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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': 18, 'characteristic': False, 'core_order': 1, 'counter': 480, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4199040.a.19440.p1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '19440.p1', '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': 19440, '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': '216.134', 'subgroup_hash': 134, 'subgroup_order': 216, 'subgroup_tex': 'C_3^2\\times \\SL(2,3)', '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': '19440.p1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '233280.n1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2160.m1', '6480.m1'], 'contains': ['58320.n1', '58320.q1', '58320.s1', '58320.t1', '58320.x1', '58320.z1', '77760.s1'], 'core': '4199040.a1', 'coset_action_label': None, 'count': 6480, 'diagramx': None, 'generators': [1858713780886442804240707563322589684619100046329074030070624797051452890800494192763967594903094035312691825850969098719, 1563242976841218170388912052257026325722161035508576379741388562951690764317394519994472373058986700218780251858696053526, 3986143740577844961930176423535058120487982404372793864730794628890350166825987334259040801764850503691607382860884679107, 5635038654180633649487133796221259796783448983560201799941937644587818137052271930051269224902436730757078355529173880362, 1908210108063097797789559030749237663854513283970246217819359794108646739176624782825828299006303214968091194733957527595, 1258487882148204228219346340905103828222063830615406954380476365484668481845312398252412366627262273663747392554706352513], 'label': '4199040.a.19440.p1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6480.m1', 'old_label': '19440.p1', 'projective_image': '4199040.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '19440.p1', 'subgroup_fusion': None, 'weyl_group': '36.11'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '27.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [6, 12], 'aut_gens': [[1, 3, 18], [115, 27, 111], [5, 174, 195]], 'aut_group': '10368.qn', 'aut_hash': 3718250448986069408, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10368, 'aut_permdeg': 13, 'aut_perms': [3991776604, 98154022], 'aut_phi_ratio': 144.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 8, 1], [3, 4, 18, 1], [4, 6, 1, 1], [6, 1, 8, 1], [6, 4, 18, 1], [12, 6, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times C_3^2:\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '432.734', 'autcent_hash': 734, 'autcent_nilpotent': False, 'autcent_order': 432, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_3^2:\\GL(2,3)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 8], [3, 4, 18], [4, 6, 1], [6, 1, 8], [6, 4, 18], [12, 6, 8]], 'center_label': '18.5', 'center_order': 18, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '8.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 134, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['24.3', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [3, 4, 2, 9], [4, 6, 1, 1], [6, 1, 2, 4], [6, 4, 2, 9], [12, 6, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 520, 'exponent': 12, 'exponents_of_order': [3, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '108.41', 'hash': 134, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[1, 57, 81], [163, 3, 126], [64, 111, 18]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 27], [2, 27], [3, 9]], 'label': '216.134', 'linC_count': 432, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 162, 'linQ_dim': 6, 'linQ_dim_count': 108, 'linR_count': 108, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C3^2*SL(2,3)', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 63, 'number_divisions': 33, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 21, 'number_subgroup_classes': 90, 'number_subgroups': 234, 'old_label': None, 'order': 216, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 1], [3, 80], [4, 6], [6, 80], [12, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 4, 2, 4, 3, 2, 6, 6], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[2, 123, 99], [14, 153, 57], [115, 15, 90], [188, 39, 213], [187, 3, 18], [43, 87, 18], [116, 123, 99], [44, 39, 27]], 'outer_group': '864.4661', 'outer_hash': 4661, 'outer_nilpotent': False, 'outer_order': 864, 'outer_permdeg': 11, 'outer_perms': [1, 4452601, 3669534, 20376841, 28079280, 11652505, 14777425, 29542375], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times C_3^2:\\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 14], [3, 1], [4, 13], [6, 4]], 'representations': {'PC': {'code': 29030609188234960449449, 'gens': [1, 2, 4], 'pres': [6, 3, 2, 3, 2, 2, 3, 685, 31, 338, 1947, 1017, 69, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101742639871487, 41624309168449642, 125101749984343022]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [23155657, 16014990, 38936315, 34196962, 23068812, 33299785]}, 'Perm': {'d': 14, 'gens': [562544787, 240, 147, 7355945520, 13986206640, 20797746480]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2\\times \\SL(2,3)', 'transitive_degree': 72, '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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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, 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[[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}
