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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '2400.bt', 'ambient_counter': 46, 'ambient_order': 2400, 'ambient_tex': '\\SL(2,5):D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 40, 'characteristic': False, 'core_order': 2, 'counter': 147, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2400.bt.120.o1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '120.o1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 120, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '20.1', 'subgroup_hash': 1, 'subgroup_order': 20, 'subgroup_tex': 'C_5:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2400.bt', 'aut_centralizer_order': 160, 'aut_label': '120.o1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '20.3', 'aut_weyl_index': 960, 'centralizer': '60.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['20.a1.a1', '24.f1.a1', '60.g1.a1', '60.i1.a1', '60.i1.b1'], 'contains': ['240.e1.a1', '600.f1.a1'], 'core': '1200.b1.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [4881, -1, 5649, -1, 6218, -1, 3985, -1], 'generators': [81420657990, 1271171980, 110599489058], 'label': '2400.bt.120.o1.a1', 'mobius_quo': None, 'mobius_sub': 10, 'normal_closure': '20.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.a1.a1', 'old_label': '120.o1.a1', 'projective_image': '1200.944', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '120.o1.a1', 'subgroup_fusion': None, 'weyl_group': '10.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', '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, 4], [1, 12], [11, 4]], 'aut_group': '40.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 40, 'aut_permdeg': 7, 'aut_perms': [361, 1855], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 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], [4, 5, 2], [5, 2, 2], [10, 2, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, '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': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 5, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 20, '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': '2.1', 'frattini_quotient': '10.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 16], [9, 4]], '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': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 4]], 'label': '20.1', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5:C4', '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': 5, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 10, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 10], [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, 8], [3, 4]], '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': 9, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2]], 'representations': {'PC': {'code': 133699, 'gens': [1, 3], 'pres': [3, -2, -2, -5, 6, 146]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [91655872471905395, 125101739906136769]}, 'Lie': [{'d': 2, 'q': 11, 'gens': [440, 6664, 7988, 13320], 'family': 'CSOPlus'}, {'d': 2, 'q': 9, 'gens': [1460, 2020, 6241], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 377]}, 'Perm': {'d': 9, 'gens': [5169, 16, 50520]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5:C_4', 'transitive_degree': 20, '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': 2, 'aut_exponent': 60, 'aut_gen_orders': [2, 20, 4, 2, 4, 2, 60, 2], 'aut_gens': [[122109387504, 77355542572, 130509168233, 147999673081, 148051235266, 116250388725], [122109387504, 77355542572, 60215928797, 147999673081, 42634393514, 116250388725], [5078828170, 77355542572, 66075318946, 147999673081, 122230481386, 151807329271], [122109387504, 113732510846, 92911416209, 75245970908, 1322734790, 43496295927], [122109387504, 77355542572, 71992911941, 147999673081, 24801160490, 116250388725], [5078828170, 29953804004, 87052026055, 111623017307, 38920487444, 115430282997], [5078828170, 40978887423, 108369567715, 38868924509, 111674579617, 42676190199], [5078828170, 12581083359, 103672680970, 147999673081, 102142146747, 151807329271], [5078828170, 77355542572, 130509168233, 147999673081, 24801160490, 151807329271]], 'aut_group': '19200.bj', 'aut_hash': 7188633043150786832, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 19200, 'aut_permdeg': 40, 'aut_perms': [253312846115475057493232496895076042579295413733, 790452548108827236261397032166563936580818014134, 398543359480943729697896347917605376056839531113, 353223867849215885972922208049971536524584138295, 407833444209382133252187284337502207718386887184, 97179909648855351984364163108684214723721121103, 134922383901032257881937333088546884049027910495, 5770753944467775514874717034639165089583836465], 'aut_phi_ratio': 30.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 150, 2, 1], [3, 20, 1, 1], [4, 5, 4, 1], [4, 30, 1, 2], [5, 2, 2, 1], [5, 12, 2, 1], [5, 24, 4, 1], [6, 20, 1, 1], [6, 20, 2, 1], [10, 2, 2, 1], [10, 2, 4, 1], [10, 12, 2, 1], [10, 12, 4, 1], [10, 24, 4, 1], [10, 24, 8, 1], [12, 100, 4, 1], [15, 40, 2, 1], [20, 60, 2, 2], [20, 60, 8, 1], [30, 40, 2, 1], [30, 40, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_4\\times F_5\\times S_5', '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': 60, 'autcentquo_group': '2400.db', 'autcentquo_hash': 3070016846447328869, 'autcentquo_nilpotent': False, 'autcentquo_order': 2400, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 150, 2], [3, 20, 1], [4, 5, 4], [4, 30, 2], [5, 2, 2], [5, 12, 2], [5, 24, 4], [6, 20, 3], [10, 2, 6], [10, 12, 6], [10, 24, 12], [12, 100, 4], [15, 40, 2], [20, 60, 12], [30, 40, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '600.146', 'commutator_count': 1, 'commutator_label': '600.54', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1', '60.5'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 46, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1200.481', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 150, 1, 2], [3, 20, 1, 1], [4, 5, 2, 2], [4, 30, 1, 2], [5, 2, 2, 1], [5, 12, 2, 1], [5, 24, 2, 2], [6, 20, 1, 3], [10, 2, 2, 3], [10, 12, 2, 3], [10, 24, 2, 6], [12, 100, 2, 2], [15, 40, 2, 1], [20, 60, 2, 2], [20, 60, 4, 2], [30, 40, 2, 3]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 57, 'exponent': 60, 'exponents_of_order': [5, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '1200.944', 'hash': 6351781673999964458, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [1, 15, 2, 5, 2, 5], 'inner_gens': [[122109387504, 77355542572, 130509168233, 147999673081, 148051235266, 116250388725], [122109387504, 77355542572, 92911416209, 147999673081, 30556644004, 116250388725], [122109387504, 40978887423, 130509168233, 38868924509, 148051235266, 7119249528], [122109387504, 77355542572, 50675370281, 147999673081, 75297533218, 116250388725], [122109387504, 21929326699, 130509168233, 38868924509, 148051235266, 7119249528], [122109387504, 77355542572, 87052026055, 147999673081, 38920487444, 116250388725]], 'inner_hash': 146, 'inner_nilpotent': False, 'inner_order': 600, 'inner_split': False, 'inner_tex': 'D_5\\times A_5', 'inner_used': [2, 3, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 12], [3, 8], [4, 16], [5, 4], [6, 12], [8, 8], [10, 4], [12, 4]], 'label': '2400.bt', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 8, 'linQ_dim': 9, 'linQ_dim_count': 8, 'linR_count': 16, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SL(2,5):D10', 'ngens': 6, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 26, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 72, 'number_divisions': 40, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 138, 'number_subgroup_classes': 232, 'number_subgroups': 5980, 'old_label': None, 'order': 2400, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 303], [3, 20], [4, 80], [5, 124], [6, 60], [10, 372], [12, 400], [15, 80], [20, 720], [30, 240]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [130509168233, 7089953025, 7089953025, 30527346876], 'outer_gens': [[122109387504, 4601837899, 130509168233, 111623017307, 148051235266, 79873342326], [122109387504, 59195525983, 71992911941, 111623017307, 24754285370, 79873342326], [5078828170, 59195525983, 71992911941, 111623017307, 148098110386, 115430282997], [5078828170, 59195525983, 118732185089, 111623017307, 42587518394, 115430282997]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [7, 23, 11543, 16567], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 31, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 6], [5, 4], [6, 4], [8, 8], [12, 6], [16, 4], [20, 2], [24, 2]], 'representations': {'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 77355542572, 130509168233, 147999673081, 148051235266, 116250388725]}, 'Perm': {'d': 31, 'gens': [292725929225504590146332350192230, 565917939345389685165176183195329, 841515720257750107397704921901215, 565917939345389685165176183193601, 565917939345389685165176183196120, 539393489113882869654087476465694]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SL(2,5):D_{10}', 'transitive_degree': 240, 'wreath_data': None, 'wreath_product': False}
