Formats: - HTML - YAML - JSON - 2026-07-24T15:45:07.449618
  • gps_subgroup_search • Show schema
    {'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': 3, 'characteristic': False, 'core_order': 243, 'counter': 16, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '52488.ky.24.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '24.b1', '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': 24, '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': '2187.8516', 'subgroup_hash': 8795916686994291310, 'subgroup_order': 2187, 'subgroup_tex': 'C_3^5:C_3^2', 'supersolvable': True, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '52488.ky', 'aut_centralizer_order': None, 'aut_label': '24.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '17496.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.a1', '12.b1'], 'contains': ['72.f1', '72.g1', '72.j1', '72.k1', '72.l1', '72.r1', '72.s1', '72.v1', '72.x1'], 'core': '216.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [9617, -1, 8139, -1], 'generators': [418829649741950486143184644, 418854380069595192529040884, 52881404179825461307730803421111878879170, 9147069168003469399049292522384, 297539845401015526942383150210693166328374, 10628907027100423092203445619434195690243, 526270820936528085475111584168119188], 'label': '52488.ky.24.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.a1', 'old_label': '24.b1', 'projective_image': '52488.ky', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.b1', 'subgroup_fusion': None, 'weyl_group': None}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '81.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 18, 'aut_gen_orders': [6, 6, 18, 6], 'aut_gens': [[1, 3, 9, 27, 81, 243], [139, 1464, 24, 945, 162, 479], [952, 168, 15, 1593, 81, 1346], [143, 3, 201, 1674, 891, 1225], [790, 1626, 1590, 945, 1620, 1745]], 'aut_group': None, 'aut_hash': 1829266181944197158, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1417176, 'aut_permdeg': 486, 'aut_perms': [23496487981240680149842595526345703091594617766249052020905969011599940886701452835486801361387478244429586698369749756961816218499647850130984742203590331086031717371541481459650293908403314045481988297464385192086106862425120959946758216524292623504913424647328924278856043011443557102620445621047714217090647091445606575724209788929019853431004727456836060225041150007565524152588510633757562990060111820859515744424177087249521836116533639787087575995988845283548455849386575871151534866111755518174548752692199841410506672608256927420811092605269478520515417100728718702227130307238936754976845710414312431394517083955402763407059174561840085538729228398498867951476508141096482664937424732652563677885604785328331285835184996539641754958445605368015631051131390315216259161717075971426770930777010804479721045863712369551815718944576538978198559159013700195674145299776279160271807887904861811361675151691760712888569994872698456174851238816628430420592786762767710709818632830462874369694212741466095652415130962475339638952194858607240155723659099917038514596323998343715818228236961728026, 11756174154560499093017633503244177836589732603195661612864188801222390363881858446811247387180536491533963075126815773020156888011565977903424257340087700795265843387263560987932483746013147426395225579907983818425735400553176583107843827456935005414742939874349345632342736551742755799380890757790047762090097428217212554969763070054328373384038757021323533574266921434824554791473367306550096682855015976336121885151093176526935998911105510153653876313347077910698759527097874274063133691508004915427167923673811672866125097963281263025664152339413845265112163075690818565635261657023319441816814729969965240342654457516286152277252525177856321549481654411406807813380905752724766432445634001977995571323473835638104716221079771653178082168900065496010630947762125908197666544661328890036188585100185635125060054842784280380022810265022183128282449834494207090206224580800346048728709293252059335254379366832084407896790973138178345607808981322483604214238231814558599395703864545127112233461034953670252055953098864346276006078104694061496904507659438373877713118153750477816615261798352669784, 12416469387535972627848109002414676191274552968022960445965891841821126743504726581902291558840150559428221418026828057302259464633210027955882303298972183135352649162777135317648522238196700557948850688941143653907407631635137400286681628808634391075038524766562297834786688400836135203331957011202357535288705021835869600302389540762201418876544713628766046255779170275595605517496340480649250012223524835002441711456705999395938739996064708862721439276661133464288843918601233586755422196922555799686410900863880838729140730201126776612086319851389646908248806013790398967886436350618339574231258595210727383187221631880341615315455887474303161203228260235424714760106145930933828036291502692333914903810860398151352703838945786633490162430871775979830205625026667964596810724741068275963505226325838571095307985958567432382530884531037240535681337966509530547653110730506893202635764407633895298489725377129832897808698594793211943438496355870974003854395962403250058259916160252653397677794613798715917262059004626419673714909447024689927148180844372910329375102097883658295401413823897355356, 580181407537071569873219113639388523337062177076779121576293902753230833158964237517443373352892974451973267458487847772059878081906936612983655262884314860399892961806362878386777189987466700589064374546391494272110262776454870428400556751931325633066521091806044976131317359985240882680174069260408687200298969718800099371944266062484045982626536579489113387480278405075627853636789302236597936780667371777310468916721342971463822355529298189259436058720140871569823195827539095492040538613633005727829812483921473314470607049332062893334289644562260654330969898655050526184266542910924701003470064448855539197020855998140455767224238834159211792040141941910726332278592668659274118717537545206847105966158467735512220212309823165046801824539074440581739154697172436503283030884779340435184485591263786783524982822429170175431073327510637825719046529877250931389073020665813384764385883326747770031343516335851499027209035111789793064928096738046347708851098946602448825781199436565605433243679165466560968554983169058025111752126948951598566856900838920964639256410289579908580187720704092608], 'aut_phi_ratio': 972.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 3, 6, 2], [3, 9, 4, 1], [3, 9, 18, 2], [3, 27, 12, 1], [3, 27, 18, 1], [9, 27, 36, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3.C_3^5.C_3^5.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '81.15', 'autcent_hash': 15, 'autcent_nilpotent': True, 'autcent_order': 81, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '17496.du', 'autcentquo_hash': 5525287510650319850, 'autcentquo_nilpotent': False, 'autcentquo_order': 17496, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^4.S_3^3', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 3, 14], [3, 9, 40], [3, 27, 30], [9, 27, 36]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '729.440', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 8516, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 7], [3, 9, 2, 20], [3, 27, 2, 15], [9, 27, 2, 18]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 9097920, 'exponent': 9, 'exponents_of_order': [7], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [[9, 0, 18]], 'familial': False, 'frattini_label': '27.5', 'frattini_quotient': '81.15', 'hash': 8516, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 3, 3, 3, 3], 'inner_gens': [[1, 3, 90, 1485, 81, 243], [1, 3, 9, 27, 81, 324], [163, 3, 9, 27, 81, 297], [730, 3, 9, 27, 81, 243], [1, 3, 9, 27, 81, 972], [1, 894, 36, 27, 1539, 243]], 'inner_hash': 440, 'inner_nilpotent': True, 'inner_order': 729, 'inner_split': False, 'inner_tex': 'C_3^4:C_3^2', 'inner_used': [1, 2, 3, 4, 6], 'irrC_degree': 9, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 81], [3, 18], [9, 24]], 'label': '2187.8516', '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': 'C3^5:C3^2', 'ngens': 4, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 10, 'number_conjugacy_classes': 123, 'number_divisions': 62, 'number_normal_subgroups': 239, 'number_subgroup_autclasses': 134, 'number_subgroup_classes': 1397, 'number_subgroups': 10079, 'old_label': None, 'order': 2187, 'order_factorization_type': 7, 'order_stats': [[1, 1], [3, 1214], [9, 972]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [6, 3, 6, 6, 3], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 732, 1611, 756, 891, 1221], [1, 3, 1551, 837, 81, 1703], [2, 6, 882, 756, 1539, 1944], [1514, 3, 927, 1512, 891, 1950], [4, 3, 9, 27, 81, 261]], 'outer_group': '1944.3564', 'outer_hash': 3564, 'outer_nilpotent': False, 'outer_order': 1944, 'outer_permdeg': 15, 'outer_perms': [39925191600, 736385569920, 195124572856, 563, 458], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:S_3^3', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 27, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 40], [6, 9], [18, 12]], 'representations': {'PC': {'code': '6880941196207637014282656', 'gens': [1, 2, 3, 4, 5, 6], 'pres': [7, 3, 3, 3, 3, 3, 3, 3, 1892, 41583, 4548, 1405, 537, 166]}, 'Perm': {'d': 27, 'gens': [7550102686338507196661760000, 2516565698315557445821374723, 806634631155472114121499574, 806634631153204606767213507, 711549292653507, 2516538717432286673398548480, 806634631153204606767248884]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3, 3], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^5:C_3^2', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'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}