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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '2592.he', 'ambient_counter': 187, 'ambient_order': 2592, 'ambient_tex': 'C_6^3.D_6', 'central': False, 'central_factor': False, 'centralizer_order': 864, 'characteristic': False, 'core_order': 1, 'counter': 1106, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '2592.he.1296.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '1296.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 1296, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2592.he', 'aut_centralizer_order': 1728, 'aut_label': '1296.b1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '1.1', 'aut_weyl_index': 5184, 'centralizer': '3.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['432.g1.a1', '432.h1.a1', '432.i1.a1', '432.j1.a1', '432.k1.a1', '432.l1.a1', '432.m1.a1', '648.b1.a1', '648.g1.a1', '648.g1.b1', '648.h1.a1', '648.i1.a1', '648.j1.a1', '648.k1.a1'], 'contains': ['2592.a1.a1'], 'core': '2592.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [6739, -1, 4548, -1, 8056, -1, 8096, -1], 'generators': [1404], 'label': '2592.he.1296.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '324.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.c1.a1', 'old_label': '1296.b1.a1', 'projective_image': '2592.he', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1296.b1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, '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': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [6, 2, 18, 6, 4], 'aut_gens': [[1, 6, 108, 216, 1296], [2077, 2190, 756, 1080, 1296], [433, 114, 108, 1080, 1296], [1129, 2562, 756, 972, 1296], [349, 462, 756, 1080, 1296], [1297, 1074, 648, 540, 1296]], 'aut_group': None, 'aut_hash': 2780599447146399266, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5184, 'aut_permdeg': 42, 'aut_perms': [413228832947684641570525075132985186008448073903358, 99428601096524005777310840416910225655302520, 238351500592957612351546040214684686180815600834078, 413228832947684641570525075132985186008423005023878, 34268272139806297945853111722330823252853018132479], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 9, 2, 1], [2, 18, 2, 1], [2, 54, 2, 1], [3, 2, 1, 2], [3, 3, 1, 2], [3, 4, 1, 1], [3, 6, 1, 2], [4, 18, 2, 1], [4, 54, 2, 1], [6, 2, 1, 2], [6, 3, 1, 6], [6, 4, 1, 1], [6, 6, 1, 14], [6, 6, 2, 1], [6, 9, 2, 4], [6, 12, 1, 6], [6, 18, 2, 5], [6, 36, 2, 3], [6, 54, 2, 2], [9, 24, 1, 3], [9, 48, 1, 3], [12, 18, 2, 2], [12, 36, 2, 3], [12, 54, 2, 2], [18, 24, 1, 3], [18, 48, 1, 3], [18, 72, 2, 3]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_6^2.(C_6\\times S_3).C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': '1296.2922', 'autcentquo_hash': 2922, 'autcentquo_nilpotent': False, 'autcentquo_order': 1296, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 9, 2], [2, 18, 2], [2, 54, 2], [3, 2, 2], [3, 3, 2], [3, 4, 1], [3, 6, 2], [4, 18, 2], [4, 54, 2], [6, 2, 2], [6, 3, 6], [6, 4, 1], [6, 6, 16], [6, 9, 8], [6, 12, 6], [6, 18, 10], [6, 36, 6], [6, 54, 4], [9, 24, 3], [9, 48, 3], [12, 18, 4], [12, 36, 6], [12, 54, 4], [18, 24, 3], [18, 48, 3], [18, 72, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '1296.2922', 'commutator_count': 1, 'commutator_label': '108.20', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 187, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['216.90', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 9, 1, 2], [2, 18, 1, 2], [2, 54, 1, 2], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 2, 1], [4, 18, 1, 2], [4, 54, 1, 2], [6, 2, 1, 2], [6, 3, 2, 3], [6, 4, 1, 1], [6, 6, 1, 6], [6, 6, 2, 5], [6, 9, 2, 4], [6, 12, 1, 2], [6, 12, 2, 2], [6, 18, 1, 2], [6, 18, 2, 4], [6, 36, 1, 2], [6, 36, 2, 2], [6, 54, 2, 2], [9, 24, 1, 1], [9, 24, 2, 1], [9, 48, 1, 1], [9, 48, 2, 1], [12, 18, 2, 2], [12, 36, 1, 2], [12, 36, 2, 2], [12, 54, 2, 2], [18, 24, 1, 1], [18, 24, 2, 1], [18, 48, 1, 1], [18, 48, 2, 1], [18, 72, 1, 2], [18, 72, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 786240, 'exponent': 36, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 2], [12, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '864.4690', 'hash': 8584545123024004071, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [12, 18, 2, 6, 1], 'inner_gens': [[1, 1038, 648, 540, 1296], [1129, 6, 756, 540, 1296], [757, 654, 108, 216, 1296], [1189, 1194, 108, 216, 1296], [1, 6, 108, 216, 1296]], 'inner_hash': 2922, 'inner_nilpotent': False, 'inner_order': 1296, 'inner_split': True, 'inner_tex': 'C_6^2.S_3^2', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 24], [2, 24], [3, 24], [4, 6], [6, 28], [12, 8]], 'label': '2592.he', 'linC_count': 108, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 12, 'linQ_dim': 8, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C6^3.D6', 'ngens': 9, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 83, 'number_characteristic_subgroups': 48, 'number_conjugacy_classes': 114, 'number_divisions': 78, 'number_normal_subgroups': 84, 'number_subgroup_autclasses': 690, 'number_subgroup_classes': 1116, 'number_subgroups': 11686, 'old_label': None, 'order': 2592, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 175], [3, 26], [4, 144], [6, 878], [9, 216], [12, 504], [18, 648]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 758], 'outer_gens': [[1297, 1302, 108, 216, 1296], [1189, 1362, 648, 540, 1296]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [3, 8], [4, 10], [6, 20], [8, 2], [12, 12], [24, 2]], 'representations': {'PC': {'code': '1163711008841031234168447556942143713659518736771783732710714861705689341618', 'gens': [1, 3, 6, 7, 9], 'pres': [9, -2, -3, -2, -3, -3, -2, 2, -3, -2, 18, 6814, 28028, 2522, 74, 28083, 12540, 138, 3244, 34997, 6827, 2948, 34026, 5694, 1734, 186, 62215, 10393]}, 'Perm': {'d': 18, 'gens': [483396864, 397539243254401, 377918372202654]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^3.D_6', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
