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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1120.784', 'ambient_counter': 784, 'ambient_order': 1120, 'ambient_tex': 'C_7\\times C_2^3.D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 280, 'characteristic': True, 'core_order': 56, 'counter': 57, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1120.784.20.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '20.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '20.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{10}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '56.8', 'subgroup_hash': 8, 'subgroup_order': 56, 'subgroup_tex': 'C_2\\times C_{28}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1120.784', 'aut_centralizer_order': 320, 'aut_label': '20.c1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '48.45', 'aut_weyl_index': 320, 'centralizer': '4.c1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.c1.a1', '10.a1.a1', '10.e1.a1', '10.e1.b1'], 'contains': ['40.a1.a1', '40.d1.a1', '140.c1.a1'], 'core': '20.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3082, 7047, 5228, 1899, 6853, 6401, 5189, 7186], 'generators': [41, 160, 4, 560], 'label': '1120.784.20.c1.a1', 'mobius_quo': 0, 'mobius_sub': -10, 'normal_closure': '20.c1.a1', 'normal_contained_in': ['4.c1.a1', '10.a1.a1'], 'normal_contains': ['40.a1.a1', '140.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '20.c1.a1', 'projective_image': '40.13', 'quotient_action_image': '4.2', 'quotient_action_kernel': '5.1', 'quotient_action_kernel_order': 5, 'quotient_fusion': None, 'short_label': '20.c1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '56.8', '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, 2], [1, 26], [29, 2], [29, 47]], 'aut_group': '48.45', 'aut_hash': 45, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 48, 'aut_permdeg': 9, 'aut_perms': [24, 10824, 120964], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1], [7, 1, 6, 1], [14, 1, 6, 1], [14, 1, 12, 1], [28, 1, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '48.45', 'autcent_hash': 45, 'autcent_nilpotent': True, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6\\times D_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, 3], [4, 1, 4], [7, 1, 6], [14, 1, 18], [28, 1, 24]], 'center_label': '56.8', 'center_order': 56, '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', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [7, 1, 6, 1], [14, 1, 6, 3], [28, 1, 12, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 24, 'exponent': 28, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '28.4', 'hash': 8, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 56]], 'label': '56.8', 'linC_count': 576, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 24, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C28', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 56, 'number_divisions': 12, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 56, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 4], [7, 6], [14, 18], [28, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 26], [29, 2], [29, 47]], 'outer_group': '48.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 9, 'outer_perms': [24, 10824, 120964], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [6, 4], [12, 2]], 'representations': {'PC': {'code': 41811557, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -7, 21, 34]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [243896, 682920]}, 'Perm': {'d': 13, 'gens': [11612160, 479001600, 4320, 3669120]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 28], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{28}', 'transitive_degree': 56, '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': '56.13', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [12, 12, 30, 10, 2, 10, 60], 'aut_gens': [[1, 2, 8, 80], [561, 562, 620, 240], [5, 615, 588, 404], [5, 50, 8, 880], [561, 626, 8, 1040], [561, 630, 636, 80], [561, 610, 8, 1040], [561, 631, 572, 724]], 'aut_group': None, 'aut_hash': 5837632494092988874, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 15360, 'aut_permdeg': 50, 'aut_perms': [18965454407775376502411152476002708222185302373968712178966887318, 7485743674254324754202192537226282782320193309141766655642104391, 12333535348248209936247138610578892690558351942575449214002203556, 25769080156806506245558185943207089584876516533567955326044649019, 29210757915746799363570092290227341582234909917908545046024080278, 14814974884060028685425149604778466768756798523693666807381427899, 27458708942417364011923412553789622964242769827031281823477263692], 'aut_phi_ratio': 40.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 4, 1, 1], [4, 4, 1, 1], [4, 10, 4, 1], [4, 20, 2, 1], [5, 2, 2, 1], [7, 1, 6, 1], [10, 2, 2, 1], [10, 2, 4, 1], [10, 4, 4, 2], [14, 1, 6, 1], [14, 1, 12, 1], [14, 2, 12, 1], [14, 4, 6, 1], [20, 4, 4, 1], [28, 4, 6, 1], [28, 10, 24, 1], [28, 20, 12, 1], [35, 2, 12, 1], [70, 2, 12, 1], [70, 2, 24, 1], [70, 4, 24, 2], [140, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{15}:(C_2^6.C_2^4)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '384.20169', 'autcent_hash': 20169, 'autcent_nilpotent': True, 'autcent_order': 384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\\times C_6', '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], [2, 4, 1], [4, 4, 1], [4, 10, 4], [4, 20, 2], [5, 2, 2], [7, 1, 6], [10, 2, 6], [10, 4, 8], [14, 1, 18], [14, 2, 12], [14, 4, 6], [20, 4, 4], [28, 4, 6], [28, 10, 24], [28, 20, 12], [35, 2, 12], [70, 2, 36], [70, 4, 48], [140, 4, 24]], 'center_label': '28.4', 'center_order': 28, 'central_product': True, 'central_quotient': '40.13', 'commutator_count': 1, 'commutator_label': '20.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 784, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['160.156', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 1], [4, 4, 1, 1], [4, 10, 2, 2], [4, 20, 1, 2], [5, 2, 2, 1], [7, 1, 6, 1], [10, 2, 2, 3], [10, 4, 2, 2], [10, 4, 4, 1], [14, 1, 6, 3], [14, 2, 6, 2], [14, 4, 6, 1], [20, 4, 4, 1], [28, 4, 6, 1], [28, 10, 12, 2], [28, 20, 6, 2], [35, 2, 12, 1], [70, 2, 12, 3], [70, 4, 12, 2], [70, 4, 24, 1], [140, 4, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 28728, 'exponent': 140, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '280.38', 'hash': 784, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 2, 10, 1], 'inner_gens': [[1, 2, 12, 80], [1, 2, 632, 80], [5, 578, 8, 80], [1, 2, 8, 80]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': False, 'inner_tex': 'C_2\\times D_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 56], [2, 154], [4, 28]], 'label': '1120.784', 'linC_count': 1536, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 160, 'linQ_dim': 16, 'linQ_dim_count': 128, 'linR_count': 576, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C7*C2^3.D10', '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': 28, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 238, 'number_divisions': 40, 'number_normal_subgroups': 66, 'number_subgroup_autclasses': 104, 'number_subgroup_classes': 156, 'number_subgroups': 416, 'old_label': None, 'order': 1120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 11], [4, 84], [5, 4], [7, 6], [10, 44], [14, 66], [20, 16], [28, 504], [35, 24], [70, 264], [140, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 4, 12], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 3, 8, 84], [565, 2, 8, 80], [565, 6, 8, 1040], [1, 2, 24, 80], [5, 3, 8, 724]], 'outer_group': '384.19882', 'outer_hash': 19882, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 15, 'outer_perms': [186810624000, 3628800, 24, 3674904, 187293254427], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4\\times C_{12}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 7], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 6], [6, 8], [8, 4], [12, 2], [24, 6], [48, 4]], 'representations': {'PC': {'code': 2784954514917482529873619117937, 'gens': [1, 2, 4, 6], 'pres': [7, -2, -2, -2, 2, -5, -2, -7, 36, 339, 8858, 80, 1131, 124]}, 'Perm': {'d': 24, 'gens': [1126433629785835941127, 87096360, 135732245, 6780385003392000, 16, 179424720, 28102451096448368640000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 14], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7\\times C_2^3.D_{10}', 'transitive_degree': 560, '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': 20, 'aut_gen_orders': [4, 10], 'aut_gens': [[1, 2], [1, 6], [7, 2]], 'aut_group': '40.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 40, 'aut_permdeg': 7, 'aut_perms': [169, 2929], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_5', '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': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 2], [5, 2, 2], [10, 2, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['10.1', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [5, 2, 2, 1], [10, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 1, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '20.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 18], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 4]], 'label': '20.4', 'linC_count': 2, 'linC_degree': 2, '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': 'D10', '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': 8, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 22, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 11], [5, 4], [10, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [1, 0], 'outer_gens': [[1, 6], [11, 2]], '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': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 2]], 'representations': {'PC': {'code': 2624387, 'gens': [1, 2], 'pres': [3, -2, -2, -5, 109, 16, 146]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7430093, 16717348]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504, 501]}, 'Perm': {'d': 7, 'gens': [126, 1, 966]}}, '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_{10}', 'transitive_degree': 10, 'wreath_data': None, 'wreath_product': False}
