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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '3888.dk', 'ambient_counter': 89, 'ambient_order': 3888, 'ambient_tex': 'C_3^3:S_3^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': True, 'core_order': 162, 'counter': 40, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '3888.dk.24.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '24.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '24.7', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 7, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 24, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_6:C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '162.54', 'subgroup_hash': 54, 'subgroup_order': 162, 'subgroup_tex': 'C_3^3:S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3888.dk', 'aut_centralizer_order': 6, 'aut_label': '24.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '10368.bd', 'aut_weyl_index': 6, 'centralizer': '1296.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.b1', '12.a1', '12.b1'], 'contains': ['48.a1', '72.bt1', '72.bu1', '72.bv1', '72.bw1', '72.bx1'], 'core': '24.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [3267, 3217, 4426, 2619], 'generators': [627, 100111334478, 12936672635, 101069337600, 56], 'label': '3888.dk.24.b1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '24.b1', 'normal_contained_in': ['8.b1', '12.a1', '12.b1'], 'normal_contains': ['48.a1'], 'normalizer': '1.a1', 'old_label': '24.b1', 'projective_image': '3888.dk', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.b1', 'subgroup_fusion': None, 'weyl_group': '1296.3497'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', '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, 3], 'aut_gens': [[24577758, 24507118, 2939606, 36731395, 14159168], [24577758, 24950012, 3035065, 37169915, 14159168], [14481270, 11495158, 20254942, 11661233, 14159168], [24577758, 24507118, 2939606, 36731395, 13835790]], 'aut_group': '1965150720.a', 'aut_hash': 6119322437363060898, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1965150720, 'aut_permdeg': 81, 'aut_perms': [657771367610839812722531985099075193581105650644659615214452318713464489102759582557145784570236080374102250639668958972, 808612898731331102202427324917560800475004880744121933655982886623123936826672862299267279681933963840112953340251939252, 707884959424340928995589841961242110118180352872775572462057266195870131286433642548161864689177486446723818580494968017], 'aut_phi_ratio': 36391680.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 40, 1]], 'aut_supersolvable': False, 'aut_tex': '\\AGL(4,3)', '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': 9360, 'autcentquo_group': '1965150720.a', 'autcentquo_hash': 6119322437363060898, 'autcentquo_nilpotent': False, 'autcentquo_order': 1965150720, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\AGL(4,3)', 'cc_stats': [[1, 1, 1], [2, 81, 1], [3, 2, 40]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '162.54', 'commutator_count': 1, 'commutator_label': '81.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 54, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 1, 40]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 31, 'exponent': 6, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 5, 13], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '162.54', 'hash': 54, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 3, 3, 2], 'inner_gens': [[24577758, 24507118, 2939606, 36731395, 8188666], [24577758, 24507118, 2939606, 36731395, 8314856], [24577758, 24507118, 2939606, 36731395, 39300942], [24577758, 24507118, 2939606, 36731395, 17350487], [20201582, 20254942, 32864349, 11225629, 14159168]], 'inner_hash': 54, 'inner_nilpotent': False, 'inner_order': 162, 'inner_split': False, 'inner_tex': 'C_3^3:S_3', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 40]], 'label': '162.54', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^3:S3', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 42, 'number_divisions': 42, 'number_normal_subgroups': 213, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 424, 'number_subgroups': 2664, 'old_label': None, 'order': 162, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 81], [3, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4680, 'outer_gen_orders': [2, 13], 'outer_gen_pows': [14408200, 14408200], 'outer_gens': [[7411244, 32705081, 20646633, 11444201, 14159168], [3035065, 11225629, 14686477, 32653905, 14159168]], 'outer_group': '12130560.c', 'outer_hash': 8461112410546189005, 'outer_nilpotent': False, 'outer_order': 12130560, 'outer_permdeg': 40, 'outer_perms': [19409857794192766541401284787497792047355199198, 743042578780903555112641255592744448909112275761], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PGL(4,3)', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 40]], 'representations': {'PC': {'code': 2271081930255, 'gens': [1, 2, 3, 4, 5], 'pres': [5, -2, -3, -3, -3, -3, 41, 182, 723, 2704]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 24507118, 2939606, 36731395, 14159168]}, 'Perm': {'d': 12, 'gens': [3633865, 43545747, 79833843, 43591200, 79879200]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3', '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': '8.2', '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': [[100115008562, 193121112394], [262492957494, 461322126948], [100115008614, 193121112822], [262975587842, 448428965521], [269198979894, 460839496548], [268240977313, 460843125354]], 'aut_group': '62208.l', 'aut_hash': 1820680360968750972, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 62208, 'aut_permdeg': 648, 'aut_perms': 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'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 2, 1], [2, 18, 2, 1], [2, 81, 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, 2], [3, 16, 4, 1], [4, 162, 4, 1], [6, 18, 2, 1], [6, 18, 4, 1], [6, 36, 2, 2], [6, 36, 4, 2], [6, 36, 8, 1], [6, 72, 2, 2], [6, 72, 4, 1], [6, 162, 1, 1], [12, 324, 4, 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, 9, 2], [2, 18, 2], [2, 81, 1], [3, 2, 1], [3, 4, 8], [3, 8, 8], [3, 16, 9], [4, 162, 4], [6, 18, 6], [6, 36, 20], [6, 72, 8], [6, 162, 1], [12, 324, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '3888.dk', 'commutator_count': 1, 'commutator_label': '486.259', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 89, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 18, 1, 2], [2, 81, 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, 3], [4, 162, 2, 2], [6, 18, 1, 2], [6, 18, 2, 2], [6, 36, 1, 8], [6, 36, 2, 6], [6, 72, 1, 4], [6, 72, 2, 2], [6, 162, 1, 1], [12, 324, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 48, 'exponent': 12, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 0, 8], [16, 0, 6]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '3888.dk', 'hash': 5387043051655160496, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 12], 'inner_gens': [[100115008562, 361729756237], [13415754873, 193121112394]], 'inner_hash': 5387043051655160496, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': True, 'inner_tex': 'C_3^3:S_3^2:C_4', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 10], [4, 32], [8, 16], [16, 9]], 'label': '3888.dk', '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': 'C3^3:S3^2:C4', 'ngens': 2, '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': 28, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 75, 'number_divisions': 54, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 378, 'number_subgroup_classes': 894, 'number_subgroups': 19656, 'old_label': None, 'order': 3888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 135], [3, 242], [4, 648], [6, 1566], [12, 1296]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 12936672713, 0, 0], 'outer_gens': [[100115043861, 193121112615], [268240977329, 361250674070], [262017549414, 193121112737], [100115043842, 193121112781]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 14], [8, 18], [16, 7], [32, 3]], 'representations': {'PC': {'code': '21836155762007731365262521521083754232986598708595860964407876720108632757517292242214866487', 'gens': [1, 3, 5, 6, 8, 9], 'pres': [9, 2, 2, 2, 3, 3, 2, 3, 3, 3, 18, 4330, 44930, 35705, 74, 57603, 23484, 38884, 562, 143213, 21398, 22379, 18014, 158, 81654, 9087, 4560, 3463, 916, 23336, 104993, 2969]}, 'Perm': {'d': 15, 'gens': [100115008562, 193121112394]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3^2:C_4', 'transitive_degree': 24, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 12], 'aut_gens': [[1, 4], [1, 20], [1, 6], [21, 6]], 'aut_group': '48.38', 'aut_hash': 38, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 7, 'aut_perms': [745, 24, 1707], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 2, 1, 1], [4, 3, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', '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, 3], [3, 2, 1], [4, 3, 4], [6, 2, 3]], 'center_label': '4.2', 'center_order': 4, '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', '2.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.1', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 3, 2, 2], [6, 2, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '12.4', 'hash': 7, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 20], [9, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 4]], 'label': '24.7', 'linC_count': 12, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6:C4', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 12, 'number_divisions': 10, 'number_normal_subgroups': 13, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 22, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 12], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 6], [15, 6]], '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': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6]], 'representations': {'PC': {'code': 122930001, 'gens': [1, 3], 'pres': [4, -2, -2, -2, -3, 8, 242, 34, 259]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26129266, 26129427]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [1497706, 72975, 1440589, 1565004]}, 'Perm': {'d': 9, 'gens': [46105, 24, 90720, 3]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6:C_4', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
