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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1760.40', 'ambient_counter': 40, 'ambient_order': 1760, 'ambient_tex': '(C_2\\times C_{22}):C_{40}', 'central': False, 'central_factor': False, 'centralizer_order': 80, 'characteristic': False, 'core_order': 4, 'counter': 85, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1760.40.220.e1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '220.e1.b1', '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': 220, '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': '8.1', 'subgroup_hash': 1, 'subgroup_order': 8, 'subgroup_tex': 'C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.40', 'aut_centralizer_order': 40, 'aut_label': '220.e1', 'aut_quo_index': None, 'aut_stab_index': 44, 'aut_weyl_group': '4.2', 'aut_weyl_index': 1760, 'centralizer': '22.b1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['20.e1.b1', '44.e1.b1', '110.b1.b1'], 'contains': ['440.c1.b1'], 'core': '440.c1.b1', 'coset_action_label': None, 'count': 22, 'diagramx': [392, -1, 8506, -1, 9394, -1, 8777, -1], 'generators': [85], 'label': '1760.40.220.e1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '10.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '22.b1.b1', 'old_label': '220.e1.b1', 'projective_image': '440.11', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '220.e1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2], 'aut_gens': [[1], [5], [3]], 'aut_group': '4.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [1, 6], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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], [4, 1, 2], [8, 1, 4]], 'center_label': '8.1', 'center_order': 8, 'central_product': False, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 8, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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, 8]], 'label': '8.1', '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': 'C8', 'ngens': 1, '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': 8, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 2], [8, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5], [3]], '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': 1, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1]], 'representations': {'PC': {'code': 323, 'gens': [1], 'pres': [3, -2, -2, -2, 6, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26315074]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [77]}, 'Perm': {'d': 8, 'gens': [40176, 16582, 5167]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_8', 'transitive_degree': 8, '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': 220, 'aut_gen_orders': [10, 10, 22, 10, 20, 10], 'aut_gens': [[1, 40, 80], [981, 40, 740], [1141, 40, 260], [1281, 40, 120], [1651, 40, 720], [601, 40, 1240], [1431, 40, 1560]], 'aut_group': None, 'aut_hash': 1531899044600894541, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7040, 'aut_permdeg': 187, 'aut_perms': [897718945735285055506706119052594538186025354032165748262790758943559118506181160513080335747674353807858688677571592149981315118505090788325469143838721738054001312133056394256246644935969739221084860321105870718506246421147860277892419527987157633318699157524154057421886424245862200633322990706398547552109209470138787367985323830036717407681, 725109253959812446286018935292812058083787320365157335019034931201116375824845962749476194498243059727408395175624890879986371746460959585594702348773884471944350576088221722395848592329023957810131609271064986905154652805540703078313152809071489369626894944849863005894207472384002968820470460429192469616750742569466157103324278921401333508921, 132929504423975498536518964875988066702413281438552841807960480188884712765857834680968060664759578236589443273263093613571526678154569979708109215463061892950886734563772822078387195388102938869049763487234949752553227461397353711593926981633275961153513927086469939424422783146663042562880312171009582118690223692621131839770173821845462923010, 1233956298548381212554505135445880466855618878451126209553623422750817632437540515946480721173065987004131650162115091860282708481301220510361505416918595601879228467508489452556067432778368250790980748192539261362823984376777661783432947605850295183864472830556763750362707728932645361969054468317188585216702112992846550405676963836153905219176, 853030652313863639391448367638161784810644050224361697122678586851591945147942431454719868740663590291725764090511717030892714699383785593239061054803936022415288132589901382734325486946246248049244820869139382424025868225698210879359577263817340672132732976800979035056185304516409708190117924709238042582010552308973626050142862625832927639674, 1335229562323103515122068813588874405042581748447219323137417273046417646174493577380487105621646032502507444541163250248840118812864125829546677546452649636613339260362501748929663135628864345112285835034007719665049554382235740093179853230554551963161408743971560522044528035476354103834102666773581118435879031768932149930349119601333102923972], 'aut_phi_ratio': 11.0, '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, 11, 1, 4], [8, 22, 8, 1], [10, 11, 1, 12], [10, 22, 2, 4], [11, 10, 1, 1], [20, 11, 4, 4], [20, 22, 2, 4], [22, 10, 1, 3], [22, 10, 4, 1], [40, 22, 8, 4], [44, 10, 4, 2]], 'aut_supersolvable': True, 'aut_tex': '(C_2^2\\times F_{11}).C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '220.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 220, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [4, 1, 4], [4, 2, 2], [5, 11, 4], [8, 22, 8], [10, 11, 12], [10, 22, 8], [11, 10, 1], [20, 11, 16], [20, 22, 8], [22, 10, 7], [40, 22, 32], [44, 10, 8]], 'center_label': '8.2', 'center_order': 8, 'central_product': False, 'central_quotient': '220.7', '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': 40, '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], [4, 1, 2, 2], [4, 2, 2, 1], [5, 11, 4, 1], [8, 22, 4, 2], [10, 11, 4, 3], [10, 22, 4, 2], [11, 10, 1, 1], [20, 11, 8, 2], [20, 22, 8, 1], [22, 10, 1, 3], [22, 10, 2, 2], [40, 22, 16, 2], [44, 10, 2, 2], [44, 10, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 440, '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': '8.2', 'frattini_quotient': '220.7', 'hash': 40, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 1, 22], 'inner_gens': [[1, 40, 1080], [1, 40, 80], [841, 40, 80]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 220, 'inner_split': True, 'inner_tex': 'C_2\\times F_{11}', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 20], [10, 16]], 'label': '1760.40', 'linC_count': 320, 'linC_degree': 11, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 24, 'linQ_dim': 16, 'linQ_dim_count': 12, 'linR_count': 80, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*C22):C40', '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': 47, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 116, 'number_divisions': 31, 'number_normal_subgroups': 45, 'number_subgroup_autclasses': 76, 'number_subgroup_classes': 100, 'number_subgroups': 546, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [4, 8], [5, 44], [8, 176], [10, 308], [11, 10], [20, 352], [22, 70], [40, 704], [44, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[41, 40, 1680], [51, 40, 80], [51, 40, 1700], [881, 40, 1700]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [7, 23, 127, 11640], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, '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, 7], [8, 4], [10, 4], [16, 3], [20, 4], [40, 1]], 'representations': {'PC': {'code': 1347729909468269191426984934998228537270303110003009, 'gens': [1, 5, 6], 'pres': [7, -2, -2, -2, -5, -2, 2, -11, 14, 36, 58, 45365, 25212, 4219, 1286, 124, 15686, 15693, 9820, 2967]}, 'GLZN': {'d': 2, 'p': 88, 'gens': [45833215, 29644075, 682177, 30666285, 6133273, 16016551, 685345]}, 'Perm': {'d': 23, 'gens': [51353481345499076527, 90842405, 109853078970681446400, 16, 135814320, 90842400, 1277531155775242368000]}}, '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_2\\times C_{22}):C_{40}', 'transitive_degree': 176, 'wreath_data': None, 'wreath_product': False}
