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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '972.713', 'ambient_counter': 713, 'ambient_order': 972, 'ambient_tex': '\\He_3:C_6^2', 'central': False, 'central_factor': False, 'centralizer_order': 36, 'characteristic': False, 'core_order': 108, 'counter': 11, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '972.713.9.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '9.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '9.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 9, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '108.30', 'subgroup_hash': 30, 'subgroup_order': 108, 'subgroup_tex': 'C_2^2\\times \\He_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '972.713', 'aut_centralizer_order': 54, 'aut_label': '9.b1', 'aut_quo_index': 4, 'aut_stab_index': 9, 'aut_weyl_group': '648.555', 'aut_weyl_index': 486, 'centralizer': '27.a1', 'complements': ['108.e1'], 'conjugacy_class_count': 9, 'contained_in': ['3.a1', '3.c1'], 'contains': ['18.b1', '27.b1', '27.f1'], 'core': '9.b1', 'coset_action_label': None, 'count': 9, 'diagramx': [7174, 2255, 6971, 6681], 'generators': [27, 324, 486, 453, 342], 'label': '972.713.9.b1', 'mobius_quo': 0, 'mobius_sub': 3, 'normal_closure': '9.b1', 'normal_contained_in': ['3.a1', '3.c1'], 'normal_contains': ['18.b1', '27.b1'], 'normalizer': '1.a1', 'old_label': '9.b1', 'projective_image': '81.12', 'quotient_action_image': '3.1', 'quotient_action_kernel': '3.1', 'quotient_action_kernel_order': 3, 'quotient_fusion': None, 'short_label': '9.b1', 'subgroup_fusion': None, 'weyl_group': '27.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '36.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [6, 6, 8], 'aut_gens': [[1, 3, 18], [38, 59, 90], [50, 11, 27], [7, 94, 90]], 'aut_group': '2592.fv', 'aut_hash': 9019849488891309561, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2592, 'aut_permdeg': 12, 'aut_perms': [135577585, 52416989, 63988105], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [3, 3, 8, 1], [6, 1, 6, 1], [6, 3, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times C_3^2:\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '54.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 54, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3\\times C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '48.29', 'autcentquo_hash': 29, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 1, 2], [3, 3, 8], [6, 1, 6], [6, 3, 24]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 30, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['27.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [3, 3, 2, 4], [6, 1, 2, 3], [6, 3, 2, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 6, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '36.14', 'hash': 30, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 1], 'inner_gens': [[1, 75, 18], [37, 3, 18], [1, 3, 18]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': True, 'inner_tex': 'C_3^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 36], [3, 8]], 'label': '108.30', 'linC_count': 108, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 6, 'linQ_dim': 7, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*He3', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 44, 'number_divisions': 24, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 15, 'number_subgroup_classes': 55, 'number_subgroups': 95, 'old_label': None, 'order': 108, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [3, 26], [6, 78]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 6, 4, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 53, 90], [12, 71, 18], [86, 76, 18], [6, 100, 81]], 'outer_group': '288.851', 'outer_hash': 851, 'outer_nilpotent': False, 'outer_order': 288, 'outer_permdeg': 11, 'outer_perms': [888600, 1103761, 19036320, 21910467], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_3\\times \\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 16], [6, 4]], 'representations': {'PC': {'code': 672742246229, 'gens': [1, 2, 4], 'pres': [5, -3, -2, -3, -2, -3, 751, 26, 632, 58]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [36728797, 2939606, 1017496, 17128748, 28816400]}, 'Perm': {'d': 13, 'gens': [479001600, 3628800, 138838, 276480, 80884]}}, 'schur_multiplier': [3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times \\He_3', 'transitive_degree': 36, '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': '108.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [6, 6, 6, 18, 6], 'aut_gens': [[1, 3, 9, 54], [325, 544, 693, 290], [650, 457, 657, 929], [650, 260, 504, 406], [649, 259, 522, 947], [650, 148, 504, 621]], 'aut_group': None, 'aut_hash': 7364595836974217985, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 314928, 'aut_permdeg': 165, 'aut_perms': [32218060770543835073954456341847712676881906071331004480992136745840874923423495680949595151568406358938617968692312103847646858043187692122777843834137936847106925244507541215748860784535576380380897314423220638866653685963873661565649327002130342448721195142035105732281225183065194307098036720, 12822949606982213640912820541338347201356032223512072096628688092670291408957994264763194984273515596800856590240258474534526610433917927268732736320492830331614951273298239845216630524360216974963785549276680091432332360503827068913250367000307807757393953822190832255648279316616270880038084825, 49979284283333650215478263496632324652915521634227419326905023931882516076896217040303064198296225963825699612080371359421023307745403902968214471295616172739433460373620895591751608412478294191541662383253163612990734925298045703891943394931916061719764430828886543762609193944808117682642895765, 34857267263517395987384997639766018094225864515462045123012623322953983728542694006725405858365206343692198182381283533380854567030239081027780019281209360809233906985188941438231921734536769009445760239565834405304803430846946692622163089344506463488131241835695255643697579203331959542382976007, 38801264168486591259699394833051557741224521317621724621544651552230255597361341904172128448381822094114538278670724939919477398410926944048204792855185702448755021483562624140229316784467516586442488842786988977300595607406235791868297944923215638977501020564154714859129103925226065447273186151], 'aut_phi_ratio': 972.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [3, 1, 6, 1], [3, 3, 2, 1], [3, 3, 4, 1], [3, 9, 18, 1], [6, 1, 6, 1], [6, 1, 18, 1], [6, 3, 6, 1], [6, 3, 12, 1], [6, 9, 54, 1], [9, 3, 18, 1], [18, 3, 54, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times C_3^4.C_3^4.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': None, 'autcent_hash': 3747242782131798215, 'autcent_nilpotent': False, 'autcent_order': 2916, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'S_3\\times C_3.\\He_3:S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '108.17', 'autcentquo_hash': 17, 'autcentquo_nilpotent': False, 'autcentquo_order': 108, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2:D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 1, 8], [3, 3, 6], [3, 9, 18], [6, 1, 24], [6, 3, 18], [6, 9, 54], [9, 3, 18], [18, 3, 54]], 'center_label': '36.14', 'center_order': 36, 'central_product': True, 'central_quotient': '27.3', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 713, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1], ['81.9', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 4], [3, 3, 2, 3], [3, 9, 2, 9], [6, 1, 2, 12], [6, 3, 2, 9], [6, 9, 2, 27], [9, 3, 6, 3], [18, 3, 6, 9]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1092, 'exponent': 18, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '108.45', 'hash': 713, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [1, 3, 3, 3], 'inner_gens': [[1, 3, 9, 54], [1, 3, 333, 738], [1, 651, 9, 54], [1, 345, 9, 54]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 27, 'inner_split': False, 'inner_tex': '\\He_3', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 108], [3, 96]], 'label': '972.713', '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': 'He3:C6^2', 'ngens': 7, 'nilpotency_class': 3, 'nilpotent': True, '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': 14, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 204, 'number_divisions': 80, 'number_normal_subgroups': 180, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 420, 'number_subgroups': 1440, 'old_label': None, 'order': 972, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [3, 188], [6, 564], [9, 54], [18, 162]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [6, 6, 6, 6, 6], 'outer_gen_pows': [0, 684, 684, 342, 0], 'outer_gens': [[649, 241, 531, 380], [326, 111, 495, 56], [326, 474, 495, 461], [649, 475, 495, 460], [1, 544, 531, 783]], 'outer_group': None, 'outer_hash': 2443853101640665261, 'outer_nilpotent': False, 'outer_order': 11664, 'outer_permdeg': 23, 'outer_perms': [5878513369995545726166, 9257704331014809936168, 7102128609344017027831, 9257704331021039101129, 3386613468511711927471], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^4.D_6^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 34, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 52], [6, 12], [18, 12]], 'representations': {'PC': {'code': 275844153375738816630974061163, 'gens': [1, 2, 3, 5], 'pres': [7, 3, 3, 2, 3, 2, 3, 3, 2340, 58, 6226, 8621, 102, 6312, 166]}, 'GLZN': {'d': 2, 'p': 54, 'gens': [8345645, 157483, 158437, 211285, 158923, 4970947, 2991835]}, 'Perm': {'d': 34, 'gens': [8683317618811886495518194401280000000, 8222838654177922817725562880000000, 9823379212595838572034159589347, 18982253052262400815558566534504, 28140211547958303057602869753107, 1840764313927677088715674302720, 28140211547958303057602869753104]}}, 'schur_multiplier': [3, 3, 3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\He_3:C_6^2', 'transitive_degree': 324, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 3], [1, 7], [4, 3]], 'aut_group': '48.29', 'aut_hash': 29, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 8, 'aut_perms': [31834, 28334], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', '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], [3, 1, 8]], 'center_label': '9.2', 'center_order': 9, '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': ['3.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '9.2', 'hash': 2, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 9]], 'label': '9.2', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 6, 'linQ_dim': 4, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 9, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 9, 'order_factorization_type': 2, 'order_stats': [[1, 1], [3, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 7], [4, 3]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [31834, 28334], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 3, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -3, 3]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16858733, 35931237]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [687, 1374]}, 'Perm': {'d': 6, 'gens': [240, 4]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2', 'transitive_degree': 9, 'wreath_data': None, 'wreath_product': False}
