires NFFT)
 # add_subdirectory(interpolation/wannier_interpolation)
diff --git a/test/unit/phys/domains/cluster/interpolation/wannier_interpolation/wannier_interpolation_test.cpp b/test/unit/phys/domains/cluster/interpolation/wannier_interpolation/wannier_interpolation_test.cpp
index 2acf8a390..b4d351d43 100644
--- a/test/unit/phys/domains/cluster/interpolation/wannier_interpolation/wannier_interpolation_test.cpp
+++ b/test/unit/phys/domains/cluster/interpolation/wannier_interpolation/wannier_interpolation_test.cpp
@@ -22,7 +22,7 @@
 #include "dca/parallel/no_threading/no_threading.hpp"
 #include "dca/profiling/null_profiler.hpp"
 #include "dca/phys/dca_step/cluster_solver/cluster_solver_name.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 #include "dca/phys/models/tight_binding_model.hpp"
 #include "dca/phys/parameters/parameters.hpp"
diff --git a/test/unit/phys/domains/cluster/symmetries/CMakeLists.txt b/test/unit/phys/domains/cluster/symmetries/CMakeLists.txt
new file mode 100644
index 000000000..1361082fb
--- /dev/null
+++ b/test/unit/phys/domains/cluster/symmetries/CMakeLists.txt
@@ -0,0 +1,5 @@
+# domains/cluster/symmetries unit tests
+
+dca_add_gtest(holohedries_2d_test
+  GTEST_MAIN
+  LIBS ${DCA_GPU_LIBS})
diff --git a/test/unit/phys/domains/cluster/symmetries/holohedries_2d_test.cpp b/test/unit/phys/domains/cluster/symmetries/holohedries_2d_test.cpp
new file mode 100644
index 000000000..a2dfc7618
--- /dev/null
+++ b/test/unit/phys/domains/cluster/symmetries/holohedries_2d_test.cpp
@@ -0,0 +1,146 @@
+// Copyright (C) 2018 UT-Battelle, LLC
+// All rights reserved.
+//
+// See LICENSE for terms of usage.
+// See CITATION.md for citation guidelines, if DCA++ is used for scientific publications.
+//
+// Author: Tyler Sax
+//
+// Completeness oracle for the 2D holohedry candidate pool. Proves the pool is a complete
+// crystallographic op set: filtering the pool by each 2D Bravais lattice reproduces that
+// lattice's full holohedry (square -> D4 order 8, hexagonal -> D6 order 12, rectangular -> D2
+// order 4, oblique -> C2 order 2). Because every one of the ten 2D crystallographic point groups
+// is a subgroup of one of these holohedries, realizing all four maximal groups proves the pool
+// misses no op.
+
+#include <array>
+#include <cmath>
+#include <vector>
+
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
+
+#include "gtest/gtest.h"
+
+#include "dca/util/type_list.hpp"
+
+namespace {
+
+// A 2x2 op stored row-major: m[i][j].
+using Mat2 = std::array<std::array<double, 2>, 2>;
+
+constexpr double kTol = 1e-10;
+
+// The op classes store their matrix column-major (m[i + 2*j] = row i, col j); convert to nested
+// row-major in order to make the algebra more readable, and easier to verify.
+Mat2 fromColumnMajor(const double* m) {
+  return {{{m[0], m[2]}, {m[1], m[3]}}};
+}
+
+// Materialize a point_group_type_list into its op matrices. Recurses over the typelist by index so
+// the same helper serves any pool/group.
+template <class List, int I, int N>
+struct Collect {
+  static void run(std::vector<Mat2>& out) {
+    using Op = typename dca::util::TypeAt<I, List>::type;
+    out.push_back(fromColumnMajor(Op::matrix()));
+    Collect<List, I + 1, N>::run(out);
+  }
+};
+template <class List, int N>
+struct Collect<List, N, N> {
+  static void run(std::vector<Mat2>&) {}
+};
+
+template <class PointGroup>
+std::vector<Mat2> collectOps() {
+  using List = typename PointGroup::point_group_type_list;
+  std::vector<Mat2> out;
+  Collect<List, 0, dca::util::Length<List>::value>::run(out);
+  return out;
+}
+
+// Note that the algebra below is implemented directly in this test, as opposed to being imported
+// via the existing linalg path. Given this is a logic-only test operating on 2x2 matrices, hand
+// rolling makes more sense than adding a dependency on LAPACK/BLAS, which is overkill here.
+bool equal(const Mat2& a, const Mat2& b) {
+  for (int i = 0; i < 2; ++i)
+    for (int j = 0; j < 2; ++j)
+      if (std::abs(a[i][j] - b[i][j]) > kTol)
+        return false;
+  return true;
+}
+
+std::vector<Mat2> dedup(const std::vector<Mat2>& ops) {
+  std::vector<Mat2> out;
+  for (const auto& o : ops) {
+    bool seen = false;
+    for (const auto& k : out)
+      seen = seen || equal(o, k);
+    if (!seen)
+      out.push_back(o);
+  }
+  return out;
+}
+
+Mat2 matmul(const Mat2& a, const Mat2& b) {
+  Mat2 c{};
+  for (int i = 0; i < 2; ++i)
+    for (int j = 0; j < 2; ++j)
+      c[i][j] = a[i][0] * b[0][j] + a[i][1] * b[1][j];
+  return c;
+}
+
+double det(const Mat2& a) {
+  return a[0][0] * a[1][1] - a[0][1] * a[1][0];
+}
+
+Mat2 inverse(const Mat2& a) {
+  const double d = det(a);
+  return {{{a[1][1] / d, -a[0][1] / d}, {-a[1][0] / d, a[0][0] / d}}};
+}
+
+bool isInteger(const Mat2& a) {
+  for (int i = 0; i < 2; ++i)
+    for (int j = 0; j < 2; ++j)
+      if (std::abs(a[i][j] - std::round(a[i][j])) > 1e-8)
+        return false;
+  return true;
+}
+
+// Order of the holohedry of the lattice whose primitive vectors are the columns of `basis`: the
+// number of pool ops that map the lattice onto itself (verified by applying the operation to the
+// basis vectors, multiplying by the inverse of the basis, and checking whether the resulting
+// matrix has integer values.)
+int holohedryOrder(const std::vector<Mat2>& ops, const Mat2& basis) {
+  const Mat2 binv = inverse(basis);
+  int count = 0;
+  for (const auto& O : ops)
+    if (isInteger(matmul(binv, matmul(O, basis))))
+      ++count;
+  return count;
+}
+
+const double kHalfRoot3 = std::sqrt(3.0) / 2.0;
+
+}  // namespace
+
+// Completeness: filtering the pool by each 2D Bravais lattice yields that lattice's full holohedry.
+// Covering all four maximal groups (D4, D6, D2, C2) covers all ten 2D point groups as subgroups, so
+// this is the load-bearing check that the pool misses no op.
+//
+// The pool lists D4 (8) + D6 (12) = 20 ops; the shared {identity, C2, mirror@0, mirror@90} collapse
+// under dedup to 16 distinct ops, which is what the per-lattice filter runs over.
+TEST(HolohedryPool2DTest, RealizesEvery2DHolohedry) {
+  const auto ops = dedup(collectOps<dca::phys::domains::holohedry_pool_2D>());
+  ASSERT_EQ(ops.size(), 16u);  // 20 listed, 16 distinct after the D4/D6 overlap collapses
+
+  const Mat2 square{{{1, 0}, {0, 1}}};
+  const Mat2 hexagonal{{{1, -0.5}, {0, kHalfRoot3}}};
+  const Mat2 rectangular{{{1, 0}, {0, 2}}};
+  const Mat2 oblique{{{1, 0}, {0.3, 1.7}}};
+
+  EXPECT_EQ(holohedryOrder(ops, square), 8);       // D4
+  EXPECT_EQ(holohedryOrder(ops, hexagonal), 12);   // D6
+  EXPECT_EQ(holohedryOrder(ops, rectangular), 4);  // D2
+  EXPECT_EQ(holohedryOrder(ops, oblique), 2);      // C2
+}
diff --git a/test/unit/phys/models/analytic_hamiltonians/bilayer_lattice_test.cpp b/test/unit/phys/models/analytic_hamiltonians/bilayer_lattice_test.cpp
index 41c0ae3d3..9e1d3b353 100644
--- a/test/unit/phys/models/analytic_hamiltonians/bilayer_lattice_test.cpp
+++ b/test/unit/phys/models/analytic_hamiltonians/bilayer_lattice_test.cpp
@@ -22,7 +22,7 @@
 #include "dca/function/function.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
 #include "dca/phys/domains/cluster/cluster_domain_initializer.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/cluster/cluster_domain_aliases.hpp"
 #include "dca/phys/parameters/model_parameters.hpp"
 
diff --git a/test/unit/phys/models/analytic_hamiltonians/square_lattice_test.cpp b/test/unit/phys/models/analytic_hamiltonians/square_lattice_test.cpp
index 2ae7c7665..04683e1f4 100644
--- a/test/unit/phys/models/analytic_hamiltonians/square_lattice_test.cpp
+++ b/test/unit/phys/models/analytic_hamiltonians/square_lattice_test.cpp
@@ -22,7 +22,7 @@
 #include "dca/function/function.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
 #include "dca/phys/domains/cluster/cluster_domain_initializer.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/cluster/cluster_domain_aliases.hpp"
 #include "dca/phys/parameters/model_parameters.hpp"
 
diff --git a/test/unit/phys/models/analytic_hamiltonians/triangular_lattice_test.cpp b/test/unit/phys/models/analytic_hamiltonians/triangular_lattice_test.cpp
index 0ef7e3431..ff9d9f8ae 100644
--- a/test/unit/phys/models/analytic_hamiltonians/triangular_lattice_test.cpp
+++ b/test/unit/phys/models/analytic_hamiltonians/triangular_lattice_test.cpp
@@ -22,14 +22,14 @@
 #include "dca/function/function.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
 #include "dca/phys/domains/cluster/cluster_domain_initializer.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_hexagonal.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/domains/cluster/cluster_domain_aliases.hpp"
 #include "dca/phys/parameters/model_parameters.hpp"
 
 using namespace dca;
 
 TEST(TriangularLatticeTest, Initialize_H_0) {
-  using PointGroup = phys::domains::C6;
+  using PointGroup = phys::domains::D6;
   using Lattice = phys::models::triangular_lattice<PointGroup>;
 
   using BandDmn = func::dmn<1, int>;
@@ -77,7 +77,7 @@ TEST(TriangularLatticeTest, Initialize_H_0) {
 }
 
 TEST(TriangularLatticeTest, Initialize_H_interaction) {
-  using PointGroup = phys::domains::C6;
+  using PointGroup = phys::domains::D6;
   using Lattice = phys::models::triangular_lattice<PointGroup>;
 
   using BandDmn = func::dmn_0<func::dmn<1, int>>;
diff --git a/test/unit/phys/models/general_interaction_test.cpp b/test/unit/phys/models/general_interaction_test.cpp
index a303fc624..07aec3601 100644
--- a/test/unit/phys/models/general_interaction_test.cpp
+++ b/test/unit/phys/models/general_interaction_test.cpp
@@ -17,7 +17,7 @@
 #include "dca/function/function.hpp"
 #include "dca/phys/domains/cluster/cluster_domain.hpp"
 #include "dca/phys/domains/cluster/cluster_domain_initializer.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/bilayer_lattice.hpp"
 #include "dca/phys/domains/cluster/cluster_domain_aliases.hpp"
 #include "dca/phys/parameters/model_parameters.hpp"
diff --git a/test/unit/phys/parameters/model_parameters/model_parameters_2d_2band_test.cpp b/test/unit/phys/parameters/model_parameters/model_parameters_2d_2band_test.cpp
index 2482eeb01..0b2f5c4e8 100644
--- a/test/unit/phys/parameters/model_parameters/model_parameters_2d_2band_test.cpp
+++ b/test/unit/phys/parameters/model_parameters/model_parameters_2d_2band_test.cpp
@@ -14,7 +14,7 @@
 #include "dca/phys/parameters/model_parameters.hpp"
 #include "gtest/gtest.h"
 #include "dca/io/json/json_reader.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 
 using PointGroup = dca::phys::domains::D4;
 
diff --git a/test/unit/phys/parameters/model_parameters/model_parameters_2d_4band_test.cpp b/test/unit/phys/parameters/model_parameters/model_parameters_2d_4band_test.cpp
index 4ff63e190..361c59c51 100644
--- a/test/unit/phys/parameters/model_parameters/model_parameters_2d_4band_test.cpp
+++ b/test/unit/phys/parameters/model_parameters/model_parameters_2d_4band_test.cpp
@@ -14,7 +14,7 @@
 #include "dca/phys/parameters/model_parameters.hpp"
 #include "gtest/gtest.h"
 #include "dca/io/json/json_reader.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 
 using PointGroup = dca::phys::domains::D4;
 
diff --git a/test/unit/phys/parameters/model_parameters/model_parameters_bilayer_hubbard_test.cpp b/test/unit/phys/parameters/model_parameters/model_parameters_bilayer_hubbard_test.cpp
index 640ff77ac..bc6fe528f 100644
--- a/test/unit/phys/parameters/model_parameters/model_parameters_bilayer_hubbard_test.cpp
+++ b/test/unit/phys/parameters/model_parameters/model_parameters_bilayer_hubbard_test.cpp
@@ -16,7 +16,7 @@
 #include "dca/phys/parameters/model_parameters.hpp"
 #include "gtest/gtest.h"
 #include "dca/io/json/json_reader.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 
 using PointGroup = dca::phys::domains::D4;
 
diff --git a/test/unit/phys/parameters/model_parameters/model_parameters_material_test.cpp b/test/unit/phys/parameters/model_parameters/model_parameters_material_test.cpp
index 8dad1373d..cc7d8f7db 100644
--- a/test/unit/phys/parameters/model_parameters/model_parameters_material_test.cpp
+++ b/test/unit/phys/parameters/model_parameters/model_parameters_material_test.cpp
@@ -16,7 +16,7 @@
 #include "dca/testing/gtest_h_w_warning_blocking.h"
 
 #include "dca/io/json/json_reader.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 
 using PointGroup = dca::phys::domains::D4;
 
diff --git a/test/unit/phys/parameters/model_parameters/model_parameters_rashba_hubbard_test.cpp b/test/unit/phys/parameters/model_parameters/model_parameters_rashba_hubbard_test.cpp
index 439579ce5..807d327f2 100644
--- a/test/unit/phys/parameters/model_parameters/model_parameters_rashba_hubbard_test.cpp
+++ b/test/unit/phys/parameters/model_parameters/model_parameters_rashba_hubbard_test.cpp
@@ -16,7 +16,7 @@
 #include "dca/phys/parameters/model_parameters.hpp"
 #include "dca/testing/gtest_h_w_warning_blocking.h"
 #include "dca/io/json/json_reader.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/rashba_hubbard.hpp"
 
 using Lattice = dca::phys::models::RashbaHubbard<dca::phys::domains::no_symmetry<2>>;
diff --git a/test/unit/phys/parameters/model_parameters/model_parameters_single_band_hubbard_test.cpp b/test/unit/phys/parameters/model_parameters/model_parameters_single_band_hubbard_test.cpp
index 7ff326ca8..a9e86574e 100644
--- a/test/unit/phys/parameters/model_parameters/model_parameters_single_band_hubbard_test.cpp
+++ b/test/unit/phys/parameters/model_parameters/model_parameters_single_band_hubbard_test.cpp
@@ -15,7 +15,7 @@
 #include "dca/phys/parameters/model_parameters.hpp"
 #include "dca/testing/gtest_h_w_warning_blocking.h"
 #include "dca/io/json/json_reader.hpp"
-#include "dca/phys/domains/cluster/symmetries/point_groups/2d/2d_square.hpp"
+#include "dca/phys/domains/cluster/symmetries/point_groups/2d/holohedries_2d.hpp"
 #include "dca/phys/models/analytic_hamiltonians/square_lattice.hpp"
 
 using PointGroup = dca::phys::domains::D4;