Formats: - HTML - YAML - JSON - 2026-07-23T18:50:05.643362
  • gps_subgroup_search • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '6480.bm', 'ambient_counter': 39, 'ambient_order': 6480, 'ambient_tex': 'C_3^4:C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 324, 'counter': 10, 'cyclic': False, 'direct': None, 'hall': 6, 'label': '6480.bm.5.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '5.a1.a1', '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': 5, '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': '1296.3499', 'subgroup_hash': 3499, 'subgroup_order': 1296, 'subgroup_tex': 'C_3^2\\wr C_2.D_4', 'supersolvable': False, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '6480.bm', 'aut_centralizer_order': None, 'aut_label': '5.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '6480.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['10.a1.a1', '10.b1.a1', '10.b1.b1', '45.a1.a1', '45.b1.a1'], 'core': '20.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [6324, -1, 7733, -1, 3624, -1, 7149, -1], 'generators': [3, 5120, 1680, 20, 40, 2160, 5040, 402], 'label': '6480.bm.5.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '5.a1.a1', 'projective_image': '6480.bm', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '5.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1296.3499'}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [8, 8, 8, 12], 'aut_gens': [[7276872962, 14464845496, 21276823680, 23, 311, 333, 107417283120, 201714508800], [697550480458, 288443519282, 218117052720, 635, 78, 627, 712287359280, 107417283120], [181753165915, 444996558741, 356976310320, 94, 627, 78, 293363849520, 351786763440], [537523701658, 379943786882, 730604896560, 635, 56, 285, 293363849520, 351786763440], [175884352640, 387560643595, 448998762240, 94, 676, 697, 293363849520, 562435014960]], 'aut_group': None, 'aut_hash': 2069019395278717008, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10368, 'aut_permdeg': 216, 'aut_perms': [5103449113146358831859751698305003756807980508936837214372161242858724813123487031723625384638213080443147124490736159881048706209357947602000544785390701407430852307082280872204841283466028192958755890858915030826927702252575547900502630641952526633463618941674549558585421651178004371869717314396041760087203579003583945287971625670818162958472324515645865450268989427943657682284915403184897663588308830288386, 532273761421187777962904892824714608783764298160388928780885487803420141158005422381133549677505450508556507173485901244900004212869673209639946141943184067040240492740940130217232339649777911316785427243212585697770472880832428460323912682328408048986705018877056980771440715072729104832863911248538416804724069371354984788410335313070602987701730236997102034632439683898671067138274049242774614291950795075974, 862710898857477586874459022698465488680558088100918774524981558518019750682323418045028090583908707830015276051792021072231662119051736275360830767382233839176858026064502604546062720238361649745803083859303295701754201500782612943620385182003737717267874932247399551399156868770292216100397448647023536734730269038706735202023725213778453080074328924384850469569744042293798174185607634742343404362949843479464, 3013763074102057804700361686828331495844890783794587149655287456862130184944793611778140534469904638706689717502826270475041138255141099210573635232589666560179303126463113828852411046480021204633099999824768583363897465111107287111960187888220771537545882023498151573616389182819437160989470390764378322810804762649146868767898290122857587882815787449706537471319309902264305645361002712851797748715879532017587], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 81, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [3, 16, 4, 1], [4, 54, 4, 1], [4, 162, 2, 1], [6, 36, 2, 1], [6, 72, 1, 1], [12, 108, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_4^2.C_2^3', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 2069019395278717008, 'autcentquo_nilpotent': False, 'autcentquo_order': 10368, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_4^2.C_2^3', 'cc_stats': [[1, 1, 1], [2, 9, 2], [2, 81, 1], [3, 4, 2], [3, 8, 1], [3, 16, 4], [4, 54, 4], [4, 162, 2], [6, 36, 2], [6, 72, 1], [12, 108, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1296.3499', 'commutator_count': 1, 'commutator_label': '162.54', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3499, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 81, 1, 1], [3, 4, 1, 2], [3, 8, 1, 1], [3, 16, 1, 4], [4, 54, 2, 2], [4, 162, 1, 2], [6, 36, 1, 2], [6, 72, 1, 1], [12, 108, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 48, 'exponent': 12, 'exponents_of_order': [4, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1296.3499', 'hash': 3499, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 4, 2, 2, 3, 3, 3, 3], 'inner_gens': [[7276872962, 45509874607, 21276823680, 23, 333, 311, 201714508800, 351786763440], [37521872661, 14464845496, 21276823680, 23, 697, 333, 293363849520, 562435014960], [7276872962, 14464845496, 21276823680, 23, 311, 333, 351786763440, 629059818240], [7276872962, 14464845496, 21276823680, 23, 697, 676, 107417283120, 201714508800], [7276873040, 14464845958, 21276823680, 713, 311, 333, 107417283120, 201714508800], [7276873014, 14464845496, 21276823680, 687, 311, 333, 107417283120, 201714508800], [270244079282, 181753165576, 356976310320, 23, 311, 333, 107417283120, 201714508800], [455654671922, 306281646856, 648696303360, 23, 311, 333, 107417283120, 201714508800]], 'inner_hash': 3499, 'inner_nilpotent': False, 'inner_order': 1296, 'inner_split': False, 'inner_tex': 'C_3^2\\wr C_2.D_4', 'inner_used': [1, 2, 5, 7], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 2], [4, 8], [8, 2], [16, 4]], 'label': '1296.3499', 'linC_count': 16, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 8, 'linQ_dim': 12, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^2wrC2.D4', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 24, 'number_divisions': 20, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 84, 'number_subgroup_classes': 166, 'number_subgroups': 4596, 'old_label': None, 'order': 1296, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 99], [3, 80], [4, 540], [6, 144], [12, 432]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 23], 'outer_gens': [[7276872981, 14464845496, 21276823680, 23, 311, 676, 351786763440, 629059818240], [14464845496, 7276872981, 21276823680, 23, 285, 78, 351786763440, 712287359280]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [7, 10], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 4], [8, 4], [16, 4]], 'representations': {'PC': {'code': 558439738114447171396777087067598257868381225520478169245484303, 'gens': [1, 3, 5, 6, 7, 8], 'pres': [8, -2, -2, -2, -2, -3, 3, 3, 3, 16, 290, 66, 17924, 8980, 188, 13829, 2317, 2325, 605, 2694, 8078, 4726, 2046, 55303, 7447, 6943]}, 'Perm': {'d': 15, 'gens': [7276872962, 14464845496, 21276823680, 23, 311, 333, 107417283120, 201714508800]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2\\wr C_2.D_4', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [4, 40, 3, 3, 3, 3], 'aut_gens': [[1, 4, 80, 240, 720, 2160], [1, 12, 5840, 6000, 2160, 4080], [4453, 4644, 6400, 80, 560, 5360], [2401, 2484, 80, 240, 720, 2160], [4081, 564, 80, 240, 720, 2160], [2401, 804, 80, 240, 720, 2160], [3601, 644, 80, 240, 720, 2160]], 'aut_group': '12960.bx', 'aut_hash': 6161210740019565112, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12960, 'aut_permdeg': 540, 'aut_perms': [763913199011282897975987456701689738685768194568125364207730244293613711798758766008125822555586747972306996185890434925474552183178752532468406951180991834476513180678899469644613953339867411827629920961567036873067028320591201795402899029827209949172198200946118777680174072716437495178152599194226802141964478794045594064697882756424785652805917000925207724418336706652747003483768650513692994583889115391797305283388856173375052416375103899615348877861040430268403070919999042898795234675457472776834082428038845868398113790626891258090613693547786544293215874913183247811251661798941393347337980214386194803397521790843930984989407928345982218599202101853002038691544226882314506642317183345393543462465683374184189194173821897315138503076958202903170816250462343069220802705661583128989012409257439280840378697264429174840043565913432005189710590453720862432034657467514813211953636186465719072467335912642856869835489648912534235343777183438284592995538963129727855396417892049492679557629124210853433049708877404200686268209832131491552265987864401885208195174787996360939201215089441757648989714804963013632540156269171771074883980336679753519575029196843134875813358223209145906693444181347874365585169504537294641475369745346185678, 4533418160875213925358258856061977413108399675994729688156794018545936909955917765694389082739930115364935872588875389484767236228485391525859125169886475806255330374807986962965266247698440955230727574277811929928360117409122647782499928457753885461392639578119310149786316199490772793250598297147986740575023686616298864895116963400116270239478438116460188400678886021460413509432589257607992495296268521698414649566577448821385346723841073875437804450144449246417432482145229539793260469234455647341979930501771443488501800475387536399976346395958573684331197569529062035754903993836521325860543527927336005811000504433886725208873313163485487638045817971868420206002109794719071305476102858989147810065339543064668268828776775584204699844903415905498853018481720968354307658446194810633061662782979552026238814964751017246067209855423239505705230435382755393406296809150886103860018141374054174255906726764130256020045090213843177162516882054761213367078016415047954162265228814717466036747621563256862327020999512495407617862298653355550294043427904412906539435722041176347589277183501628977227440715199915406167757905601093249274230749155770797928296850065831415170510294966435736820669106008407163334742724490940652179906777352562478549, 1990807203749333026124266963251809687179796703018061912472982563755903732739877450511896980794892625841390724601871489515422668842759049329889248957944836355841184361889027626412956916151753119930875141285541997349103503902213423870174872575324399752735069541780881153807398837384746263337129483929013134751049337733214399399512318714858573756791879837336938799854174227966521540627651132721754237980584989190054631421994992045169623208206312378096233244054102609223824989067478417895340976465775875424449034736070614567856514291912588129348562339426054302691717089810430225151242818107695335479641165825004537509416634967297433994155677843473367739654665090685861314958419224655043743492042129400843830153947505694446760052603376349338517541153971636319658627517528243305976476264283861395901245439994654569794435537441464013287987824442101976152762283189359397539236757282582759133924031888010447944936623579843445811866075038623852259662922464725688476102703958704329932339792139401686976657879370674865607475498624524352141089381541410895166055696486063394270540745342387394292891016239828557334372921923032861107588451864017372758956690718236961001152582743155400322364180241351712540609742024624885745532513404249698335318747105843190622, 513284944662217373637499348036023142335530775728717794088740348371881754599011156518510829808763244888254953337859754258726542846739214936737884621385493155466824041488773343303257886002385406771907121469017268924057906015341521604324552822216537162929877430176459531682233459720257120066302539961597231131490254239610158395975844120657072176853402207354612513180137216511444424143639610241915904883002875883096993372249889036555816588409962876966238861227073637353513169778005476981542694662259008374139342449233260590829663816743858411953705208845397703985093278783536275058478140602798959702831055374488567344136433158715355550502694804954693821310119198831399391927916229179152028929212895914532772109400286169629087466825973212099428178874365222384093685158189646727659277522942302297996148876065674154090675968348988507952934338895819623912691981584290801903105988163455646967130373183504614496032831057462387967376826212913002950276811304738777388802689016174635031500479665330831893060510430227829066280238027207552348424637225942387224674473216778382230073297512417669010174470532968862060525644568677151775169795554883732085034059186236428231357069644663727076508040631427443541346815411306710912263603041053528128223372742307719807, 6966490035582152484249993339377001368749718876260969648394028636385795880885192573504349267402003283631644300140812140229518299218506860616461814603521124430892652615453949257686158183867449910042522250733041163191084849548400256721655915085023088940081624562222090857838605591267090470215445914918556681674026667267369489805810652962771260809724659062418704436757985330935443638825265434938150670416674734325762740757531110870529322238468423000522201595723335640983696054379105067638070619789032111936553550231255813668500402961868176097662516344471520285693772716336400816016182673949495725644203188054408972207790911848758162935954874765812287875113935319700982603718271303224281957696458684966413458224281190419719911768638223098591572707194106176413475549758221688885093797432704053002228422851326415950562521058129801560415138499020102390026675711203663036925513073954496375417501999943844811148110412293615280762558627657684954613802688809247273048412634352799674741791764047156921739071870204217974565905344099596626874796607089943508080432728070068422098577337358834220909924826672437254331787113966512622606409307405383208332748479781318475211679168464958993249814550046163454400360058728154415427751175808897055297753976501534715, 687447759621627694420930450942228918737749627303540285288795224658615500792345877200615210554073583207631367214645982647834815269941923560195153300141269452094055778698599285292304498906662933876732021385680923315010202725027263911572288967136291323701430776681520295896327568892576475949815159362903035673813123161305376088263340780510462744206376009015352935357618317483108638868114212411879804391319537755659185928620419380063560289088620536630083665088824464556853763455496521841880095966000894464922307646875178497505778197553009195295870713519508458093907055159063715845944225950791728266938921984020154616804988315899409358760908115497234652240011164171501728478670167752954776504328004363130214956299907487402643080580204008196290521378206598188421882232576958860859391633600582674497784267193230280447870907396722454543023983910912774822552060723400275178658927565893395786908705304394680711205780982071563132954758934137666999538477371766167881399256586787286306132104150912504682162916017143964292959663972675596276899273947399576577428076908938838545378742902016253693851806978074573467506973148299611317486625526056690516706649202546394769314896671182369894301487228472694048105555270735273312080545406201272114060887320081893665], 'aut_phi_ratio': 7.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 45, 1, 2], [2, 81, 1, 1], [3, 20, 2, 1], [3, 40, 1, 1], [4, 162, 1, 1], [4, 270, 2, 2], [4, 810, 1, 1], [5, 324, 1, 1], [6, 180, 2, 1], [6, 360, 1, 1], [10, 324, 1, 1], [12, 540, 2, 2], [20, 324, 1, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4:C_{40}:C_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': 120, 'autcentquo_group': '12960.bx', 'autcentquo_hash': 6161210740019565112, 'autcentquo_nilpotent': False, 'autcentquo_order': 12960, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4:C_{40}:C_4', 'cc_stats': [[1, 1, 1], [2, 45, 2], [2, 81, 1], [3, 20, 2], [3, 40, 1], [4, 162, 1], [4, 270, 4], [4, 810, 1], [5, 324, 1], [6, 180, 2], [6, 360, 1], [10, 324, 1], [12, 540, 4], [20, 324, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6480.bm', 'commutator_count': 1, 'commutator_label': '810.102', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '5.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 39, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 45, 1, 2], [2, 81, 1, 1], [3, 20, 1, 2], [3, 40, 1, 1], [4, 162, 1, 1], [4, 270, 2, 2], [4, 810, 1, 1], [5, 324, 1, 1], [6, 180, 1, 2], [6, 360, 1, 1], [10, 324, 1, 1], [12, 540, 2, 2], [20, 324, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 960, 'exponent': 60, 'exponents_of_order': [4, 4, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[20, 0, 4], [20, 1, 4], [40, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '6480.bm', 'hash': 6378546528877111736, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [12, 20, 3, 3, 3, 3], 'inner_gens': [[1, 1228, 3200, 3360, 2400, 720], [5097, 4, 1680, 3040, 2160, 2480], [6241, 1284, 80, 240, 720, 2160], [6241, 6084, 80, 240, 720, 2160], [5521, 5044, 80, 240, 720, 2160], [3601, 644, 80, 240, 720, 2160]], 'inner_hash': 6378546528877111736, 'inner_nilpotent': False, 'inner_order': 6480, 'inner_split': False, 'inner_tex': 'C_3^4:C_{20}:C_4', 'inner_used': [1, 2], 'irrC_degree': 20, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 20, 'irrep_stats': [[1, 8], [2, 2], [4, 4], [20, 8], [40, 2]], 'label': '6480.bm', 'linC_count': 8, 'linC_degree': 20, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 4, 'linQ_dim': 20, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 20, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^4:C20:C4', 'ngens': 9, '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, 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': 18, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 24, 'number_divisions': 19, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 105, 'number_subgroup_classes': 146, 'number_subgroups': 12866, 'old_label': None, 'order': 6480, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 171], [3, 80], [4, 2052], [5, 324], [6, 720], [10, 324], [12, 2160], [20, 648]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [60], 'outer_gens': [[421, 4, 3680, 5760, 4720, 1760]], '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': 6, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 2], [8, 1], [20, 4], [40, 4]], 'representations': {'PC': {'code': '110829151074325681295770853968059856867267907704670189136417235756716836195105409334790726389012657512896620083300185260256256590677', 'gens': [1, 3, 6, 7, 8, 9], 'pres': [9, -2, -2, -2, -2, -5, -3, 3, 3, 3, 18, 3610, 33158, 13457, 74, 148899, 79068, 102, 221044, 26653, 172805, 8654, 22703, 5972, 13541, 211686, 15135, 47904, 40353, 21777, 172807, 146896, 38905, 22354, 18403, 58328, 97217, 50246, 48635, 31229]}, 'Perm': {'d': 30, 'gens': [182832086518341409550528277131396, 45156560520507221447563222863810]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:C_{20}:C_4', 'transitive_degree': 30, 'wreath_data': None, 'wreath_product': False}