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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1760.292', 'ambient_counter': 292, 'ambient_order': 1760, 'ambient_tex': 'C_{11}:(D_4\\times C_{20})', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': True, 'core_order': 176, 'counter': 43, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1760.292.10.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '10.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '10.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 10, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{10}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '176.35', 'subgroup_hash': 35, 'subgroup_order': 176, 'subgroup_tex': 'C_2^2.D_{22}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.292', 'aut_centralizer_order': 4, 'aut_label': '10.c1', 'aut_quo_index': 4, 'aut_stab_index': 1, 'aut_weyl_group': '1760.1272', 'aut_weyl_index': 4, 'centralizer': '220.a1.a1', 'complements': ['176.e1.b1', '176.e1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1', '5.a1.a1'], 'contains': ['20.a1.a1', '20.e1.a1', '20.g1.a1', '20.i1.a1', '20.i1.b1', '110.c1.a1'], 'core': '10.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [9062, 6762, 8265, 7625, 8240, 4949, 7085, 6513], 'generators': [441, 880, 900, 450, 160], 'label': '1760.292.10.c1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '10.c1.a1', 'normal_contained_in': ['2.c1.a1', '5.a1.a1'], 'normal_contains': ['20.a1.a1', '20.e1.a1', '20.g1.a1'], 'normalizer': '1.a1.a1', 'old_label': '10.c1.a1', 'projective_image': '440.42', 'quotient_action_image': '10.2', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '10.c1.a1', 'subgroup_fusion': None, 'weyl_group': '220.7'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 660, 'aut_gen_orders': [2, 2, 11, 2, 2, 2, 10, 2], 'aut_gens': [[1, 2, 8], [1, 6, 168], [93, 94, 168], [1, 114, 8], [5, 6, 168], [88, 2, 81], [1, 90, 168], [1, 2, 152], [1, 91, 168]], 'aut_group': '21120.c', 'aut_hash': 7504208320117401807, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21120, 'aut_permdeg': 19, 'aut_perms': [368209, 13516122268016209, 28288846, 87657661009, 110031936905809, 13516309078640209, 6423384155685265, 27121166392688209], 'aut_phi_ratio': 264.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [4, 11, 8, 1], [11, 2, 5, 1], [22, 2, 5, 1], [22, 2, 30, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3:S_4\\times F_{11}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '192.1493', 'autcent_hash': 1493, 'autcent_nilpotent': False, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3:S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 7], [4, 11, 8], [11, 2, 5], [22, 2, 35]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '22.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 35, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['44.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 11, 2, 4], [11, 2, 5, 1], [22, 2, 5, 7]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 84, 'exponent': 44, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '88.11', 'hash': 35, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [1, 2, 11], 'inner_gens': [[1, 2, 8], [1, 2, 168], [1, 18, 8]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 22, 'inner_split': True, 'inner_tex': 'D_{11}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 40]], 'label': '176.35', 'linC_count': 1680, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 48, 'linQ_dim': 13, 'linQ_dim_count': 48, 'linR_count': 240, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2.D22', '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': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 56, 'number_divisions': 20, 'number_normal_subgroups': 43, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 54, 'number_subgroups': 164, 'old_label': None, 'order': 176, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [4, 88], [11, 10], [22, 70]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [2, 5, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[92, 6, 101], [1, 2, 120], [89, 2, 101], [5, 2, 168], [1, 2, 172], [1, 3, 172], [5, 94, 168], [1, 6, 168]], 'outer_group': '960.11113', 'outer_hash': 11113, 'outer_nilpotent': False, 'outer_order': 960, 'outer_permdeg': 13, 'outer_perms': [2087286480, 64, 1049489280, 482630400, 3669120, 3133236960, 1037968560, 482671440], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_5\\times C_2^3:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [10, 8]], 'representations': {'PC': {'code': 80897543527400009, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -2, -2, -11, 26, 1688, 58, 2009]}, 'GLZN': {'d': 2, 'p': 44, 'gens': [1959255, 1789877, 85361, 1788885, 86153]}, 'Perm': {'d': 19, 'gens': [357001369764489, 5160, 11536, 16, 7116376444894080]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2.D_{22}', 'transitive_degree': 176, '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': '80.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 110, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 10, 11], 'aut_gens': [[1, 2, 40], [901, 2, 840], [881, 2, 840], [1, 902, 840], [1, 882, 840], [1, 2, 1740], [1, 2, 1720], [1, 2, 520], [1, 802, 40]], 'aut_group': None, 'aut_hash': 3808563084807252545, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7040, 'aut_permdeg': 52, 'aut_perms': [54658446580341305259158774732577075919647126988015774641278544619041, 15850063654835410077145769026262780536448476988473945779568132001121, 991220515658632449972445224724343475720568643457746690142706606357, 743579679339339086795009600662485398391342198639079381015540500517, 55311411933828687810130619682466995633592637053855253017093191844842, 16254767212215277561467413020873937960800855633418971024426341728708, 127913999465956866624082339792435569561084143453679573859013791005, 342640591249109697622689083705480948027192518002272020276600272051], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 22, 2, 1], [4, 2, 2, 2], [4, 11, 4, 1], [4, 22, 2, 2], [5, 11, 1, 4], [10, 11, 1, 12], [10, 22, 2, 8], [11, 10, 1, 1], [20, 11, 4, 4], [20, 22, 2, 16], [22, 10, 1, 3], [22, 20, 2, 1], [44, 20, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_{11}:C_5.C_2^6.C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 22, 2], [4, 2, 4], [4, 11, 4], [4, 22, 4], [5, 11, 4], [10, 11, 12], [10, 22, 16], [11, 10, 1], [20, 11, 16], [20, 22, 32], [22, 10, 3], [22, 20, 2], [44, 20, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '440.42', 'commutator_count': 1, 'commutator_label': '22.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 292, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 22, 1, 2], [4, 2, 2, 2], [4, 11, 2, 2], [4, 22, 1, 2], [4, 22, 2, 1], [5, 11, 4, 1], [10, 11, 4, 3], [10, 22, 4, 4], [11, 10, 1, 1], [20, 11, 8, 2], [20, 22, 4, 2], [20, 22, 8, 3], [22, 10, 1, 3], [22, 20, 1, 2], [44, 20, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 249984, 'exponent': 220, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '440.42', 'hash': 292, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [2, 10, 22], 'inner_gens': [[1, 22, 60], [21, 2, 520], [21, 1282, 40]], 'inner_hash': 42, 'inner_nilpotent': False, 'inner_order': 440, 'inner_split': True, 'inner_tex': 'C_2^2\\times F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 20], [10, 8], [20, 2]], 'label': '1760.292', 'linC_count': 80, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 32, 'linQ_dim': 14, 'linQ_dim_count': 32, 'linR_count': 160, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C11:(D4*C20)', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 62, 'number_characteristic_subgroups': 49, 'number_conjugacy_classes': 110, 'number_divisions': 38, 'number_normal_subgroups': 75, 'number_subgroup_autclasses': 144, 'number_subgroup_classes': 188, 'number_subgroups': 1258, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 51], [4, 140], [5, 44], [10, 484], [11, 10], [20, 880], [22, 70], [44, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[901, 2, 840], [1, 902, 840], [1, 882, 840], [1, 2, 1740]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 9], [8, 6], [10, 4], [16, 1], [20, 4]], 'representations': {'PC': {'code': 25863261335148596411015522276222911495916963753616084670630003009, 'gens': [1, 2, 5], 'pres': [7, 2, 2, 2, 5, 2, 2, 11, 6300, 309, 36, 58, 2104, 9111, 12968, 900, 102, 21852, 12619, 2126, 124, 7853, 7860, 4927]}, 'GLZN': {'d': 2, 'p': 88, 'gens': [682177, 30666285, 31010913, 6133273, 44637427, 16016551, 685345]}, 'Perm': {'d': 19, 'gens': [357001416581160, 727, 11537, 756024112529280, 7, 16680, 7469717690131200]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}:(D_4\\times C_{20})', 'transitive_degree': 176, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [7]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', '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], [5, 1, 4], [10, 1, 4]], 'center_label': '10.2', 'center_order': 10, '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': ['2.1', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 10, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '10.2', 'hash': 2, 'hyperelementary': 10, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 10]], 'label': '10.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C10', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 10, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 10, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [5, 4], [10, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[7]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2]], 'representations': {'PC': {'code': 83, 'gens': [1], 'pres': [2, -2, -5, 4]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16717348]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504]}, 'Perm': {'d': 7, 'gens': [720, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}', 'transitive_degree': 10, 'wreath_data': None, 'wreath_product': False}
