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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '7776.cc', 'ambient_counter': 55, 'ambient_order': 7776, 'ambient_tex': 'S_3^4:S_3', 'central': False, 'central_factor': False, 'centralizer_order': 216, 'characteristic': False, 'core_order': 3, 'counter': 647, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '7776.cc.216.cf1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '216.cf1', 'outer_equivalence': True, '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': 216, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '36.12', 'subgroup_hash': 12, 'subgroup_order': 36, 'subgroup_tex': 'C_6\\times S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7776.cc', 'aut_centralizer_order': None, 'aut_label': '216.cf1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '36.bg1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['72.a1', '72.bx1', '72.by1', '108.s1', '108.bc1'], 'contains': ['432.b1', '432.z1', '432.bp1', '648.r1', '648.be1'], 'core': '2592.a1', 'coset_action_label': None, 'count': 12, 'diagramx': [3377, -1, 5229, -1], 'generators': [713, 38, 80640, 14], 'label': '7776.cc.216.cf1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '72.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.e1', 'old_label': '216.cf1', 'projective_image': '7776.cc', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '216.cf1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 2, 6], 'aut_gens': [[1, 6], [5, 6], [1, 30], [31, 6]], 'aut_group': '24.14', 'aut_hash': 14, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [7, 127, 856], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_6', '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': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 1, 2], [3, 2, 3], [6, 1, 2], [6, 2, 3], [6, 3, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '36.12', 'hash': 12, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 30], [13, 6]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 12], [2, 6]], 'label': '36.12', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 7, 'linQ_dim': 4, 'linQ_dim_count': 7, 'linR_count': 7, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6*S3', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 18, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 22, 'number_subgroups': 36, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 8], [6, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[19, 6], [5, 6]], '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': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 2]], 'representations': {'PC': {'code': 403784533, 'gens': [1, 3], 'pres': [4, -2, -3, -2, -3, 8, 362, 34, 387]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16858573, 35931481]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 687, 1718]}, 'Perm': {'d': 8, 'gens': [720, 1, 30, 5760]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times S_3', 'transitive_degree': 12, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [6, 3, 2, 2, 12], 'aut_gens': [[45400, 100670169616, 193995653230], [81078, 94446778014, 280255858070], [45400, 362647756816, 200226302830], [81078, 455131354044, 286486588209], [81348, 94446778313, 280255858436], [435971370240, 180587232449, 448904338190]], 'aut_group': '62208.l', 'aut_hash': 1820680360968750972, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 62208, 'aut_permdeg': 120, 'aut_perms': [1479522355132891209411357193329702257298515365201155791327356901550291750817479479150538147647415451798341012204555615049465780164912541128080553077706727915110327503492874124121550823066800881606605, 2439527021283243582685753518965774791697106563508758015631298245175603214165892007355985319207275437614016467124153135091017866712707459753846027480567895686575907881878698816818379500241625417623968, 754102703057515092932449827170910387445330156280021358574726765077445850539955636193785801751298224976998162732380944576917800018231415458805885117217119488276993333312216025655320510295090043817323, 47649121642189374792018036198103517022163477659517170361320668472930687152958600973485615969801112534208561917597763609956608074203750024518023899831019491532981976686923016867115918354540072969065, 6039764507282031436977055445061035915213980421114013456869207402693321338536948106542039117825519900337682567352855987793799622851737917184275429055023126466682170409401719290406504353629668364458454], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 6, 2, 1], [2, 9, 2, 1], [2, 18, 2, 1], [2, 54, 2, 1], [2, 81, 1, 1], [2, 108, 1, 1], [3, 2, 1, 1], [3, 4, 2, 2], [3, 4, 4, 1], [3, 8, 2, 2], [3, 8, 4, 1], [3, 16, 1, 1], [3, 16, 2, 1], [3, 16, 4, 1], [3, 32, 1, 1], [4, 324, 2, 1], [4, 972, 1, 1], [6, 6, 4, 1], [6, 12, 2, 1], [6, 12, 4, 2], [6, 12, 8, 1], [6, 18, 2, 1], [6, 18, 4, 1], [6, 24, 2, 1], [6, 24, 4, 2], [6, 24, 8, 1], [6, 36, 2, 2], [6, 36, 4, 2], [6, 36, 8, 1], [6, 48, 2, 1], [6, 48, 4, 1], [6, 54, 4, 1], [6, 72, 2, 2], [6, 72, 4, 1], [6, 108, 2, 1], [6, 108, 4, 1], [6, 162, 1, 1], [6, 216, 2, 1], [6, 432, 1, 1], [12, 648, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3^5.D_4', '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': 24, 'autcentquo_group': '62208.l', 'autcentquo_hash': 1820680360968750972, 'autcentquo_nilpotent': False, 'autcentquo_order': 62208, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^5.D_4', 'cc_stats': [[1, 1, 1], [2, 6, 2], [2, 9, 2], [2, 18, 2], [2, 54, 2], [2, 81, 1], [2, 108, 1], [3, 2, 1], [3, 4, 8], [3, 8, 8], [3, 16, 7], [3, 32, 1], [4, 324, 2], [4, 972, 1], [6, 6, 4], [6, 12, 18], [6, 18, 6], [6, 24, 18], [6, 36, 20], [6, 48, 6], [6, 54, 4], [6, 72, 8], [6, 108, 6], [6, 162, 1], [6, 216, 2], [6, 432, 1], [12, 648, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '7776.cc', 'commutator_count': 1, 'commutator_label': '972.888', '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', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 55, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 6, 1, 2], [2, 9, 1, 2], [2, 18, 1, 2], [2, 54, 1, 2], [2, 81, 1, 1], [2, 108, 1, 1], [3, 2, 1, 1], [3, 4, 1, 4], [3, 4, 2, 2], [3, 8, 1, 4], [3, 8, 2, 2], [3, 16, 1, 3], [3, 16, 2, 2], [3, 32, 1, 1], [4, 324, 1, 2], [4, 972, 1, 1], [6, 6, 2, 2], [6, 12, 1, 6], [6, 12, 2, 6], [6, 18, 1, 2], [6, 18, 2, 2], [6, 24, 1, 6], [6, 24, 2, 6], [6, 36, 1, 8], [6, 36, 2, 6], [6, 48, 1, 2], [6, 48, 2, 2], [6, 54, 2, 2], [6, 72, 1, 4], [6, 72, 2, 2], [6, 108, 1, 2], [6, 108, 2, 2], [6, 162, 1, 1], [6, 216, 1, 2], [6, 432, 1, 1], [12, 648, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2150400, 'exponent': 12, 'exponents_of_order': [5, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 0, 16], [16, 0, 8], [32, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '7776.cc', 'hash': 4809026425870751764, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 6, 12], 'inner_gens': [[45400, 100670169682, 193995733943], [45436, 100670169616, 281177573261], [81041, 449822419847, 193995653230]], 'inner_hash': 4809026425870751764, 'inner_nilpotent': False, 'inner_order': 7776, 'inner_split': True, 'inner_tex': 'S_3^4:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 22], [4, 56], [8, 34], [16, 14], [32, 1]], 'label': '7776.cc', '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': None, 'name': 'S3^4:S3', 'ngens': 3, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 48, 'number_characteristic_subgroups': 17, 'number_conjugacy_classes': 135, 'number_divisions': 99, 'number_normal_subgroups': 55, 'number_subgroup_autclasses': 1122, 'number_subgroup_classes': 2706, 'number_subgroups': 77912, 'old_label': None, 'order': 7776, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 363], [3, 242], [4, 1620], [6, 4254], [12, 1296]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [647, 193077562247], 'outer_gens': [[46015, 100670170257, 193995653376], [94446822960, 25909632449, 449826053541]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 30], [8, 30], [16, 16], [32, 5]], 'representations': {'PC': {'code': '5667211035620131757192221860287065683835816772621057732795915491870627895447678681785286314736251250568506539', 'gens': [1, 2, 4, 6, 8, 10], 'pres': [10, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 69420, 6441, 51, 242, 723, 6973, 113, 7204, 1214, 169925, 10815, 1835, 175, 3366, 10096, 1716, 397447, 5337, 237, 233288, 2218, 86409, 7259, 1279]}, 'Perm': {'d': 15, 'gens': [45400, 100670169616, 193995653230]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3^4:S_3', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
