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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '800.315', 'ambient_counter': 315, 'ambient_order': 800, 'ambient_tex': 'C_{10}^2:C_8', 'central': False, 'central_factor': False, 'centralizer_order': 400, 'characteristic': True, 'core_order': 80, 'counter': 23, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '800.315.10.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '10.b1.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': False, 'standard_generators': False, 'stem': False, 'subgroup': '80.45', 'subgroup_hash': 45, 'subgroup_order': 80, 'subgroup_tex': 'C_2^2\\times C_{20}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '800.315', 'aut_centralizer_order': 80, 'aut_label': '10.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '64.260', 'aut_weyl_index': 80, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '5.a1.a1'], 'contains': ['20.b1.a1', '20.d1.a1', '20.f1.a1', '20.k1.a1', '20.k1.b1', '50.a1.a1'], 'core': '10.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2231, 8305, 8557, 6906, 7175, 6538, 2379, 6690], 'generators': [400, 40, 160, 2, 4], 'label': '800.315.10.b1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '10.b1.a1', 'normal_contained_in': ['2.a1.a1', '5.a1.a1'], 'normal_contains': ['20.b1.a1', '20.d1.a1', '20.f1.a1', '50.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '10.b1.a1', 'projective_image': '100.14', 'quotient_action_image': '2.1', 'quotient_action_kernel': '5.1', 'quotient_action_kernel_order': 5, 'quotient_fusion': None, 'short_label': '10.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '80.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [2, 4, 2, 3, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [3, 2, 78], [1, 2, 68], [1, 2, 36], [2, 3, 7], [1, 2, 5], [1, 2, 6], [1, 42, 45], [41, 2, 6], [1, 2, 44]], 'aut_group': '768.1090064', 'aut_hash': 1090064, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 768, 'aut_permdeg': 12, 'aut_perms': [41505136, 40723329, 16, 1216200, 85800, 766200, 167410080, 136095240, 40723320], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [4, 1, 8, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 1, 24, 1], [20, 1, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times C_2^3:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '768.1090064', 'autcent_hash': 1090064, 'autcent_nilpotent': False, 'autcent_order': 768, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_4\\times C_2^3:S_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, 7], [4, 1, 8], [5, 1, 4], [10, 1, 28], [20, 1, 32]], 'center_label': '80.45', 'center_order': 80, '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', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 45, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['4.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 1, 2, 4], [5, 1, 4, 1], [10, 1, 4, 7], [20, 1, 8, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 217, 'exponent': 20, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '40.14', 'hash': 45, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 80]], 'label': '80.45', 'linC_count': 27776, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 96, 'linQ_dim': 7, 'linQ_dim_count': 96, 'linR_count': 192, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C20', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 80, 'number_divisions': 24, 'number_normal_subgroups': 54, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 54, 'number_subgroups': 54, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [4, 8], [5, 4], [10, 28], [20, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 4, 2, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[3, 2, 78], [1, 2, 68], [1, 2, 36], [2, 3, 7], [1, 2, 5], [1, 2, 6], [1, 42, 45], [41, 2, 6], [1, 2, 44]], 'outer_group': '768.1090064', 'outer_hash': 1090064, 'outer_nilpotent': False, 'outer_order': 768, 'outer_permdeg': 12, 'outer_perms': [41505136, 40723329, 16, 1216200, 85800, 766200, 167410080, 136095240, 40723320], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times C_2^3:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 8], [8, 4]], 'representations': {'PC': {'code': 1426075651, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -5, 42, 58]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 128232072316, 82502651532, 26013099178, 26013244590]}, 'Perm': {'d': 13, 'gens': [131040, 479001600, 3628800, 96, 41040]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 20], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{20}', 'transitive_degree': 80, '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': '80.23', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 4, 4, 2, 4, 4, 20], 'aut_gens': [[1, 8, 80], [47, 56, 724], [247, 8, 280], [281, 56, 600], [247, 72, 760], [321, 8, 284], [365, 8, 564], [561, 8, 124]], 'aut_group': None, 'aut_hash': 4438983215857966134, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5120, 'aut_permdeg': 84, 'aut_perms': [2106638782796144454231068709918250543740650780228422245830642836542497241985030334257254135029943850481816622692690215186519405, 1027460998373226693589501759328386682696213628428594831837276978536058634747272970280885462439126931275357287987627796660644112, 260006143546308361623976720889830701539865699837449785794465368064775932929789856704966046377465920041208146984325589590734717, 199125585467300245930524688625777564968502017027669289805656640981399935918009561645541633714214236957037977300096848606837511, 20693757586355416229050663430135103728468179552410567426148925574594128874557853387733995810510422964251939438261213424111384, 977717310136719078417782704643174830489264910712588207440508614499560242298564959519981885495483221943022195014728497238649454, 528316067678735832583599712116078365153986372186415408896349262006461051955943468352633795680479103478490504313189375610629472], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [4, 1, 4, 1], [4, 2, 2, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [8, 10, 8, 1], [10, 1, 4, 3], [10, 2, 2, 3], [10, 2, 8, 5], [10, 2, 32, 1], [20, 1, 16, 1], [20, 2, 8, 3], [20, 2, 32, 2], [40, 10, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_5:(C_2^4.C_2^6)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.2319', 'autcent_hash': 2319, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [4, 1, 4], [4, 2, 2], [5, 1, 4], [5, 2, 10], [8, 10, 8], [10, 1, 12], [10, 2, 78], [20, 1, 16], [20, 2, 88], [40, 10, 32]], 'center_label': '40.9', 'center_order': 40, 'central_product': True, 'central_quotient': '20.4', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 315, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['160.37', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [4, 1, 2, 2], [4, 2, 2, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [8, 10, 4, 2], [10, 1, 4, 3], [10, 2, 2, 3], [10, 2, 4, 18], [20, 1, 8, 2], [20, 2, 4, 2], [20, 2, 8, 10], [40, 10, 16, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 40, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '100.14', 'hash': 315, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 1, 10], 'inner_gens': [[1, 8, 760], [1, 8, 80], [201, 8, 80]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'D_{10}', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 180]], 'label': '800.315', 'linC_count': 3072, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 24, 'linQ_dim': 12, 'linQ_dim_count': 24, 'linR_count': 960, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C10^2:C8', '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': 30, 'number_characteristic_subgroups': 38, 'number_conjugacy_classes': 260, 'number_divisions': 55, 'number_normal_subgroups': 58, 'number_subgroup_autclasses': 92, 'number_subgroup_classes': 154, 'number_subgroups': 300, 'old_label': None, 'order': 800, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 7], [4, 8], [5, 24], [8, 80], [10, 168], [20, 192], [40, 320]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 4, 4], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[3, 72, 120], [5, 72, 124], [3, 24, 560], [5, 56, 80], [447, 72, 84]], 'outer_group': '256.26979', 'outer_hash': 26979, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 14, 'outer_perms': [6267300487, 39916800, 16577, 6267300497, 19639070760], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^3:C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 8, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 11], [8, 24], [16, 12]], 'representations': {'PC': {'code': 2228382235772938764103106611, 'gens': [1, 4, 6], 'pres': [7, 2, 2, 2, 2, 5, 2, 5, 14, 36, 80, 31925, 124, 31366]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [898450, 21967, 35962, 826574, 359380, 48302, 22363]}, 'Perm': {'d': 22, 'gens': [51212609590287767040, 21009968179200, 33, 44462014702560, 44460928512000, 1611308160, 53767134380703744000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 40], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}^2:C_8', 'transitive_degree': 80, '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}
