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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '972.474', 'ambient_counter': 474, 'ambient_order': 972, 'ambient_tex': 'C_3^3:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 81, 'characteristic': True, 'core_order': 81, 'counter': 19, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '972.474.12.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '12.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '12.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 12, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '81.15', 'subgroup_hash': 15, 'subgroup_order': 81, 'subgroup_tex': 'C_3^4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '972.474', 'aut_centralizer_order': 81, 'aut_label': '12.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '432.601', 'aut_weyl_index': 81, 'centralizer': '12.b1', 'complements': ['81.c1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1', '6.b1', '6.e1'], 'contains': ['36.a1', '36.e1', '36.f1', '36.i1'], 'core': '12.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [5200, 2911, 2217, 3283], 'generators': [24577758, 24507118, 11495158, 11225629], 'label': '972.474.12.b1', 'mobius_quo': 0, 'mobius_sub': -6, 'normal_closure': '12.b1', 'normal_contained_in': ['4.a1', '6.b1'], 'normal_contains': ['36.a1'], 'normalizer': '1.a1', 'old_label': '12.b1', 'projective_image': '972.474', 'quotient_action_image': '12.4', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '12.b1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '81.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 9360, 'aut_gen_orders': [2, 13], 'aut_gens': [[1, 3, 9, 27], [2, 3, 9, 27], [29, 2, 6, 18]], 'aut_group': None, 'aut_hash': 1045913143454417265, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24261120, 'aut_permdeg': 80, 'aut_perms': [894910464770715304112463402887197477343442096610712847651341310429022571428803639963392602465820278981730644545159864, 1812326269711177385513856665846033904155014172742552368476102969481397534672420762470589026159121368093405030544787282], 'aut_phi_ratio': 449280.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [3, 1, 80, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2.\\PSL(4,3).C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 9360, 'autcent_group': None, 'autcent_hash': 1045913143454417265, 'autcent_nilpotent': False, 'autcent_order': 24261120, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2.\\PSL(4,3).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, 80]], 'center_label': '81.15', 'center_order': 81, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 4]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 40]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3, 5, 13], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '81.15', 'hash': 15, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 3, 9, 27], [1, 3, 9, 27], [1, 3, 9, 27], [1, 3, 9, 27]], '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, 81]], 'label': '81.15', 'linC_count': None, 'linC_degree': 4, '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^4', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 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': 81, 'number_divisions': 41, 'number_normal_subgroups': 212, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 212, 'number_subgroups': 212, 'old_label': None, 'order': 81, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 9360, 'outer_gen_orders': [2, 13], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 3, 9, 27], [29, 2, 6, 18]], 'outer_group': None, 'outer_hash': 1045913143454417265, 'outer_nilpotent': False, 'outer_order': 24261120, 'outer_permdeg': 80, 'outer_perms': [894910464770715304112463402887197477343442096610712847651341310429022571428803639963392602465820278981730644545159864, 1812326269711177385513856665846033904155014172742552368476102969481397534672420762470589026159121368093405030544787282], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2.\\PSL(4,3).C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 40]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4], 'pres': [4, -3, 3, 3, 3]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 24507118, 2939606, 36731395]}, 'Perm': {'d': 12, 'gens': [79833600, 80640, 240, 4]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3, 3], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4', 'transitive_degree': 81, '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': '4.2', '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, 12, 3, 3, 3, 3, 3], 'aut_gens': [[24577758, 24507118, 1305911, 26172358, 2939606, 36731395, 7355200], [20201582, 20327769, 29656017, 37063598, 32864349, 32552046, 36728741], [20201582, 24454001, 5504967, 20278105, 11495158, 36678278, 7284560], [7138662, 32758225, 35469088, 29361018, 11495158, 20573833, 43020291], [24577758, 20327769, 1305911, 39825765, 2939606, 20479832, 3156141], [24577758, 24507118, 9861463, 16281068, 2939606, 36731395, 7284560], [24577758, 24507118, 27786090, 29363677, 2939606, 36731395, 14798938], [24577758, 24507118, 1305911, 39825765, 2939606, 36731395, 3156141], [24577758, 24507118, 1305911, 13089749, 2939606, 36731395, 14683796]], 'aut_group': '34992.ns', 'aut_hash': 5395324004515259423, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 34992, 'aut_permdeg': 54, 'aut_perms': [3712920747491381539271848168628321544262425095592444298056406002891079, 206653413735992926053826615306920349455402054225125505197905169082796603, 91126428485545547953406994036919776647294851743040947173220815182227407, 2559428772521509727383113445146803286418846233889145396940941448209761, 12151667977233322930379765979898793761135779522084465742095068876131626, 152040991600193293799936360556781407144742421772048712080003542702179065, 115079840247327961487724591316632538892366518730680481482667023916596386, 115080077724465208049533810494857347819003894589289583199820453721032672], 'aut_phi_ratio': 108.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 2, 1], [2, 81, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 6, 6, 1], [3, 12, 3, 1], [3, 18, 1, 1], [3, 36, 2, 2], [6, 54, 2, 1], [6, 54, 6, 1], [6, 162, 1, 1]], 'aut_supersolvable': False, 'aut_tex': '\\He_3^2:(S_3\\times 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': 12, 'autcentquo_group': '34992.ns', 'autcentquo_hash': 5395324004515259423, 'autcentquo_nilpotent': False, 'autcentquo_order': 34992, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\He_3^2:(S_3\\times D_4)', 'cc_stats': [[1, 1, 1], [2, 27, 2], [2, 81, 1], [3, 2, 2], [3, 4, 1], [3, 6, 6], [3, 12, 3], [3, 18, 1], [3, 36, 4], [6, 54, 8], [6, 162, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '972.474', 'commutator_count': 1, 'commutator_label': '243.37', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 474, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 81, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 6], [3, 12, 1, 3], [3, 18, 1, 1], [3, 36, 1, 2], [3, 36, 2, 1], [6, 54, 1, 8], [6, 162, 1, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 3]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '108.40', 'hash': 474, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 3, 6, 3, 3, 2], 'inner_gens': [[24577758, 24507118, 1305911, 13089749, 2939606, 36731395, 11534549], [24577758, 24507118, 5504967, 8640151, 2939606, 36731395, 11607619], [24577758, 20327769, 1305911, 41400411, 2939606, 11537230, 20477175], [43014227, 6973453, 29656017, 26172358, 32864349, 11225629, 24676207], [24577758, 24507118, 1305911, 39825765, 2939606, 36731395, 20477175], [24577758, 24507118, 41439590, 29363677, 2939606, 36731395, 32870443], [20201582, 20254942, 31289712, 8733253, 32864349, 11225629, 7355200]], 'inner_hash': 474, 'inner_nilpotent': False, 'inner_order': 972, 'inner_split': True, 'inner_tex': 'C_3^3:S_3^2', 'inner_used': [1, 2, 3, 4, 7], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 6], [4, 5], [6, 12], [12, 3]], 'label': '972.474', '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:S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 30, 'number_divisions': 29, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 117, 'number_subgroup_classes': 349, 'number_subgroups': 6241, 'old_label': None, 'order': 972, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 135], [3, 242], [6, 594]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 3, 3], 'outer_gen_pows': [24676207, 43014227, 14408200, 14408200], 'outer_gens': [[7138662, 6973453, 9861463, 26270739, 11495158, 2721277, 42522093], [20201582, 20254942, 9861463, 39333749, 11495158, 32479219, 24676207], [24577758, 2888673, 1305911, 26172358, 2939606, 20479832, 7355200], [24577758, 20327769, 1305911, 26172358, 2939606, 20479832, 7355200]], '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': 6, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 3], [6, 12], [8, 1], [12, 3]], 'representations': {'PC': {'code': 239504451333840078632361135505769980191, 'gens': [1, 2, 4, 5, 6, 7], 'pres': [7, -2, -2, -3, -3, -3, 3, -3, 141, 36, 170, 346, 2524, 9077, 11352, 4555, 656, 15889]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 24507118, 1305911, 26172358, 2939606, 36731395, 7355200]}, 'Perm': {'d': 18, 'gens': [3423317058650635, 3402106385166720, 174400128148, 1845584622926787, 2267507354388480, 376703666922484, 711549292688884]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3^2', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [1, 10], [3, 2]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', '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': 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, 2, 1], [6, 2, 1]], 'center_label': '2.1', 'center_order': 2, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 10], [5, 2]], '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.4', 'linC_count': 1, 'linC_degree': 2, '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': 'D6', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 16, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[7, 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': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 43377, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 61, 16, 74]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [17, 35]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 55, 56]}, 'Perm': {'d': 5, 'gens': [6, 1, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_6', 'transitive_degree': 6, 'wreath_data': None, 'wreath_product': False}
