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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '52488.ky', 'ambient_counter': 285, 'ambient_order': 52488, 'ambient_tex': 'C_3^6:(C_3\\times \\SL(2,3))', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 1, 'counter': 73, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '52488.ky.108.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '108.f1', 'outer_equivalence': True, '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': 108, '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': '486.167', 'subgroup_hash': 167, 'subgroup_order': 486, 'subgroup_tex': 'C_3^4:C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '52488.ky', 'aut_centralizer_order': None, 'aut_label': '108.f1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '52488.a1', 'complements': None, 'conjugacy_class_count': 6, 'contained_in': ['36.f1'], 'contains': ['216.e1', '324.s1', '324.bb1', '324.bc1', '324.bd1', '324.bl1'], 'core': '52488.a1', 'coset_action_label': None, 'count': 648, 'diagramx': [8118, -1, 4727, -1], 'generators': [319413682596130149075079329234295447191093, 295495930182248274248230215974766629066, 526270821324360038037651578902662963, 271371359530713312458782619439168003, 10628907027100035260250640835378253888148, 9147069167998603594939490304244], 'label': '52488.ky.108.f1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '108.f1', 'old_label': '108.f1', 'projective_image': '52488.ky', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '108.f1', 'subgroup_fusion': None, 'weyl_group': '486.167'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', '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, 9, 6], 'aut_gens': [[24577758, 24507118, 2939606, 36731395, 14159168, 35376673], [37010401, 14582400, 14686477, 2939606, 8188666, 8713305], [2939606, 7245169, 24577758, 37010401, 14159168, 16261147], [32864349, 2794104, 43014227, 36731395, 14254600, 27828129], [7138662, 20533219, 2939606, 36731395, 39177185, 39805856]], 'aut_group': None, 'aut_hash': 4498663051369005763, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 52488, 'aut_permdeg': 99, 'aut_perms': [49410327038682731861394074394490360637917649589600472698633996958542500820323723614149787225714446876746986706978381504474535886869705454622212312453396247, 702633686408406593583143930767045084291589554751342187071878290914740020847601430061047439300901632723924433728976829055041947730887902562010539604221830861, 303765419944567440006671528531398863766962877437220680464921838048834100478005134913523025291962641551797014476963862504885897762886802750002724398763928541, 754348079244638627494869490799900887144471134261816561678599479270820165807350259049269381696527036600535202282661197304643092051474808218057748993188515208], 'aut_phi_ratio': 324.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 1, 1], [3, 2, 3, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 6, 9, 1], [3, 9, 2, 1], [3, 18, 2, 1], [6, 81, 2, 1], [9, 18, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^4.C_3^4.C_2^3', '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': 18, 'autcentquo_group': None, 'autcentquo_hash': 4498663051369005763, 'autcentquo_nilpotent': False, 'autcentquo_order': 52488, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^4.C_3^4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 81, 1], [3, 2, 4], [3, 6, 12], [3, 9, 2], [3, 18, 2], [6, 81, 2], [9, 18, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '486.167', 'commutator_count': 1, 'commutator_label': '81.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 167, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 1, 4], [3, 6, 1, 12], [3, 9, 2, 1], [3, 18, 2, 1], [6, 81, 2, 1], [9, 18, 2, 3]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 1092, 'exponent': 18, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '54.13', 'hash': 167, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [3, 3, 3, 3, 2, 3], 'inner_gens': [[24577758, 24507118, 2939606, 36731395, 8188666, 9651067], [24577758, 24507118, 2939606, 36731395, 8314856, 1305905], [24577758, 24507118, 2939606, 36731395, 39300942, 17937568], [24577758, 24507118, 2939606, 36731395, 17350487, 17937568], [20201582, 20254942, 32864349, 11225629, 14159168, 17835718], [7138662, 42750636, 20201582, 11537230, 17392559, 35376673]], 'inner_hash': 167, 'inner_nilpotent': False, 'inner_order': 486, 'inner_split': True, 'inner_tex': 'C_3^4:C_6', 'inner_used': [1, 2, 3, 4, 5, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 12], [6, 12]], 'label': '486.167', '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': 'C3^4:C6', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, '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': 11, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 30, 'number_divisions': 24, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 67, 'number_subgroup_classes': 204, 'number_subgroups': 3032, 'old_label': None, 'order': 486, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 81], [3, 134], [6, 162], [9, 108]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [3, 3, 2, 2, 6], 'outer_gen_pows': [14408200, 14408200, 14408200, 14408200, 14408200], 'outer_gens': [[32864349, 7245169, 43014227, 11537230, 14159168, 17937568], [24577758, 7245169, 2939606, 36731395, 14159168, 35376673], [14686477, 14633333, 11225629, 32864349, 14159168, 4514276], [36731395, 36804465, 14729278, 24577758, 14159168, 23409962], [11537230, 20327769, 14729278, 24577758, 14159168, 21835283]], 'outer_group': '108.28', 'outer_hash': 28, 'outer_nilpotent': False, 'outer_order': 108, 'outer_permdeg': 11, 'outer_perms': [24520008, 18235470, 1, 184680, 5171761], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:D_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [4, 4], [6, 12]], 'representations': {'PC': {'code': 1803166355202868771024991, 'gens': [1, 3, 4, 5, 6], 'pres': [6, -2, -3, -3, -3, -3, -3, 12, 866, 1844, 8643, 4113, 3244, 11669]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 24507118, 2939606, 36731395, 14159168, 35376673]}, 'Perm': {'d': 12, 'gens': [1502665, 47917560, 11294643, 4, 15040224, 92348664]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:C_6', 'transitive_degree': 27, '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': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [6, 18, 12], 'aut_gens': [[308167971258442996497287516118302064401733, 319413682596130149075079329234295447191093], [95734918039542775698251394395430273981813, 31919340963195754053668288226087299844475], [297539845401016333550030864885963197649730, 276256227384111403551168748266428183299679], [7868451686399369052959624808794101196, 361659092457999576172530167800944939889925]], 'aut_group': None, 'aut_hash': 3765517972909412070, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 314928, 'aut_permdeg': 270, 'aut_perms': [5830208806454223043795038035148358952396312121319765389673791935302297705802755705115764874066173745829918743524399027152489347740665761782506154839084978662841796721559959368234246737142310877643471718702321287609599588444311888954313076783465570594037142634869477389965593381744539112174294920515732416588952183652772852236422877948336430286369448850060832485285124076681581258674165386549965992304192590478553710636001193716576838205766375986078192675156902383504189190102074744557690878609132708442010650460754182568798239279767597794073, 2386351689076275969315435276537276236853135185438974245350301691614100667991328946740136026700068053870214188567619009735976268101912519926587418307760466183216347955756493387514637270315791602823500403580325326901836621517969866372458383323473091392079165130044554196380389157582845388445281372297208528135356659825797890966286954889206553346215132656514522821355222129981775297900099331863817265872300685371643492666966330625211124835858008723231032721950234035085833460116281188615670588766698830282178569998824451071945301967239155017107, 2609845301786568570225254615897483136159623348898766793633483781062060784974673979059037783930896147897904279313662904318835747141601111452227298308906787985466718446912055810996246910444320680879572315034771107509078687625394974352759947310876821246673803503589574244544823081212804978610733779287903865524716328779571318068945934293130993032912697077108403611697258686855591336728422734870039052103063438603296935950191547594710469056355444188114655413681513797968886164795105944127245138449673477789507139441208502554317714772984237056493], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 729, 1, 1], [3, 8, 1, 1], [3, 24, 1, 1], [3, 24, 2, 1], [3, 24, 3, 1], [3, 72, 1, 2], [3, 72, 6, 1], [3, 81, 2, 1], [3, 108, 1, 2], [3, 216, 1, 2], [3, 324, 2, 2], [3, 648, 1, 2], [4, 4374, 1, 1], [6, 729, 2, 1], [6, 2916, 1, 2], [6, 2916, 2, 2], [9, 648, 2, 3], [9, 648, 3, 2], [9, 1944, 2, 2], [12, 4374, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^6.C_2.C_6^2.C_6', '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': 36, 'autcentquo_group': None, 'autcentquo_hash': 3765517972909412070, 'autcentquo_nilpotent': False, 'autcentquo_order': 314928, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^6.C_2.C_6^2.C_6', 'cc_stats': [[1, 1, 1], [2, 729, 1], [3, 8, 1], [3, 24, 6], [3, 72, 8], [3, 81, 2], [3, 108, 2], [3, 216, 2], [3, 324, 4], [3, 648, 2], [4, 4374, 1], [6, 729, 2], [6, 2916, 6], [9, 648, 12], [9, 1944, 4], [12, 4374, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '52488.ky', 'commutator_count': 1, 'commutator_label': '5832.nf', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 285, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 729, 1, 1], [3, 8, 1, 1], [3, 24, 1, 6], [3, 72, 1, 8], [3, 81, 2, 1], [3, 108, 2, 1], [3, 216, 2, 1], [3, 324, 2, 2], [3, 648, 2, 1], [4, 4374, 1, 1], [6, 729, 2, 1], [6, 2916, 2, 3], [9, 648, 2, 6], [9, 1944, 2, 2], [12, 4374, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3456, 'exponent': 36, 'exponents_of_order': [8, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 0, 6], [24, 1, 3], [72, 1, 8]], 'familial': False, 'frattini_label': '81.15', 'frattini_quotient': '648.702', 'hash': 7346670298140495151, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [9, 6], 'inner_gens': [[308167971258442996497287516118302064401733, 138275050892995429164872667745638279217892], [95734391769332034713762239915646708730838, 319413682596130149075079329234295447191093]], 'inner_hash': 7346670298140495151, 'inner_nilpotent': False, 'inner_order': 52488, 'inner_split': True, 'inner_tex': 'C_3^6:(C_3\\times \\SL(2,3))', 'inner_used': [1, 2], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 9], [2, 9], [3, 3], [8, 9], [24, 18], [72, 8]], 'label': '52488.ky', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'C3^6:(C3*SL(2,3))', 'ngens': 2, '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': 32, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 56, 'number_divisions': 37, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 1144, 'number_subgroup_classes': 4089, 'number_subgroups': 2216466, 'old_label': None, 'order': 52488, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 729], [3, 4130], [4, 4374], [6, 18954], [9, 15552], [12, 8748]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[297539590493017635207024329445865931960278, 64117625538144909400944921531176633372012]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': None, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 5], [3, 1], [4, 4], [6, 1], [8, 1], [16, 4], [24, 6], [48, 6], [72, 8]], 'representations': {'PC': {'code': '2516317908057645493927253817115674445047226421572935515119078962725175931692064437576803588145016844673356038836981003765291231929928217760298209168765182771523261156647543440498984352910163032625012311060720593698111', 'gens': [1, 2, 4, 6, 7, 8, 9, 10, 11], 'pres': [11, 3, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 262152, 674653, 56, 705872, 46158, 1881399, 178214, 311149, 124, 956344, 13875, 457076, 389669, 278800, 380187, 119894, 313, 3914070, 410273, 180208, 9587, 974, 4371847, 1132050, 161597, 247496, 3219, 1568168, 1504027, 531066, 239225, 10744, 3888729, 982100, 502951, 63402, 35693, 4068514, 145221, 137972, 254143, 117666]}, 'Perm': {'d': 36, 'gens': [319413682596130149075079329234295447191093, 308167971258442996497287516118302064401733]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^6:(C_3\\times \\SL(2,3))', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
