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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1728.7359', 'ambient_counter': 7359, 'ambient_order': 1728, 'ambient_tex': 'C_2^2.(D_6\\times C_{36})', 'central': False, 'central_factor': False, 'centralizer_order': 72, 'characteristic': True, 'core_order': 288, 'counter': 24, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1728.7359.6.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '6.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '6.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 6, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '288.265', 'subgroup_hash': 265, 'subgroup_order': 288, 'subgroup_tex': 'C_6^2.D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.7359', 'aut_centralizer_order': None, 'aut_label': '6.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '24.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1', '3.a1.a1'], 'contains': ['12.a1.a1', '12.e1.a1', '12.e1.b1', '18.d1.a1', '18.h1.a1'], 'core': '6.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [37, 864, 72, 432, 576, 48, 866], 'label': '1728.7359.6.c1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '6.c1.a1', 'normal_contained_in': ['2.c1.a1', '3.a1.a1'], 'normal_contains': ['12.a1.a1', '12.e1.b1', '12.e1.a1', '18.d1.a1'], 'normalizer': '1.a1.a1', 'old_label': '6.c1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.c1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '48.20', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 4, 2, 6, 2, 12, 6], 'aut_gens': [[1, 4, 24], [85, 4, 134], [73, 4, 182], [3, 4, 36], [97, 20, 180], [13, 20, 278], [25, 20, 24], [111, 4, 24]], 'aut_group': '1536.230951259', 'aut_hash': 1816206915461081621, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 50, 'aut_perms': [29188989920586137727917876942828020377681358867957357261357055305, 1273741910028983713985358057812551280604654771364131658140728130, 23595442486623578134563839034689859843564632642349177503066229120, 638800657014953061594173668691573640568899167649476640168147717, 27327663028295354664366084805385044971429829223171641770988689865, 18633695939239827888614416217862349571300358983902111915309472067, 20504338980239292813495851315354031495562901759419336401229168731], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 2, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 2, 4, 1], [4, 6, 8, 1], [6, 1, 2, 3], [6, 1, 4, 2], [6, 2, 1, 3], [6, 2, 2, 5], [6, 2, 4, 2], [12, 2, 8, 2], [12, 2, 16, 1], [12, 6, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^7:D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 1, 2], [3, 2, 3], [4, 2, 4], [4, 6, 8], [6, 1, 14], [6, 2, 21], [12, 2, 32], [12, 6, 16]], 'center_label': '24.15', 'center_order': 24, 'central_product': True, 'central_quotient': '12.4', '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', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 265, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['96.38', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 2, 2, 2], [4, 6, 2, 4], [6, 1, 2, 7], [6, 2, 1, 7], [6, 2, 2, 7], [12, 2, 4, 8], [12, 6, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 12, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '36.12', 'hash': 265, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 1, 6], 'inner_gens': [[1, 4, 132], [1, 4, 24], [205, 4, 24]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'D_6', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48], [2, 60]], 'label': '288.265', 'linC_count': 9984, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 436, 'linQ_dim': 8, 'linQ_dim_count': 112, 'linR_count': 224, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6^2.D4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 38, 'number_conjugacy_classes': 108, 'number_divisions': 50, 'number_normal_subgroups': 90, 'number_subgroup_autclasses': 104, 'number_subgroup_classes': 179, 'number_subgroups': 346, 'old_label': None, 'order': 288, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 7], [3, 8], [4, 56], [6, 56], [12, 160]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[147, 20, 180], [1, 4, 36], [157, 20, 182], [1, 4, 38], [3, 4, 180], [73, 4, 38]], 'outer_group': '128.2163', 'outer_hash': 2163, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 32, 'outer_perms': [18040040231674517819654488857952561, 26613839300677899158918793270129230, 372916560599383498841542738, 103475840684034083709650924727492151, 135368630734863229607901061423635538, 229628048794621410655124709513915057], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 24], [8, 4]], 'representations': {'PC': {'code': 9877226824890250212874889, 'gens': [1, 3, 5], 'pres': [7, 2, 2, 2, 3, 2, 2, 3, 14, 58, 4624, 102, 10085, 124, 9414]}, 'GLZN': {'d': 2, 'p': 28, 'gens': [548825, 21977, 285389, 22345, 158599, 329295, 22553]}, 'Perm': {'d': 18, 'gens': [380813497308856, 401737205726640, 325, 795246639264000, 1175606442223920, 1552384829318400, 435]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2.D_4', 'transitive_degree': 96, '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': '144.47', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [12, 6, 12, 12, 6, 12, 6, 4, 6], 'aut_gens': [[1, 4, 144], [1299, 676, 1082], [1, 332, 792], [433, 1462, 218], [1371, 990, 1008], [865, 668, 1080], [435, 966, 1080], [939, 676, 1656], [433, 1230, 1586], [937, 1486, 1008]], 'aut_group': None, 'aut_hash': 3153643385379992926, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 36864, 'aut_permdeg': 46, 'aut_perms': [4233930828821242017723067170115693396903744777791030404183, 4550340395588648309266139590203304906533462944270664616740, 3744336512779932039064855730988868682180384743951632099694, 4960169656498849912227514637049124203521938223818356023374, 3740894720049663341819797342968406054420271369584762567793, 4205219136667814726265982234733200744367922394784995853175, 4207666245303859364914588512029769407665460796492367675449, 31997702545817432370757198826114087483365929111329533314, 4584988425766764193968398239581726469713146258958559625098], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 2, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 2, 4, 1], [4, 4, 4, 1], [4, 6, 4, 2], [4, 12, 4, 1], [6, 1, 2, 3], [6, 1, 4, 2], [6, 2, 1, 3], [6, 2, 2, 5], [6, 2, 4, 2], [9, 1, 6, 1], [9, 2, 6, 1], [12, 2, 8, 1], [12, 4, 4, 1], [12, 4, 8, 3], [12, 4, 16, 1], [12, 6, 8, 2], [12, 12, 8, 1], [18, 1, 6, 3], [18, 1, 12, 2], [18, 2, 6, 3], [18, 2, 12, 2], [36, 2, 24, 1], [36, 4, 24, 2], [36, 4, 48, 1], [36, 6, 24, 2], [36, 12, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6.(C_2^5\\times C_6).C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': None, 'autcent_hash': 1957374279625110835, 'autcent_nilpotent': True, 'autcent_order': 3072, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^9\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 1, 2], [3, 2, 3], [4, 2, 4], [4, 4, 4], [4, 6, 8], [4, 12, 4], [6, 1, 14], [6, 2, 21], [9, 1, 6], [9, 2, 6], [12, 2, 8], [12, 4, 44], [12, 6, 16], [12, 12, 8], [18, 1, 42], [18, 2, 42], [36, 2, 24], [36, 4, 96], [36, 6, 48], [36, 12, 24]], 'center_label': '72.18', 'center_order': 72, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '12.5', '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', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 7359, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['192.210', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 2, 2, 2], [4, 4, 2, 2], [4, 6, 2, 4], [4, 12, 2, 2], [6, 1, 2, 7], [6, 2, 1, 7], [6, 2, 2, 7], [9, 1, 6, 1], [9, 2, 6, 1], [12, 2, 4, 2], [12, 4, 2, 2], [12, 4, 4, 10], [12, 6, 4, 4], [12, 12, 4, 2], [18, 1, 6, 7], [18, 2, 6, 7], [36, 2, 12, 2], [36, 4, 12, 8], [36, 6, 12, 4], [36, 12, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 39312, 'exponent': 36, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '24.15', 'frattini_quotient': '72.48', 'hash': 7359, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 6], 'inner_gens': [[1, 940, 216], [937, 4, 720], [73, 1156, 144]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 144], [2, 252], [4, 36]], 'label': '1728.7359', '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': 'C2^2.(D6*C36)', 'ngens': 9, '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': 57, 'number_characteristic_subgroups': 75, 'number_conjugacy_classes': 432, 'number_divisions': 94, 'number_normal_subgroups': 189, 'number_subgroup_autclasses': 296, 'number_subgroup_classes': 490, 'number_subgroups': 1160, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 7], [3, 8], [4, 120], [6, 56], [9, 18], [12, 384], [18, 126], [36, 1008]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [12, 6, 6, 6, 6, 12, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 576], 'outer_gens': [[1299, 380, 722], [3, 1246, 720], [867, 102, 794], [1371, 404, 216], [75, 1534, 216], [1299, 630, 218], [939, 580, 146]], 'outer_group': '1536.7540510', 'outer_hash': 3687838209913611258, 'outer_nilpotent': True, 'outer_order': 1536, 'outer_permdeg': 192, 'outer_perms': [138252656801427759514382568566681619239295869144401725473608738095279950735328669466959462002966393243884459498417263416164850704252709585126276703377983288149977961137806962211862491876543399153678256960152674743702320877468255976576159704830026112058454565897435137179079830601093876567839535696868910266153949157243376663863850186429289948951866841262981, 101591077510537118230336015404093943125442205177252892066595007545299590693589586302798698511212031787894222326713334038803693135320447754322910733140571780752034649703672791672575258958258334562096198531926990277778109149551086604516155612636448763949134979388813578046141511692560277810726386953212656225373977950753581719316439755250892993626350780861039, 317579435134348904913771297533758049260952379583959876070991126241779789074818966340851764431213867348518874267772724893186922244219188961925614093901348949005443617937471873462213707864723021927398219909259905164438770139000751883202904526262125942485550814949867428977108912887011119072403102669077572751399436262406604700904893754495658006589603072162066, 188278453353764419397246377117547022898798255324588407896261257863323151378937563018892410500810375306086829570864534318230904372222086596071713633165640789972248050432521576466557916650269785593069051877483411848832249266492503873941763912901321312381726219451072360385187334880629679873468155038814169944388665959998958684855784426475618374568506365075925, 295190435489715866034156075573765397673862127169876536549576409533500848388637644428103609576761386935422360445845088220736117308399475058565312418966324993668256733357453141198232341760782070583487514029299679224271179632541708691894270778298376279234947842894939610928952064943203827542165727936668943000566082843718923547760149936078533167156153276865732, 146609149867370616102683619202384819328776711585078058626611930704540481458121533108585856733159513640469358010769775355551711296045875854737036541792402469543016333126485296594534190974463983783329410328410157505283095214073325069812395606835694963812969851931336230963825863644011561093379647461891356652292640410424312728557819502775148673524440058763367, 1045575122488048718812613208886585464662594063170351352037775333133476760318019973185566463300086014169027528138342515044107890261720500293062753955290352098238842791232742598596261368615569788132561635370194671635603796911765905337495220923853919184314397966495641667906644234950706714956026420841870892874853072820950058493471296610021312366659162990800], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times C_2^7.C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 32, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 9], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 20], [6, 8], [8, 18], [12, 8], [24, 16]], 'representations': {'PC': {'code': 2140571185342227017457896437329793425006454487577, 'gens': [1, 3, 7], 'pres': [9, 2, 2, 2, 2, 3, 3, 2, 2, 3, 18, 25382, 74, 102, 175, 13614, 11364, 186, 25945, 214, 23354]}, 'Perm': {'d': 32, 'gens': [8771659467755021889550324118092801, 17260056661135556709326500309094400, 35705409564847182898631781888000, 47952144, 25903107681056686080000, 25764927116365842750596579328000000, 25764927116365843142568048112896000, 91944960, 3]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 36], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2.(D_6\\times C_{36})', 'transitive_degree': 576, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [5]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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], [3, 1, 2], [6, 1, 2]], 'center_label': '6.2', 'center_order': 6, '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', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 6]], 'label': '6.2', 'linC_count': 2, 'linC_degree': 1, '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': 'C6', 'ngens': 2, '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': 6, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2]], 'representations': {'PC': {'code': 21, 'gens': [1], 'pres': [2, -2, -3, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [73]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 56]}, 'Perm': {'d': 5, 'gens': [24, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6', 'transitive_degree': 6, 'wreath_data': None, 'wreath_product': False}
