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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1944.2451', 'ambient_counter': 2451, 'ambient_order': 1944, 'ambient_tex': '\\He_3:(C_3\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 972, 'characteristic': True, 'core_order': 3, 'counter': 277, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1944.2451.648.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '648.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '648.572', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 572, 'quotient_metabelian': False, 'quotient_nilpotent': False, 'quotient_order': 648, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'C_3^3:S_4', 'simple': True, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1944.2451', 'aut_centralizer_order': 1944, 'aut_label': '648.a1', 'aut_quo_index': 18, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 1944, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['216.a1.a1', '216.b1.a1', '216.c1.a1', '216.c1.b1', '216.d1.a1', '216.e1.a1', '216.f1.a1', '216.f1.b1', '216.g1.a1', '216.g1.b1', '216.h1.a1', '216.i1.a1', '324.a1.a1', '324.m1.a1'], 'contains': ['1944.a1.a1'], 'core': '648.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6724, 2764, 6899, 3692, 3927, 4191, 4005, 4076], 'generators': [648], 'label': '1944.2451.648.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '648.a1.a1', 'normal_contained_in': ['162.a1.a1', '216.a1.a1', '216.b1.a1', '216.c1.a1', '216.c1.b1'], 'normal_contains': ['1944.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '648.a1.a1', 'projective_image': '1944.2451', 'quotient_action_image': '2.1', 'quotient_action_kernel': '324.135', 'quotient_action_kernel_order': 324, 'quotient_fusion': None, 'short_label': '648.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [12, 6, 12], 'aut_gens': [[1, 6, 18, 108, 648], [43, 1578, 1710, 1440, 1296], [1567, 96, 1314, 756, 648], [1885, 408, 1044, 954, 1296]], 'aut_group': None, 'aut_hash': 7837221543919875162, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3888, 'aut_permdeg': 54, 'aut_perms': [13493339075529514361578545336210800329962218527450205985987792245968814, 43709856919485060810192446956490523112785020894142931998129266956610536, 31291324045015865550686016687959818416735500113570131399368028140887900], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 54, 1, 1], [3, 2, 1, 1], [3, 3, 1, 2], [3, 3, 2, 3], [3, 18, 1, 3], [3, 72, 2, 3], [4, 54, 1, 1], [6, 3, 1, 2], [6, 6, 1, 3], [6, 9, 2, 3], [6, 18, 1, 3], [6, 18, 2, 3], [6, 54, 1, 2], [6, 54, 2, 3], [9, 72, 1, 3], [12, 54, 1, 2], [12, 54, 2, 3]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.C_3^3.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 7837221543919875162, 'autcentquo_nilpotent': False, 'autcentquo_order': 3888, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.C_3^3.C_2^2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 54, 1], [3, 2, 1], [3, 3, 8], [3, 18, 3], [3, 72, 6], [4, 54, 1], [6, 3, 2], [6, 6, 3], [6, 9, 6], [6, 18, 9], [6, 54, 8], [9, 72, 3], [12, 54, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1944.2451', 'commutator_count': 2, 'commutator_label': '324.54', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2451, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 54, 1, 1], [3, 2, 1, 1], [3, 3, 2, 4], [3, 18, 1, 1], [3, 18, 2, 1], [3, 72, 1, 2], [3, 72, 2, 2], [4, 54, 1, 1], [6, 3, 2, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 9, 2, 3], [6, 18, 1, 1], [6, 18, 2, 4], [6, 54, 2, 4], [9, 72, 1, 1], [9, 72, 2, 1], [12, 54, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 884520, 'exponent': 36, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[18, 0, 2], [18, 1, 1]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '216.164', 'hash': 2451, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 3, 6, 6, 3], 'inner_gens': [[1, 12, 18, 1242, 1296], [13, 6, 1692, 126, 648], [1, 1032, 18, 756, 648], [1567, 744, 1314, 108, 648], [1297, 6, 18, 108, 648]], 'inner_hash': 2451, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': True, 'inner_tex': '\\He_3:(C_3\\times S_4)', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 18, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 6], [2, 12], [3, 30], [6, 9], [18, 4]], 'label': '1944.2451', 'linC_count': 3, 'linC_degree': 18, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 1, 'linQ_dim': 18, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'He3:(C3*S4)', 'ngens': 8, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 43, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 61, 'number_divisions': 36, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 220, 'number_subgroup_classes': 290, 'number_subgroups': 3700, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 57], [3, 512], [4, 54], [6, 672], [9, 216], [12, 432]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [38], 'outer_gens': [[37, 1770, 90, 540, 648]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 31, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [3, 2], [4, 4], [6, 15], [12, 4], [18, 2], [36, 1]], 'representations': {'PC': {'code': '366333628811933168886800999391030898678361727106285106558213368171', 'gens': [1, 3, 4, 6, 8], 'pres': [8, 2, 3, 3, 2, 3, 2, 3, 3, 16, 290, 9043, 91, 4580, 59621, 18157, 1029, 2045, 1909, 141, 96774, 42350, 8422, 4734, 1382, 82951]}, 'Perm': {'d': 31, 'gens': [1917425416707642908724769203245, 293073972352247921312728928967960, 576645604848771247223566776687003, 815362678031561605210424338737480, 1063189376375803223040780493108320, 1381834815760255401384760593560880, 7, 16]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\He_3:(C_3\\times S_4)', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 6, 6, 6, 6], 'aut_gens': [[1, 6, 18, 108], [451, 24, 90, 540], [433, 228, 18, 558], [551, 186, 360, 558], [253, 150, 18, 108], [307, 420, 18, 108]], 'aut_group': '2592.jb', 'aut_hash': 6898096242996517164, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2592, 'aut_permdeg': 24, 'aut_perms': [69690952200231007002212, 76534936139814304535949, 160948946215292178707253, 13541764093129214228105, 10169710627794712883484], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 18, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 24, 3, 1], [3, 24, 6, 1], [4, 18, 1, 1], [6, 3, 2, 1], [6, 3, 6, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 6, 6, 1], [6, 18, 2, 1], [6, 18, 6, 1], [12, 18, 2, 1], [12, 18, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times C_6^2:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '432.523', 'autcentquo_hash': 523, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2:D_6', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 18, 1], [3, 1, 8], [3, 6, 3], [3, 24, 9], [4, 18, 1], [6, 3, 8], [6, 6, 9], [6, 18, 8], [12, 18, 8]], 'center_label': '9.2', 'center_order': 9, 'central_product': True, 'central_quotient': '72.43', 'commutator_count': 2, 'commutator_label': '108.22', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 572, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['216.95', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 18, 1, 1], [3, 1, 2, 4], [3, 6, 1, 1], [3, 6, 2, 1], [3, 24, 1, 3], [3, 24, 2, 3], [4, 18, 1, 1], [6, 3, 2, 4], [6, 6, 1, 1], [6, 6, 2, 4], [6, 18, 2, 4], [12, 18, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 49140, 'exponent': 12, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '216.164', 'hash': 572, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3, 2, 6], 'inner_gens': [[1, 12, 18, 558], [13, 6, 342, 450], [1, 330, 18, 108], [307, 420, 18, 108]], 'inner_hash': 43, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'C_3:S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 12], [3, 30], [6, 9]], 'label': '648.572', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3:S4', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 19, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 57, 'number_divisions': 33, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 96, 'number_subgroup_classes': 196, 'number_subgroups': 1160, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 21], [3, 242], [4, 18], [6, 222], [12, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 3, 3], 'outer_gen_pows': [0, 0, 3, 0], 'outer_gens': [[5, 6, 18, 108], [5, 12, 90, 198], [1, 300, 18, 558], [37, 6, 18, 108]], 'outer_group': '36.10', 'outer_hash': 10, 'outer_nilpotent': False, 'outer_order': 36, 'outer_permdeg': 6, 'outer_perms': [316, 341, 378, 373], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [3, 2], [4, 4], [6, 15], [12, 4]], 'representations': {'PC': {'code': 3222718250755765020368323440170165, 'gens': [1, 3, 4, 6], 'pres': [7, 2, 3, 3, 2, 3, 2, 3, 14, 254, 1613, 80, 23441, 3169, 124, 22938, 2078]}, 'Perm': {'d': 16, 'gens': [87178293556, 43597501, 93405315013, 98356, 123858, 1313901388800, 2789705318400]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_4', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
