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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '4199040.a', 'ambient_counter': 1, 'ambient_order': 4199040, 'ambient_tex': 'C_3^4:\\Sp(4,3)', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 1, 'counter': 176, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4199040.a.2916.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '2916.a1', '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': 2916, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': False, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '1440.4591', 'subgroup_hash': 4591, 'subgroup_order': 1440, 'subgroup_tex': '\\SL(2,9):C_2', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4199040.a', 'aut_centralizer_order': None, 'aut_label': '2916.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2099520.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['36.a1', '81.a1'], 'contains': ['5832.a1', '17496.a1', '17496.b1', '29160.b1', '43740.c1', '43740.e1'], 'core': '4199040.a1', 'coset_action_label': None, 'count': 2916, 'diagramx': None, 'generators': [3479720417261455776370939850309397083307932241909640881041636562041091276544056154129803489547383802779021072225471255625, 3763315116773011649646134517607907476491964740303190813128160050243920792659609450454002295645230418256753692474167434400, 5046644971563614294711892034272501818557365648048139472670191453394251270521794052680594306853063678343027882917114841353], 'label': '4199040.a.2916.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2916.a1', 'old_label': '2916.a1', 'projective_image': '4199040.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2916.a1', 'subgroup_fusion': None, 'weyl_group': '720.763'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 60, 'aut_gen_orders': [6, 6], 'aut_gens': [[28816400, 31762786, 12299789], [28816400, 26324703, 815724], [28816400, 21590161, 8100010]], 'aut_group': '1440.5842', 'aut_hash': 5842, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1440, 'aut_permdeg': 30, 'aut_perms': [14374165695140193260620905471722, 166424867513097134144225017663495], 'aut_phi_ratio': 3.75, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 30, 1, 1], [3, 40, 1, 2], [4, 30, 1, 1], [4, 90, 1, 1], [5, 144, 1, 1], [6, 40, 1, 2], [6, 120, 2, 1], [8, 180, 1, 2], [10, 144, 1, 1], [12, 120, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_6', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '720.763', 'autcentquo_hash': 763, 'autcentquo_nilpotent': False, 'autcentquo_order': 720, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 30, 1], [3, 40, 2], [4, 30, 1], [4, 90, 1], [5, 144, 1], [6, 40, 2], [6, 120, 2], [8, 180, 2], [10, 144, 1], [12, 120, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '720.763', 'commutator_count': 1, 'commutator_label': '720.409', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '360.118'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4591, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 30, 1, 1], [3, 40, 1, 2], [4, 30, 1, 1], [4, 90, 1, 1], [5, 144, 1, 1], [6, 40, 1, 2], [6, 120, 2, 1], [8, 180, 1, 2], [10, 144, 1, 1], [12, 120, 2, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 636, 'exponent': 120, 'exponents_of_order': [5, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, -1, 2], [4, 0, 2], [16, -1, 1], [20, -1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '720.763', 'hash': 4591, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [1, 5, 2], 'inner_gens': [[28816400, 31762786, 12299789], [28816400, 31762786, 8100010], [28816400, 22585011, 12299789]], 'inner_hash': 763, 'inner_nilpotent': False, 'inner_order': 720, 'inner_split': True, 'inner_tex': 'S_6', 'inner_used': [2, 3], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 2], [4, 4], [5, 4], [9, 2], [10, 2], [16, 2], [20, 1]], 'label': '1440.4591', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SL(2,9):C2', 'ngens': 3, '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': 15, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 17, 'number_divisions': 15, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 65, 'number_subgroup_classes': 65, 'number_subgroups': 1772, 'old_label': None, 'order': 1440, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 31], [3, 80], [4, 120], [5, 144], [6, 320], [8, 360], [10, 144], [12, 240]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [14408200], 'outer_gens': [[28816400, 31762786, 8630074]], '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': None, 'perfect': False, 'permutation_degree': 80, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [5, 4], [8, 2], [9, 2], [10, 2], [16, 2], [20, 1]], 'representations': {'GLFp': {'d': 4, 'p': 3, 'gens': [28816400, 31762786, 12299789]}, 'Perm': {'d': 80, 'gens': [2764721950057380917935708056250507687596307644341085643156406519261132649944827297831522101226939288274031451856619410, 962586680643911999628574441475018823907216490396524236602810233104824562895726967804885801030127522093263694105006536]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SL(2,9):C_2', 'transitive_degree': 80, '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': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 360, 'aut_gen_orders': [3, 3, 3, 3, 18, 24], 'aut_gens': [[894620098577469322175519985020681194881012085306020847924401451838941698632743523764516155091325981942145124541054665, 72464222328264089297489695031869689110933567064632866522034925580676309902221639007165937290297183227419990860884679107, 8255416410225549539474188160789099456940975010261873257173331522788699932363771849249387076626303064303536109658336300], [3914574138348158341954862248488209112571929849393467019056684104761512798622398438775639380629644646446129537124541054665, 72464222328264089297489695031869689110933567064632866522034925580676309902221639007165937290297183227419990860884679107, 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72464222328264089297489695031869689110933567064632866522034925580676309902221639007165937290297183227419990860884679107, 515550266210317542333643794413761467150944826119603047485761637089251187957473797478979200433834034340029233889553138716], [1289269376081644324911775837888167218356978465257472496758953511601083096700536294866362697962419131783405561614924864517, 2464156378696176367517571567920336775669681315641606605711662818143587573685133496925650223342562229294780525223499227587, 4005974152576718963995149895439119818024414843965603701710249976617243527690891927116203206135792300788723564972171566047], [1968726748988237643769861219816066435809745945309739162056377746244544873821001143217621077070576356352261791138220533364, 1956839759124790436316343364251594215688524418654198141441931863786863366443785819253875741769036531033568313737216000000, 5671533454485053518982651777987842099158894965567816545911746536491289540546745243516042230374697007455027525466041161880]], 'aut_group': '8398080.j', 'aut_hash': 1287654720189540806, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 8398080, 'aut_permdeg': 240, 'aut_perms': [3774143783706953992616996963712254001389828694673822779832611497774080164884174615137262023254306808333215777271485436633092439512505971326932746921592270452025011946770826492052472023643700473616402173605488430321044666190283141431849530653342893799134597040583035822300545525307412460678171686544438766825373002125823299433258379233647970397589646211658399768733603152097972751621197458267880153237764962228076306981865891224742099203834522283587625574335509794134294, 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3774143796120107954171510387136158410672520495520179674538183087973575390140069345632293505036277575596067651814439009850025587647978561571088176474127591703821589717770093038316963764933804768641970526999428508136008512917846689071744493918788281956468678484132443449938609167276180872513594651079551171357146355093046345766255411406122046503425518539712203455609175510585194985472420268283904353604382433862668780414455943450485088483271518856738367154269152883665600, 647400743023819641288986364696131819106738389959367901539775745053170237903088780255470165262860444339865451139030685356902573774634472052697868097479017481469820170600722627981454558776748864263523759174294323148082983410222760023101365413462713301443105126720285609196534260873825398649864666379302405757111591274052230673990667317046447959402092351280161146875340823653679006791790337659787369427137315942033662768058621130107658297495970230474386784894806392801413, 3780492737606606555206403997371766945174042793737126827732812530177897849929360244196926623195993107590057907561268017937419856541326141009028554137862519368817168131872929208985062319735690663299739140981511860009381683698763532207173914911166541957303041353129842279349147441594041415415738540898241607246679418384081757295542056274064715042426156682211302011680166536721930743638364279101367986971460197579331314313233182126588431119749947362133352060477300161692067], 'aut_phi_ratio': 7.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [2, 810, 1, 1], [3, 80, 1, 1], [3, 120, 2, 1], [3, 960, 2, 1], [3, 2160, 1, 1], [3, 2160, 2, 1], [3, 4320, 1, 1], [3, 8640, 2, 2], [3, 17280, 1, 1], [4, 4860, 1, 1], [4, 43740, 1, 2], [5, 419904, 1, 1], [6, 3240, 2, 2], [6, 6480, 1, 1], [6, 9720, 2, 1], [6, 19440, 1, 1], [6, 19440, 2, 1], [6, 25920, 2, 1], [6, 38880, 1, 1], [6, 38880, 2, 2], [6, 77760, 2, 2], [8, 524880, 1, 1], [9, 77760, 2, 1], [9, 155520, 2, 1], [10, 419904, 1, 1], [12, 38880, 1, 1], [12, 58320, 2, 1], [12, 116640, 2, 1], [12, 174960, 2, 1], [12, 349920, 1, 1], [18, 233280, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3:S_3.\\SO(5,3)', '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': 360, 'autcentquo_group': '8398080.j', 'autcentquo_hash': 1287654720189540806, 'autcentquo_nilpotent': False, 'autcentquo_order': 8398080, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^3:S_3.\\SO(5,3)', 'cc_stats': [[1, 1, 1], [2, 81, 1], [2, 810, 1], [3, 80, 1], [3, 120, 2], [3, 960, 2], [3, 2160, 3], [3, 4320, 1], [3, 8640, 4], [3, 17280, 1], [4, 4860, 1], [4, 43740, 2], [5, 419904, 1], [6, 3240, 4], [6, 6480, 1], [6, 9720, 2], [6, 19440, 3], [6, 25920, 2], [6, 38880, 5], [6, 77760, 4], [8, 524880, 1], [9, 77760, 2], [9, 155520, 2], [10, 419904, 1], [12, 38880, 1], [12, 58320, 2], [12, 116640, 2], [12, 174960, 2], [12, 349920, 1], [18, 233280, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '4199040.a', 'commutator_count': 1, 'commutator_label': '4199040.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '25920.a'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [2, 810, 1, 1], [3, 80, 1, 1], [3, 120, 2, 1], [3, 960, 2, 1], [3, 2160, 1, 1], [3, 2160, 2, 1], [3, 4320, 1, 1], [3, 8640, 2, 2], [3, 17280, 1, 1], [4, 4860, 1, 1], [4, 43740, 1, 2], [5, 419904, 1, 1], [6, 3240, 2, 2], [6, 6480, 1, 1], [6, 9720, 2, 1], [6, 19440, 1, 1], [6, 19440, 2, 1], [6, 25920, 2, 1], [6, 38880, 1, 1], [6, 38880, 2, 2], [6, 77760, 2, 2], [8, 524880, 1, 1], [9, 77760, 2, 1], [9, 155520, 2, 1], [10, 419904, 1, 1], [12, 38880, 1, 1], [12, 58320, 2, 1], [12, 116640, 2, 1], [12, 174960, 2, 1], [12, 349920, 1, 1], [18, 233280, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1840800, 'exponent': 360, 'exponents_of_order': [8, 7, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[80, 0, 2], [80, 1, 1], [160, -1, 1], [160, 0, 2], [240, 0, 6], [240, 1, 1], [480, 0, 6], [640, 0, 2], [640, 1, 1], [720, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4199040.a', 'hash': 8308765179899110006, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 360, 'inner_gen_orders': [2, 3, 9], 'inner_gens': [[894620098577469322175519985020681194881012085306020847924401451838941698632743523764516155091325981942145124541054665, 143139204459373239950628350093698015832105109958888976023325072521727612370983734222679461333931456093904926947119248884, 16216544311235618793790674162452165401346658498794159431436354518255557706422276201719890988693391078523239583515817645], [144033824557950709272803870078718697026986122044194996871249473973566554069616477746443977489022782075847072071660303549, 72464222328264089297489695031869689110933567064632866522034925580676309902221639007165937290297183227419990860884679107, 1289260412850871382195807821582069101976107443877529887430820225693279199045746737322133993507102127773114840915257607154], [441280187620916090287104560003671022849053765358592926185629817250266748348625516646932736572058812064009663363444, 724744142036527568069604149782400806466301879838398003591891865359342456772857411045702957080295442029216521228000725827, 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False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [6, 1], [8, 1], [10, 1], [15, 2], [20, 3], [24, 1], [30, 1], [40, 2], [60, 3], [64, 2], [72, 1], [80, 3], [81, 1], [90, 1], [120, 1], [160, 2], [240, 1], [320, 1], [480, 3], [640, 1], [960, 3], [1280, 1], [1440, 1]], 'representations': {'Lie': [{'d': 4, 'q': 3, 'gens': [894620098577469322175519985020681194881012085306020847924401451838941698632743523764516155091325981942145124541054665, 72464222328264089297489695031869689110933567064632866522034925580676309902221639007165937290297183227419990860884679107, 8255416410225549539474188160789099456940975010261873257173331522788699932363771849249387076626303064303536109658336300], 'family': 'ASp'}, {'d': 4, 'q': 3, 'family': 'ASigmaSp'}], 'Perm': {'d': 81, 'gens': [894620098577469322175519985020681194881012085306020847924401451838941698632743523764516155091325981942145124541054665, 72464222328264089297489695031869689110933567064632866522034925580676309902221639007165937290297183227419990860884679107, 8255416410225549539474188160789099456940975010261873257173331522788699932363771849249387076626303064303536109658336300]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:\\Sp(4,3)', 'transitive_degree': 81, 'wreath_data': None, 'wreath_product': False}
