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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '960.10151', 'ambient_counter': 10151, 'ambient_order': 960, 'ambient_tex': 'S_3\\times Q_8\\times C_{20}', 'central': False, 'central_factor': True, 'centralizer_order': 240, 'characteristic': True, 'core_order': 160, 'counter': 24, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '960.10151.6.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '6.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '6.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 6, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'S_3', 'simple': False, 'solvable': True, 'special_labels': ['U0'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '160.180', 'subgroup_hash': 180, 'subgroup_order': 160, 'subgroup_tex': 'Q_8\\times C_{20}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.10151', 'aut_centralizer_order': 24, 'aut_label': '6.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1536.408640968', 'aut_weyl_index': 24, 'centralizer': '4.a1', 'complements': ['160.d1'], 'conjugacy_class_count': 1, 'contained_in': ['2.e1', '3.a1'], 'contains': ['12.a1', '12.b1', '12.c1', '30.a1'], 'core': '6.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8694, 545, 7248, 9523], 'generators': [1, 480, 720, 192, 4, 488], 'label': '960.10151.6.a1', 'mobius_quo': 0, 'mobius_sub': 3, 'normal_closure': '6.a1', 'normal_contained_in': ['2.e1'], 'normal_contains': ['12.a1', '12.b1', '12.c1', '30.a1'], 'normalizer': '1.a1', 'old_label': '6.a1', 'projective_image': '24.14', 'quotient_action_image': '1.1', 'quotient_action_kernel': '6.1', 'quotient_action_kernel_order': 6, 'quotient_fusion': None, 'short_label': '6.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '80.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2, 4, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 8], [41, 82, 152], [41, 2, 55], [81, 82, 72], [5, 6, 72], [1, 2, 136], [5, 2, 12], [5, 2, 8], [1, 2, 88], [81, 2, 8], [1, 2, 72]], 'aut_group': '1536.408640968', 'aut_hash': 807845205999895426, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 16, 'aut_perms': [9529301738903, 6656768697600, 16, 7, 11663, 6719953726080, 5786825950080, 7094612810880, 4103646986880, 5160], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 4, 1], [4, 2, 6, 2], [5, 1, 4, 1], [10, 1, 4, 3], [20, 1, 16, 1], [20, 2, 24, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_2^7.D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.56082', 'autcent_hash': 56082, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 1, 4], [4, 2, 12], [5, 1, 4], [10, 1, 12], [20, 1, 16], [20, 2, 48]], 'center_label': '40.9', 'center_order': 40, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 180, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [4, 2, 1, 6], [4, 2, 2, 3], [5, 1, 4, 1], [10, 1, 4, 3], [20, 1, 8, 2], [20, 2, 4, 6], [20, 2, 8, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 868, 'exponent': 20, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '40.14', 'hash': 180, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 1, 2], 'inner_gens': [[1, 2, 88], [1, 2, 8], [81, 2, 8]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 20]], 'label': '160.180', 'linC_count': 768, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 64, 'linQ_dim': 10, 'linQ_dim_count': 104, 'linR_count': 208, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'Q8*C20', 'ngens': 6, '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': 14, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 100, 'number_divisions': 30, 'number_normal_subgroups': 64, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 70, 'number_subgroups': 76, 'old_label': None, 'order': 160, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 28], [5, 4], [10, 12], [20, 112]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 6, 2, 2, 2, 4], 'outer_gen_pows': [1, 0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 91], [41, 82, 51], [85, 86, 8], [1, 2, 92], [85, 2, 8], [1, 2, 136]], 'outer_group': '384.20048', 'outer_hash': 20048, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 12, 'outer_perms': [362880, 7262760, 11527, 40279680, 87091200, 17], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^5.D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 9], [8, 6], [16, 1]], 'representations': {'PC': {'code': 19721404171464323075, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -2, -2, -2, -5, 504, 31, 2115, 69, 88]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [73926346121, 122109387504, 128232072316, 16699014870, 82502651532, 26013099178]}, 'Perm': {'d': 17, 'gens': [23906015527680, 45968909101920, 46080, 96, 68436399225600, 68436399316320]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 20], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_8\\times C_{20}', 'transitive_degree': 160, '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': '160.228', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 6, 12, 12, 12, 2, 6, 6], 'aut_gens': [[1, 2, 4, 16], [481, 650, 244, 176], [9, 2, 244, 77], [481, 170, 732, 784], [489, 810, 249, 701], [9, 802, 721, 453], [481, 490, 484, 184], [489, 482, 244, 269], [9, 482, 241, 797]], 'aut_group': None, 'aut_hash': 390762952444785250, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 36864, 'aut_permdeg': 32, 'aut_perms': [108831772660835944258404973736292127, 67968685111372362481948742579574454, 113131491537124122504534895137154021, 141269868417641375103398203170131472, 71701032663418973692719386743754759, 109741252739562287684822048972855964, 142239300856765761028116496071290394, 67708632011838291637977587757389431], 'aut_phi_ratio': 144.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 4, 1], [3, 2, 1, 1], [4, 1, 4, 1], [4, 2, 6, 2], [4, 3, 4, 1], [4, 6, 6, 2], [5, 1, 4, 1], [6, 2, 1, 3], [10, 1, 4, 3], [10, 3, 16, 1], [12, 2, 4, 1], [12, 4, 6, 2], [15, 2, 4, 1], [20, 1, 16, 1], [20, 2, 24, 2], [20, 3, 16, 1], [20, 6, 24, 2], [30, 2, 4, 3], [60, 2, 16, 1], [60, 4, 24, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times C_4\\times C_2^2:S_4.C_2^2\\times S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 1718285292446712972, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [3, 2, 1], [4, 1, 4], [4, 2, 12], [4, 3, 4], [4, 6, 12], [5, 1, 4], [6, 2, 3], [10, 1, 12], [10, 3, 16], [12, 2, 4], [12, 4, 12], [15, 2, 4], [20, 1, 16], [20, 2, 48], [20, 3, 16], [20, 6, 48], [30, 2, 12], [60, 2, 16], [60, 4, 48]], 'center_label': '40.9', 'center_order': 40, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10151, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1], ['6.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [3, 2, 1, 1], [4, 1, 2, 2], [4, 2, 1, 6], [4, 2, 2, 3], [4, 3, 2, 2], [4, 6, 1, 6], [4, 6, 2, 3], [5, 1, 4, 1], [6, 2, 1, 3], [10, 1, 4, 3], [10, 3, 4, 4], [12, 2, 2, 2], [12, 4, 1, 6], [12, 4, 2, 3], [15, 2, 4, 1], [20, 1, 8, 2], [20, 2, 4, 6], [20, 2, 8, 3], [20, 3, 8, 2], [20, 6, 4, 6], [20, 6, 8, 3], [30, 2, 4, 3], [60, 2, 8, 2], [60, 4, 4, 6], [60, 4, 8, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6814080, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '240.206', 'hash': 10151, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 2, 6], 'inner_gens': [[1, 2, 4, 16], [1, 2, 4, 656], [1, 2, 4, 496], [1, 322, 484, 16]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 160], [2, 120], [4, 20]], 'label': '960.10151', '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': 'S3*Q8*C20', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 36, 'number_characteristic_subgroups': 44, 'number_conjugacy_classes': 300, 'number_divisions': 90, 'number_normal_subgroups': 338, 'number_subgroup_autclasses': 192, 'number_subgroup_classes': 596, 'number_subgroups': 1136, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 15], [3, 2], [4, 112], [5, 4], [6, 6], [10, 60], [12, 56], [15, 8], [20, 448], [30, 24], [60, 224]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [4, 6, 2, 12, 6, 12], 'outer_gen_pows': [640, 0, 320, 0, 0, 0], 'outer_gens': [[489, 802, 249, 261], [1, 650, 241, 29], [1, 802, 241, 269], [1, 162, 721, 597], [481, 2, 721, 797], [489, 642, 732, 133]], 'outer_group': '1536.408640917', 'outer_hash': 5803646388267248920, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 256, 'outer_perms': [238657759802658054715592441147181242890624495015003180697109759079124460246064777849862546689618186446679957028558474667223397717148272214087563927973121262220044114172210440703467523365025167010915748812796965780924828277202120749424814044657196541854431372553328199738940374150976922059717603850907567232347449736272831972299493355371701578638008620806205176899241560522469728772195193521970610136114146802844411663189789533417692478130873406844841207024374821469719071449422604084559217330142190930591884, 311905835680779531855065255235903400957792142682970778667153927145973418581960682463201916104563363687203822422418999050407405245903254114771321390710077462406682052397090300161032321201422347799856108211212123522575021504143702386471628012252383749695247285691421860672932185231697891204605052600308089134958267241760805013092720625645297651447904108946460866626276190479346273957790852039384798080385072073632815497066766068143271491237456232901845385928816956294434569769778360010117207473389925101984496, 700410051012903102905696829040268943829944091767124767444930438021306357839059745017237843671565303790603782385394528483640147176487121531894413060043033221798227862920831356015916847840678641528047415955703840549165230612005409373756502843382195335734830369829932801707850039297912649349411385737861081174169318605372515425261209492730926413415103968704700176315752742865699593764753079608905660538075651855737878379364079914544573119789805029049833193505769308407892919528329410335905749642491856726907689, 521838214597843666421006337017898252140226401534536553767697821089663184492048606327042287187340091047184108230095026802259765804270957269328592540534506710425327464087939362729615982814262612315158473198295861885095005394814664761918473054040697971151897336083901331780456497901051754651272552282374510875678573374853629753503141857693694788328547757717182616800365577679881050132527225176867778053202796304996843997330304507821639195651743244713907882580054946631908477494473341859807063222701806891583935, 570837447974163608583238536626942377247651884123395229402733711023262651436577541341576083432783661051875810530678818196670519036453755250601902257478346953841249797056167361835323929080614346320987555711735188688648420452583989950804103988587271399310929022134873346604587011967037315086301077295407854797671477026793486352966696467858037314117794050349599669285895655850506653341831161791124285066027690668660696021569859252174759572797426957449850390809328674403927463391015057549792611834729671327059767, 348937057156585005540224866638954901476553165755137465486207377232576102370181384125309787852428818504533156983217981836129895527223990031454214678813725431005833012415643382719331028709957167567605422078646796084039342548459604710659447004060413147299775382358799915218508337401192824359758105562452347093781170932368414180806482163226850261435828526119842863216861743213093689029367331561952723413963288653781564048833902239763678129712765570750856421032425438978337228152569067811786989084166983490149630], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4\\times C_4\\times S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 24], [8, 21], [16, 8], [32, 1]], 'representations': {'PC': {'code': 35750904507878323799870479032716558139399, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -5, 3904, 66, 13132, 4980, 116, 8461, 141, 19726, 222]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [189007, 40966, 639613, 297011, 27451, 27181, 630313, 513019]}, 'Perm': {'d': 20, 'gens': [122398327071821520, 262936874919358080, 391992397842084480, 87454080, 33, 40279680, 122398327071820800, 5760]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times Q_8\\times C_{20}', 'transitive_degree': 480, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 4], [1, 3], [2, 4]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 2, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 3], [2, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 1]], 'label': '6.1', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'S3', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 3, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 6, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 3], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 3, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 25, 'gens': [1, 2], 'pres': [2, -2, -3, 17]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [28, 45]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'GL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'SL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGL'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'SO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'SU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PSO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'PSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'GO'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'Omega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PGO'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'POmega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CSO'}, {'d': 2, 'q': 4, 'gens': [20, 194], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'CSOMinus'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'CSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CO'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'COMinus'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGammaL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSigmaL'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGammaU'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGL'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11, 6]}, 'Perm': {'d': 3, 'gens': [1, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
