Formats: - HTML - YAML - JSON - 2026-07-24T16:00:36.587300
  • gps_subgroup_search • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '254561089305600.b', 'ambient_counter': 2, 'ambient_order': 254561089305600, 'ambient_tex': 'C_3^{11}.A_{12}.C_6', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 531441, 'counter': 657, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '254561089305600.b.15400.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '15400.A', 'outer_equivalence': False, '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': 15400, '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': '16529940864.x', 'subgroup_hash': None, 'subgroup_order': 16529940864, 'subgroup_tex': 'C_3^4.(C_3^8.C_6\\wr S_4)', 'supersolvable': False, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '254561089305600.b', 'aut_centralizer_order': None, 'aut_label': '15400.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 15400, 'diagramx': None, 'generators': [526270821324335312575476111777792004, 43545603, 2555258576846976244, 80880, 93448857600, 106069177607087835418268191867726968754706, 806609900825601745961088144, 376610217984147, 45360, 42245689556817692416611299374839859818387, 80644, 10628907036247073405826846134827754157604, 26976017466662584320003, 52872984129466453354891027484664136400643, 9146650338353971073702918246644, 106069432497398831379284679758487364304788], 'label': '254561089305600.b.15400.a1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '15400.A', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '15400.a1.a1', 'subgroup_fusion': None, 'weyl_group': None}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[234012956951058319580561230402052613314160, 212138610077008645853570363129260831897108], [351030160038952644907661914452661754611894, 212141598150474430266927227504181985803024], [170187058154911303341667821851890174293714, 371694826093321477971179758944702699874533], [223384049923959509730636894864710843525040, 212138338723333864592010782724757565170803], [148319484969879000648961289176626039502227, 84489206839231618018955404571202118137478], [371696456036459682508756795340301761973714, 74459440313001773093425365005744163957123], [361655026441765672762107681184653353660853, 212141869504150824234801964514075918762307], [42540659234172853295063207678098566601557, 74452666376102241059737153649150994230134], [148313509527136926740358790480215140558880, 361653125219467753660977010076153846485078], [361651495753150238173227862793457129834114, 74452666540932042963425358802888144372533], [244050617210406580826796896170726943813284, 212138864985006925396153730088198259720464], [244050872118404860394108279475710895170068, 212138864985006925396153730088291744865923], [223384321277634290994439610585907266766720, 212138338723333057982117377850921073855604], [244050872118404860342404245663176303682163, 212138864985006925369177712621629160591424], [116406103094392477914202448612086858051440, 212136420387184988742471295413044392861314], [116406103085855604292050477608003670787347, 212136420378648115145045029540038743117493], [234012702043060040037977863442940669079920, 212138610077008645905274396606949754073104], [106368187918509482305902615193355813059200, 339780934393305233536672098007631459484210], [371696201046046454171679151392033631889250, 319120597366024043133724413873577733389890], [244044913176894203460856240803502568092323, 74452395068828723139591693636914574314308], [95733576874975043337441170630189134781040, 329743274308573415389040948984625740279014], [223384576185632989285742112924925791976563, 212138338723333476784786236865225235471424], [244050872118405666977030532352217345128564, 202100949799977667227500696158312124066163], [244050345856731411809714248837701428350323, 212138338714796990915909331711161490575283], [234012956951058319580561230402052613314160, 212138610077008645853570363129260831897108], [223384321269098223901255546979391657566788, 212138610077008645853570363129260831897108], [223384576167949046495591539413900678913920, 212138338714796990915909331711161490575283], [159558677407168271648717008515794265000244, 223383761155475905958270031084198875084757]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 99179645184, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': None, 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 36, 1], [2, 486, 1], [2, 972, 1], [2, 2916, 1], [2, 6561, 1], [2, 17496, 1], [2, 78732, 2], [3, 1, 2], [3, 4, 6], [3, 6, 3], [3, 12, 21], [3, 24, 3], [3, 36, 36], [3, 54, 18], [3, 72, 15], [3, 81, 6], [3, 108, 81], [3, 144, 1], [3, 216, 72], [3, 324, 120], [3, 432, 22], [3, 486, 15], [3, 648, 102], [3, 864, 3], [3, 972, 60], [3, 1296, 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[18, 1944, 54], [18, 3888, 54], [18, 11664, 324], [18, 17496, 168], [18, 23328, 54], [18, 34992, 564], [18, 52488, 126], [18, 69984, 207], [18, 104976, 666], [18, 157464, 174], [18, 209952, 288], [18, 314928, 483], [18, 419904, 27], [18, 629856, 233], [18, 944784, 228], [18, 1259712, 11], [18, 1417176, 18], [18, 1889568, 146], [18, 2834352, 78], [18, 3779136, 11], [18, 5668704, 182], [18, 8503056, 12], [18, 11337408, 26], [18, 17006112, 74], [18, 25509168, 2], [18, 34012224, 35], [18, 51018336, 3], [24, 229582512, 8], [27, 3779136, 6], [27, 11337408, 12], [27, 22674816, 2], [27, 68024448, 6], [36, 314928, 54], [36, 944784, 108], [36, 1889568, 18], [36, 2834352, 180], [36, 5668704, 27], [36, 25509168, 18], [36, 459165024, 2], [54, 34012224, 18]], 'center_label': '3.1', 'center_order': None, 'central_product': None, 'central_quotient': None, 'commutator_count': None, 'commutator_label': '1377495072.ft', 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 24, 'conjugacy_classes_known': False, 'counter': 24, 'cyclic': False, 'derived_length': 6, 'dihedral': False, 'direct_factorization': None, 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 36, 1, 1], [2, 486, 1, 1], [2, 972, 1, 1], [2, 2916, 1, 1], [2, 6561, 1, 1], [2, 17496, 1, 1], [2, 78732, 1, 2], [3, 1, 2, 1], [3, 4, 2, 3], [3, 6, 1, 1], [3, 6, 2, 1], [3, 12, 1, 1], [3, 12, 2, 10], [3, 24, 1, 1], [3, 24, 2, 1], [3, 36, 2, 18], [3, 54, 2, 9], [3, 72, 1, 1], [3, 72, 2, 7], [3, 81, 2, 3], [3, 108, 1, 3], [3, 108, 2, 39], [3, 144, 1, 1], [3, 216, 1, 4], [3, 216, 2, 34], [3, 324, 2, 60], [3, 432, 1, 2], [3, 432, 2, 10], [3, 486, 1, 3], [3, 486, 2, 6], [3, 648, 2, 51], [3, 864, 1, 1], [3, 864, 2, 1], [3, 972, 2, 30], [3, 1296, 1, 5], [3, 1296, 2, 40], [3, 1944, 1, 4], [3, 1944, 2, 37], [3, 2592, 1, 1], [3, 2592, 2, 4], [3, 3888, 1, 7], [3, 3888, 2, 38], [3, 5832, 2, 15], [3, 7776, 1, 1], [3, 7776, 2, 10], [3, 11664, 2, 22], [3, 15552, 1, 1], [3, 17496, 2, 3], [3, 23328, 1, 5], [3, 23328, 2, 11], [3, 34992, 1, 3], [3, 34992, 2, 6], [3, 69984, 1, 1], [3, 69984, 2, 6], [3, 104976, 1, 1], [3, 139968, 1, 1], [3, 209952, 1, 1], [3, 209952, 2, 1], [3, 629856, 2, 3], [3, 1259712, 1, 1], [3, 3779136, 1, 1], [4, 8748, 1, 1], [4, 157464, 1, 1], [4, 708588, 1, 1], [4, 1417176, 1, 1], [4, 6377292, 1, 1], [4, 25509168, 1, 1], [6, 36, 2, 13], [6, 108, 2, 27], [6, 216, 1, 1], [6, 216, 2, 4], [6, 324, 2, 81], [6, 486, 2, 13], [6, 648, 1, 1], [6, 648, 2, 94], [6, 972, 1, 5], [6, 972, 2, 177], [6, 1296, 1, 1], [6, 1296, 2, 13], [6, 1944, 1, 9], [6, 1944, 2, 270], [6, 2916, 2, 238], [6, 3888, 1, 2], [6, 3888, 2, 107], [6, 4374, 2, 27], [6, 5832, 1, 2], [6, 5832, 2, 491], [6, 6561, 2, 4], [6, 7776, 1, 1], [6, 7776, 2, 4], [6, 8748, 1, 3], [6, 8748, 2, 318], [6, 11664, 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11337408, 1, 2], [18, 11337408, 2, 12], [18, 17006112, 2, 37], [18, 25509168, 2, 1], [18, 34012224, 1, 1], [18, 34012224, 2, 17], [18, 51018336, 1, 1], [18, 51018336, 2, 1], [24, 229582512, 2, 4], [27, 3779136, 2, 3], [27, 11337408, 2, 6], [27, 22674816, 2, 1], [27, 68024448, 2, 3], [36, 314928, 2, 27], [36, 944784, 2, 54], [36, 1889568, 2, 9], [36, 2834352, 2, 90], [36, 5668704, 1, 1], [36, 5668704, 2, 13], [36, 25509168, 2, 9], [36, 459165024, 2, 1], [54, 34012224, 2, 9]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 216, 'exponents_of_order': [17, 7], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': None, 'familial': False, 'frattini_label': '6561.1396077', 'frattini_quotient': '2519424.a', 'hash': 4449120529947405189, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': None, 'inner_split': None, 'inner_tex': None, 'inner_used': None, 'irrC_degree': None, 'irrQ_degree': None, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': None, 'label': '16529940864.x', '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': 'C3^4.(C3^8.C6wrS4)', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 4, 1, 1, 3, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': None, 'number_characteristic_subgroups': 38, 'number_conjugacy_classes': 18975, 'number_divisions': 9630, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 16529940864, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 185931], [3, 13548464], [4, 34178436], [6, 1356701040], [8, 229582512], [9, 794773296], [12, 3454200288], [18, 5391882288], [24, 1836660096], [27, 612220032], [36, 2193788448], [54, 612220032]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': None, 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 18, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': None, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': None, 'representations': {'PC': {'code': '3689969078104521763206052061421388881953123700110516111925663131065157071435537960289165993316281079370675581044042027138028688679163682518804158952289987568188504955784993536839974626406550064821769601080888579439136781546810163437313521384464821354918644548472208797134255076776758992780325495050541046536536953317746571446658314905109959775880482983120859035599480954742458052012256122186409573431633321747345727821479377143289993932018364252264873218730373581927945007938440593879296634714549568120409883646005985607719013391167962470619438379222475545863851229134180796947580742830436314381693828769743283259498363975373196174761789877559589360548108278957544000532348703487206504926246585543653401043212399116162594183753965531832190690307026335155241614032610789868320827449916622848628056769849848138290420699870799657201163664072002997500910032544139924770944769770528071495238936906851056570400765862130617857555100145417180560419793181322717427079936437386364830158961984680761850602714071773483040601117093621352589654594107458732881712571464437744168198165752185579469803963030164531836432520532204507990793983920466839749301710648614535505127394390184017877389377777895843991920990102068124899349769717811528992304631806677582996703823798104224324679627049239446330777972185061532045408935212432107579819522674045196960823669959509531816318385921802458930686165798684239831450891354976559649911148375092865659431769041512761354181438765031058249156605942207195170036816779023539612902095404471717026196513641895784937401877826334220582753020396237978816631790932749046795125231965808227643898596732230447416530496341932908828735318565975744705704408257991531486467937084986196050355224535990609887939316803564265338766706660578938975307100634315953175104011473773708030142885868454475299445511354928541773504307598861802697710152728189675025407237193788645609761347800891311078827434973439920313286066198844979550925472735171331072670617023831929015653870538659075994860022197037592395578647930107477080560805212658537679742003771365066786976537838380850913370132848491741557605234220132073290650763592366387377596941714651741777856578927012921355296382370757033249764012697470796151579592027860053591894123955007135851906830363066598060905503628046964141739084424855556009360454299019358684233556002994934979897171176559626142573029406508784048286292612595355260169585751962423005346937415698370239653752459491142336716957155272999132831847961197741584062925156351', 'gens': [1, 3, 5, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], 'pres': [24, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 48, 16318667833, 216385268402, 182869521938, 194, 100080713091, 557941128219, 47806898763, 722423361604, 497269292428, 229067802772, 109250001796, 18368294500, 893460891461, 484884076637, 252566977877, 167246759741, 35955399557, 413, 932440190982, 270258823326, 409882588470, 14693301582, 739841190, 2142770262, 750379686919, 494388665887, 317857872439, 219098320207, 20772506983, 37554788863, 322486999, 559, 1027596160520, 310090799264, 588151321400, 180410359184, 43493864, 20324105408, 5577959528, 2119014812169, 225447079713, 597498877497, 147418194321, 5591635305, 1512138369, 2041411833, 2166688977, 912937881, 705, 4322502264970, 274362536482, 334893297082, 269515059346, 10727621098, 4641221506, 415657594, 23098066, 770215910411, 899443325987, 272807025467, 350057759699, 45983220587, 21970279427, 885810971, 2618635571, 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1867417970706, 666620970, 1157911096386, 102015459162, 34899102834, 20994225546, 5840024994, 3901295418, 135629202, 904997898, 665106, 119719434, 33575130, 2216490, 739074, 111186, 669354, 12738, 2461958737939, 1229695395883, 641404293187, 180775203931, 98077132915, 15120898699, 366820003, 17002414267, 4142603731, 2076825835, 878849539, 455596123, 114424147, 8437339, 471307, 157411, 121210048532, 418497601580, 45560890436, 79527764828, 141425657972, 34314803852, 7327744868, 4602951548, 255721748, 1035532748, 24984548, 120564140, 30128420, 122804, 41180, 272540, 776060, 30668, 110852, 4620044335125, 4171600300077, 848183187525, 255893603421, 9603754101, 53213428365, 6758484645, 3789397053, 2246156565, 45537069, 47925765, 118223709, 76982709, 2452365, 2789781, 2851581, 171501, 57477, 6837, 1828799493142, 2485220299822, 884499222598, 152399957854, 138067794550, 2471373545495, 3351519203375, 1352023695431, 205947795551, 251861778551, 42648809615]}, 'Perm': {'d': 36, 'gens': [212138610077008645853570363129260831897108, 234012956951058319580561230402052613314160]}}, 'schur_multiplier': None, 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 96, 'supersolvable': False, 'sylow_subgroups_known': False, 'tex_name': 'C_3^4.(C_3^8.C_6\\wr S_4)', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': None}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[201808976437226955712436576492760431457868, 275957998380027379648953596564741855093043], [74455925849650716130004561311212118616253, 106073778244894121943192919086847222968932], [159560035794840439185415767468216154977988, 371991170549410902717032533981186684718643], [159558125499003422685449397544292979838054, 84488688809235650428134495691199250774962], [159564636102978670099669635093995191138291, 371984651841334282200911460866988767586747], [201802465924518469517380371161757121377868, 275958278679025239225082372546530828779490], [32210499451146737699747604337613221161478, 116700817955651416681849717540496216673867], [159564636322755836278837511212222368090214, 371984651758913896793140159960788094039533], 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'rational_characters_known': False, 'ratrep_stats': None, 'representations': {'Perm': {'d': 36, 'gens': [275957998380027379648953596564741855093043, 201808976437226955712436576492760431457868]}}, 'schur_multiplier': None, 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 26400, 'supersolvable': False, 'sylow_subgroups_known': False, 'tex_name': 'C_3^{11}.A_{12}.C_6', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': None}