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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': 9, 'counter': 65, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '52488.ky.81.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '81.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': 81, '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': '648.730', 'subgroup_hash': 730, 'subgroup_order': 648, 'subgroup_tex': 'C_3^4:Q_8', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '52488.ky', 'aut_centralizer_order': None, 'aut_label': '81.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '52488.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['9.a1', '27.a1'], 'contains': ['162.a1', '729.b1', '729.d1'], 'core': '5832.a1', 'coset_action_label': None, 'count': 27, 'diagramx': [95, -1, 4493, -1], 'generators': [17683942790150574222493454400144, 418802673725195279366246643, 295767301540553574986382245002549928905, 191472314162874761265933150517437963964224, 10628907027100423092203445619434195690243, 526270820936528085475111584168119188, 106074066766426974599538264118697495335925], 'label': '52488.ky.81.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '9.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.a1', 'old_label': '81.b1', 'projective_image': '52488.ky', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '81.b1', 'subgroup_fusion': None, 'weyl_group': '1944.2295'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 72, 'aut_gen_orders': [6, 24], 'aut_gens': [[11445301, 24577758, 24507118, 14850479, 2939606, 36731395, 7355200], [42961672, 24577758, 14481270, 2793523, 32864349, 14582400, 20258849], [24734080, 14633333, 2721277, 14634003, 14686477, 6973453, 32703882]], 'aut_group': '93312.jq', 'aut_hash': 4164635616307602250, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 93312, 'aut_permdeg': 486, 'aut_perms': [5437964660965527585979868445704484624832474997271145096148891209700100478376184576002795419461316555390849901037825186759105777071549839590085117592194816475757418798367695604316141897118558141162596806297221485843253356115885107937381030258848521611473345481693221798260630503776151587480845045243553131787257518117875170737019066223168012784351183610493649370229479820121881448464577765734517223924906958819366969569678769001616790702829234004784940839608963844625215951610902000412492153578174584950344287138620583164860370809385327847571885845775519412080808887555478910110485668491891620559393102016701008506746686647802795064390841591650024235219026942795086518419254772525711387960052227199831433463686208762975604450593557139925892295926662917731731660161922955295834036863534252400980534072706359582048863890101314086042085805665568744042377895891536995302071140808896763805714162406763315529342434886837120502260096792717374925110349083685154209584621083444423565648883784302658210130068930176485080500757198955401448076983927584406283993653086861600484142388618949629829279954550580109, 4671539892863792140978471744711147415712497110236979247827686932387157431185795985268929338445370328164386257734597354321555878679051763108087549378896797274829102549564664355775716199284556102160430639442520843796295569223823450073255465799917906718823696473895736607508298251758970555113531066278236991488173970166235024879537674830565528397341924909691082627862029053641388336441574243319634679573383157770625478624081510214596326867389604616380850692143325423133675324956813447866301761554942286179505669065496702137455850813815960425708922843778866956972953073670836237193618495649903843399903089212030081128451121869914723315259868055778605683604012779619792409417536071861923092336661561675440825708553878909669703971156733210937179291207310155995700691582521663076378848627627960273277388892491341304058347204275384391331452083963139192685783913521696616394232040984245395477561108529312098808819590408337013678378487073718442609780039635441944388326791498774939957756211730781929542633127537374954156423900416185697484529269304734122840349343946152019067557665094161463515822942251588858], 'aut_phi_ratio': 432.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 8, 4, 1], [3, 8, 6, 1], [4, 162, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4:\\GL(2,3):S_4', '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': 72, 'autcentquo_group': '93312.jq', 'autcentquo_hash': 4164635616307602250, 'autcentquo_nilpotent': False, 'autcentquo_order': 93312, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4:\\GL(2,3):S_4', 'cc_stats': [[1, 1, 1], [2, 81, 1], [3, 8, 10], [4, 162, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '648.730', 'commutator_count': 1, 'commutator_label': '162.54', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 730, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 8, 1, 10], [4, 162, 1, 3]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 1, 'exponent': 12, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 6]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '648.730', 'hash': 730, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 3, 3, 4, 3, 3, 2], 'inner_gens': [[11445301, 11495158, 11225629, 20327188, 43014227, 24507118, 20207646], [24508308, 24577758, 24507118, 11169773, 2939606, 36731395, 11534549], [24734080, 24577758, 24507118, 11659160, 2939606, 36731395, 11607619], [42746117, 32864349, 32479219, 14850479, 24577758, 6973453, 11501225], [14358460, 24577758, 24507118, 6970741, 2939606, 36731395, 20477175], [20481904, 24577758, 24507118, 3156995, 2939606, 36731395, 32870443], [42652886, 20201582, 20254942, 24511696, 32864349, 11225629, 7355200]], 'inner_hash': 730, 'inner_nilpotent': False, 'inner_order': 648, 'inner_split': False, 'inner_tex': 'C_3^4:Q_8', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 1], [8, 10]], 'label': '648.730', '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': True, 'name': 'C3^4:Q8', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 15, 'number_divisions': 15, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 29, 'number_subgroup_classes': 157, 'number_subgroups': 3298, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 81], [3, 80], [4, 486]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 3, 2, 2], 'outer_gen_pows': [14408200, 2790694, 36905383, 14408200, 2790694, 2790694], 'outer_gens': [[42894928, 20201582, 20254942, 32752764, 11495158, 32479219, 11660736], [20695177, 32864349, 14800134, 24511696, 24577758, 14582400, 37006991], [20695177, 32864349, 14633333, 24582579, 24577758, 14853281, 36902914], [42894928, 11495158, 11225629, 14413702, 2939606, 6973453, 20477175], [11445301, 20254942, 24577758, 20206376, 32479219, 37043725, 24782714], [11445301, 14481270, 24454001, 24781216, 14633333, 37010401, 32703882]], 'outer_group': '144.183', 'outer_hash': 183, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 7, 'outer_perms': [1, 120, 144, 3, 744, 1680], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_3\\times S_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [8, 10]], 'representations': {'PC': {'code': 39726950185964376922945666064459919893945921282447, 'gens': [1, 2, 4, 5, 6, 7], 'pres': [7, -2, -2, -2, -3, 3, -3, 3, 28, 85, 36, 1795, 346, 129, 1404, 291, 438, 21173, 6060, 1531, 28230, 12361, 5312]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [11445301, 24577758, 24507118, 14850479, 2939606, 36731395, 7355200]}, 'Perm': {'d': 12, 'gens': [267361200, 135354691, 3633865, 43545748, 43590963, 45507, 79879200]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:Q_8', 'transitive_degree': 12, '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}
