-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '11664.ek', 'ambient_counter': 115, 'ambient_order': 11664, 'ambient_tex': 'C_3^4.F_9:C_2', 'central': False, 'central_factor': False, 'centralizer_order': 81, 'characteristic': True, 'core_order': 27, 'counter': 85, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '11664.ek.432.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '432.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '432.520', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 520, 'quotient_metabelian': False, 'quotient_nilpotent': False, 'quotient_order': 432, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': '\\He_3:\\SD_{16}', 'simple': False, 'solvable': True, 'special_labels': ['C6'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '27.5', 'subgroup_hash': 5, 'subgroup_order': 27, 'subgroup_tex': 'C_3^3', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '11664.ek', 'aut_centralizer_order': None, 'aut_label': '432.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '144.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['144.a1.a1', '144.b1.a1', '216.c1.a1', '216.h1.a1'], 'contains': ['1296.b1.a1', '1296.g1.a1'], 'core': '432.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1094, 5001, 1307, 5001, 1099, 5000, 1389, 5000], 'generators': [1728, 5184, 3888], 'label': '11664.ek.432.a1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '432.a1.a1', 'normal_contained_in': ['144.a1.a1'], 'normal_contains': ['3888.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '432.a1.a1', 'projective_image': '11664.ek', 'quotient_action_image': '144.182', 'quotient_action_kernel': '3.1', 'quotient_action_kernel_order': 3, 'quotient_fusion': None, 'short_label': '432.a1.a1', 'subgroup_fusion': None, 'weyl_group': '144.182'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '27.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 312, 'aut_gen_orders': [2, 13], 'aut_gens': [[1, 3, 9], [6, 2, 9], [5, 17, 11]], 'aut_group': '11232.a', 'aut_hash': 778507202365856770, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11232, 'aut_permdeg': 15, 'aut_perms': [6452863345, 297839648544], 'aut_phi_ratio': 624.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [3, 1, 26, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(3,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 312, 'autcent_group': '11232.a', 'autcent_hash': 778507202365856770, 'autcent_nilpotent': False, 'autcent_order': 11232, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(3,3)', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [3, 1, 26]], 'center_label': '27.5', 'center_order': 27, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 13]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '27.5', 'hash': 5, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 3, 9], [1, 3, 9], [1, 3, 9]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 27]], 'label': '27.5', 'linC_count': 1872, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 234, 'linQ_dim': 6, 'linQ_dim_count': 234, 'linR_count': 234, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^3', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 27, 'number_divisions': 14, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 28, 'number_subgroups': 28, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 26]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 312, 'outer_gen_orders': [2, 13], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 2, 9], [5, 17, 11]], 'outer_group': '11232.a', 'outer_hash': 778507202365856770, 'outer_nilpotent': False, 'outer_order': 11232, 'outer_permdeg': 15, 'outer_perms': [6452863345, 297839648544], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(3,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 13]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -3, 3, 3]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41624334336939899, 125101725607167719, 125101750005091125]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [10475581, 9534373, 23068812]}, 'Perm': {'d': 9, 'gens': [80640, 240, 4]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'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': 4, 'aut_exponent': 72, 'aut_gen_orders': [12, 24, 9, 3, 9, 3, 3], 'aut_gens': [[1, 2, 16, 48, 144, 1296, 3888], [721, 9606, 10160, 9120, 4912, 6048, 7776], [2645, 5042, 10336, 1824, 7904, 3024, 3888], [1153, 5938, 4384, 4368, 9696, 9072, 3888], [3025, 962, 9520, 48, 2304, 1296, 3888], [705, 2786, 8272, 6528, 4992, 9072, 3888], [5617, 4322, 3904, 48, 4032, 1296, 3888], [3889, 2, 16, 48, 144, 1296, 3888]], 'aut_group': '11664.ek', 'aut_hash': 1447588491907944804, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11664, 'aut_permdeg': 810, 'aut_perms': [51157548181307723313888153752251349173796309636065369667582536345740851995860651702848159654787554518999985838401462295732849552702695105805043392136268632707143162571668589411916528435994476766530646379037060833341372144719888011513920805274097783513706674555314384083189829125023112048499886605847591377000173721619450211173292675422393818454878230456465294261719993079764490662382524949791114177977700193858366755752929912617715930527088113601555251754420395451260536858753275418544371113125768056734575358958903445537044540033778582440649874630462584204745169786774697326616951285492836401802503480312046494345733988450045072760614975511291386252344185125769835411283462601500452042157765840958533399546313007843379652933854911893169969443559942377422369587441952353160306561828994887401877543162147864461732709627052706610522567487874306946621122468239521178068201855078695789587569285161799887980696095553102008740697385834220435173916612736466386554011844412222997708266837728918500796200965057763075048751837145055820122089465720450223398403512837291090930007684645290476089937382661646929126627847699388814862464901310295087523936896898257158488049913111960414442925676979573005374326420650609772767713485971072997105393788485277045148081433549849561345013030370158399357272250836911932402644200580819785903718458131845246235896807829608015430968135316496333913950202827220058486765119082105217071032059296929223981240868218931557869351822304003735187001674186762030125372073266359662837371230678151946160830361853820362191960775680833147267913444847336066141375409274568037164360568875383124661206288277368180637072505937288149887738541400879255211118941444376266068803763110611067718477719103997599584413056109241664352673219840973343925235130456807231889907888836085456694591024824686144338555697498209502120992262540596218216997833137940928641795600506537825908359949514218275563499231948138735295014698953381615413010548542064364167430190812981602589470344307912117300200403555319203691277359, 24861282268306924376503136612123587323495434894823975818150442112665699119405698076501007124810203892497268590441632407846114148756621062085148125456925103825793668011784321574360696459196038905039134835503454214088503157501762986887451595532387620755740387011382232162953254103553608509286543660653686543783144364611307240145161675177949107152864146248183483419309963885673709163230323169608803061134397883182370714769078014121260554968641802404204994076092425040602218751022606770636297429745536332770353414875495136439246736465934628806708001690336044680615843242163704041122143016431673162356969648637538045173643306848964956108333450104213323436028383692045830487299251985193541954112537062629991648137738097053239597316776754486775344999565053395677529310574159976048800402791695038491705219705637088620110371075831074205765376670380725288754427505523078921026433335351207329085219694414925706978435001297947853698584961647017960016211849166469809662925548921464238518288865511115327164996886318241777445224862777736057827809984336623438838225787698103561623482676581582694399581094011471206688613825036362940721318421357716224965991095908200409820484019578947156809177273761191373794595269021229410148074075270339552794582828158982590246377453925060368773309135073059790838024211660428759759820736163260604387479016089453114531709225343048438915047302202675863407675444369945365116758625322616320330214008873178803823205230556715058595601447573762692244155838991728980076002256400531495818984664697018453294516940611579426359701304118943913024102739468950758179825798555455758471711701652971291500898940513903436732012317031552648639038590459919179529531379193502238712723198384949887802948687080114347120164773413810121342610598338808622317479698850001820084071181278252553142639149192337346394865917229978409576014423684287064776440924180508126124179583581979915925734942809667218030406746542099935883305537066260135194223204941485306069263495940300684064561202292666766449166767116711984311865874, 22123867997931196422531974572744450736353580060705176766113041649656888394219728738190385268141002040901819725839679889979278555469411900550120716430003065740686641000632261928695332462797113827855698857755579167249761758226408432326510381400036505106311756296035131892202830788714240530819377254307785309373442632805461699430473215205974870083771104763621546395265948971414650813295923991068275141029747586470633793657310726317063882196420093593469175491007880033525968457581167144400531227921024416775283928095190060998204282827318492136538294587291103998066029263137479917725718812100404308225359259120296107465428724777513044504555324930982131772813654223800050380289071952836893392744784956833150350813361899019919302274706159261493496194462624753002051363083129095910418090431169871615265546078080100422123028632365930787645131450386091861070025033532240922255081800123225597388844557108741761860344071976212354978470721547273153309856636609756309252800260985943884274930052856424358062142165582643513722383574566588551670771062278313011671026697794000137045307125143414119944341228684441330394622384409275320457629387388314071140790606835089590590729373703502559248084099005878453461741129988469810971099525268484217072150117362283809623337017610226934193540364762896388463886232387037808853962684490955445719628033959066876595650177553738976873975219583293951618264950724934234693838794491281166248927896911734357608357410724949599340415220207035218566575956283795439487923911199448950195375514409923879301925994324904553596842365179591065155602667137588144972142542963741526551900292833613788798579351850155801195465803572381028581625436775105059873527621382711808368323125321795853883713435514884420112977740500290081264718509339195119627217404296377023161099237018031118892934995151864220382977331645029104912171906097058025627102369113607645325917591555064122663189383420557382066812585676614135643622537575861426753151845252386207432258535145096956819849900098041470163890044354289583206382030, 60076397900467455212020806964249144208204247438646750402582473484116918925036879004844506160166393377762047956284825471190566391565658287343983076615084962759924874816208180622586681805321365282293597038632763834534280607140774943294833526257544816122964165702562774343874751283212469172581342863482180199403998189913731203459490951442951927510618097945452652954044304573103767995204578364253158227036060857015273349426709484948749025137309500804854669927426900586078810261681526609181643682074943568761744169052865578698940321286726211515566879187312278626014242205943877484951117399928228899934366419440119445632064580617582086640775241923282487957672385451894078235206170557711137372559985019460571390464321466088795539887406615165048111314234893137749171698547669728318688135519598361100546882456563815152381186014728021840420076781247668553289114681977968692783096380234578008971589244626476679201639686244293785032385499509972811051216243120534289589746695384870105393408946351859451442171023652605326045340095895981199190709110087885898940074744504356082526163759398250631412814848080786405561765033218922329295289100277330354211618342160111398713037379787194466766830876404705127785502955546629173458185626407319255305296459861754782793877529254060469485265931155813106389108330359386369553624117490068489900686560617250001354619245734590432579625116115567734527795453488308297065079848806250639561539705341679072351736627539521421358876639002789021437963906239969477703287654967694075240510731032582578673711396728454330383784726978489434067272476768509406249252560354648080227933472405349517126500156714065187604427139764808916463267137409416318026050494790535502437826103156258310832774418419912907221165747616891254686759034689524209324242984421901516953833978069768132417533582004115846240234647409200643532877898756994027679172929912015550453240085492140620522963875174210785974811652769113250579241557574622844786438944267655942757241463026598080889043943110086622292756837943306481645940799, 17911691878013008863583682371273132005952579313827794893575974444307692736979452864678451002646026146706051327189061064567846972487377562395014408434248427763146247656646682756086019349240625429486458690964450014483244311721941609289731903439542183747077879118255516740282303659092014842427773511387510642711778792005975661777733566287926751044760552309357497592967607173782749941906473155176054485583240598285177064824493499314941334454148968228059818524145352680387238567356596458640476699500977784875167175778719018916436270856142191825367806373360434940548069083112011793096858881953837834357025755876415027678813139176261836641526644263882451645980738770931591841874448088040103206100306562709098231249875808713247536386917485476983033560008634532687690552917775023819053120046697148582436494134679918944913195043433968726922623516727868900561474582235831911583506691718037088696732751569383872090471139670108438260581119524351678951591571075965473029440742323248552359585214723378626085781533273553671337553834683607668485080760962498848849383358831448416708578933546760910756141954791756303926184776182331664007823134366809656442644654890920098632460204418239273557776890673845209792848631179824952692207744085670822183707034419100588339610239384891001174571401351835025400655407809868927052668047034421242159450056513285649519062766485466645524849476054242159538353687593999277094570995260152128236248116496234121290828232069375163236893597294282195132485622922385999568852579965009126210493750198125487770806580456737436485015147276259559690050952813226582293161614158243256084004170714680391283572244249588636864861078807890715386523789281356153045449508720519083189960192356882233858607715798915590675025818013797966481592879639215944559585823065108408518411841880153167887050969130181290988615287729762838039567868518686718297366302024419755681368761580402908667159230644645305285474483699491780331492360935853809997130912423369397605587756059631905135304762347451531500560930937321688060846208, 44251033862267283766793191978012117468898608293602779617880465945618286172632999529083625401760983875810977949521968507238852642232568070621524765452956010534564085346197901392745346639650147078858520236592898286085130794344570289125615549681089825050178164264399983035027783939585520348419620269803106024769231842276593406129383747457756975728221462419137915795577754799525383670800365532972005419211899433415848664313592397761415533238782277983885330594323143102769378180156039032995308139961412156912375512135550753498579920030053502396013645597312432386150735028978230695888482326316798303538475850415277935633527352965233899228800887167027854783133060196366557481994279258964569226142474576188375887103637539985719981013289312177472111028011503416840348076793298403630820749779384880444572736954944523600783007520760776280165287895529596878508081530297625671965970569334746344743096022882672105721988756817241611193313766296460223350562543361349472118115523336366380112217009583369087832317349219429343825277643532454220896663700959688295954029075784804108547682361563814394561591867195294608205579104482569769179803391513959069603482827580372096686487196110955321978239077915383509372313717038124598850835131679086284626736767312425643225137510734224714504815466667417593451250952818186573110582724394247319934029452742130767586777119104043060296497309653966346952495606443627327722586127411203800227683906144755062783282219666055524401160184311857776741837002014600402263677777384959770448259675587425167094020597504970332876833451266811460922981526226505522695816466738758585186094000462499552606278909044650786876908012370979689586815553853101091697321314637233138411885705134652055674729962874295550589582090153254611627118079973765757278455796046152649396393603998224692237870640863606681720681198998102211165859822610970465006090669218545562046046243511116600124341177512568842150712863140055923151625280356305573027491599260320164645854564210784157818930097105051845010798748793733370083216867, 232166781949680208864844481391037383858974827700404952523639812455588349068716422877077443204387672463926379617378914912753614031500020858408636650238369813723667693541088341650156398109766811769493064741253001994794252063590106483813968287981395346411318648595542609755074488293116897045597697742032626756600989760108992350548057861506625735951739447272877384988099672940340373642132839119411978862584619805949865259410247622583662007294099681719733414706294320806855906580029640437383333851980746626765300060319664042091200790339133919542537577256442310982446020192755149590321164357414246850690954396908476088401119970379742246992377962357158467729597662150067905350430640312466332319993551060866714332027968728853699130992607549243245212874237926630605557190886090259546579478844482989705382380730351474115247517577340795845455428240272810255638645708335707066695104543914216711583100683221054869224647661812684581136557856848575480920413897251689547152508046946028243996073362553179090575644081240269887809686883598891679963885465463460857207462983636247510216299111755542898839547911983104992437154048422033007606002318630631210988540654593346126282631078643254111377461903456046181913936512521016134605729003595410734566686365280426824106776493335733560501114267738065931196142285704835602022170195642675006424750003830663494447999256826490078155825842985682511854195943915385972290405331808928422351688975825397760626984601986085634057390286578952688454094430156386974901889731729534357202943941580273342687222354090832783868170166211073910915612343351769920549937504167175661959257902781948388186070402493190031954966779670737474735203210899785689711745757015195914911218353799268201560240017156319136463289399426541022185594562979024173554471267011187082295323539419123503323859740448049810138302921510438258904579720981128928699234782207162221619540711471823041363297272194378427269951832816105651733920650772820814883316033669304189318456292487953562973154398019985051489827644249], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [2, 324, 1, 1], [3, 2, 1, 1], [3, 18, 1, 1], [3, 24, 1, 1], [3, 36, 1, 1], [4, 162, 1, 1], [4, 972, 1, 1], [6, 162, 1, 2], [6, 324, 1, 1], [6, 648, 1, 1], [8, 486, 1, 2], [9, 216, 1, 3], [12, 162, 1, 2], [12, 324, 1, 3], [12, 972, 1, 2], [18, 648, 1, 3], [24, 486, 1, 4]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.F_9:C_2', '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': '11664.ek', 'autcentquo_hash': 1447588491907944804, 'autcentquo_nilpotent': False, 'autcentquo_order': 11664, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.F_9:C_2', 'cc_stats': [[1, 1, 1], [2, 81, 1], [2, 324, 1], [3, 2, 1], [3, 18, 1], [3, 24, 1], [3, 36, 1], [4, 162, 1], [4, 972, 1], [6, 162, 2], [6, 324, 1], [6, 648, 1], [8, 486, 2], [9, 216, 3], [12, 162, 2], [12, 324, 3], [12, 972, 2], [18, 648, 3], [24, 486, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '11664.ek', 'commutator_count': 1, 'commutator_label': '2916.ez', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 115, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [2, 324, 1, 1], [3, 2, 1, 1], [3, 18, 1, 1], [3, 24, 1, 1], [3, 36, 1, 1], [4, 162, 1, 1], [4, 972, 1, 1], [6, 162, 1, 2], [6, 324, 1, 1], [6, 648, 1, 1], [8, 486, 2, 1], [9, 216, 3, 1], [12, 162, 2, 1], [12, 324, 1, 1], [12, 324, 2, 1], [12, 972, 2, 1], [18, 648, 3, 1], [24, 486, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3888, 'exponent': 72, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[18, 0, 4], [18, 1, 4], [36, 1, 4]], 'familial': False, 'frattini_label': '81.15', 'frattini_quotient': '144.182', 'hash': 1447588491907944804, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 72, 'inner_gen_orders': [12, 24, 9, 3, 9, 3, 3], 'inner_gens': [[1, 7030, 2384, 6096, 3664, 6048, 7776], [4541, 2, 5760, 528, 1072, 4752, 3888], [6801, 9474, 16, 7392, 2784, 1296, 3888], [10801, 962, 9520, 48, 2304, 1296, 3888], [1761, 466, 3568, 3072, 144, 5184, 3888], [8209, 9506, 16, 48, 7920, 1296, 3888], [7777, 2, 16, 48, 144, 1296, 3888]], 'inner_hash': 1447588491907944804, 'inner_nilpotent': False, 'inner_order': 11664, 'inner_split': True, 'inner_tex': 'C_3^4.F_9:C_2', 'inner_used': [1, 2], 'irrC_degree': 18, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 4], [2, 3], [6, 4], [8, 2], [12, 1], [18, 8], [24, 6], [36, 4]], 'label': '11664.ek', 'linC_count': 8, 'linC_degree': 18, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 2, 'linQ_dim': 18, 'linQ_dim_count': 2, 'linR_count': 4, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C3^4.F9:C2', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 32, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 32, 'number_divisions': 21, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 156, 'number_subgroup_classes': 156, 'number_subgroups': 20767, 'old_label': None, 'order': 11664, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 405], [3, 80], [4, 1134], [6, 1296], [8, 972], [9, 648], [12, 3240], [18, 1944], [24, 1944]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1], [6, 2], [8, 2], [12, 2], [18, 2], [36, 3], [72, 4]], 'representations': {'PC': {'code': '34635088984181124347571049162115670457423824307550967904158290126449118099689404929847887289386496545121948904050411332463319023629611853467532695377517575231', 'gens': [1, 2, 5, 6, 7, 9, 10], 'pres': [10, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 49680, 140601, 51, 35882, 82, 64323, 38913, 119204, 144014, 132224, 57234, 20294, 365765, 15855, 46105, 55475, 27765, 256486, 37536, 187626, 90336, 12226, 3416, 276, 276487, 138257, 198747, 47557, 41087, 544328, 213858, 126388, 72938, 3308, 777609]}, 'Perm': {'d': 27, 'gens': [6710230576803116411098242934, 6307127140457963767328554172]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4.F_9:C_2', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'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': 4, 'aut_exponent': 24, 'aut_gen_orders': [4, 2, 3, 3, 12, 6, 3], 'aut_gens': [[20189134, 32004859, 22023521, 20206118, 16262215, 20438021, 1305911], [20189134, 782163, 26377746, 16225603, 41053625, 7191393, 1305911], [25834994, 32004859, 20206118, 22023521, 26337246, 5741406, 29656017], [29098928, 689175, 22023521, 33839929, 1239572, 26089305, 1305911], [27355766, 17144144, 7307223, 20206118, 33950737, 39336014, 1305911], [2492512, 8374639, 20206118, 28136091, 16262215, 26089305, 1305911], [8092706, 18902489, 42832787, 9863731, 23632259, 12783622, 1305911], [20189134, 18902489, 22023521, 20206118, 16262215, 20438021, 1305911]], 'aut_group': '432.520', 'aut_hash': 520, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 432, 'aut_permdeg': 72, 'aut_perms': [55575407439779946399003919067509480074341849024646174945925587897591177438453020799099139846537792347957, 10702176858838368127771858964961402679665095462879280464093191878499183687181718712586872678822827180043, 32867214233208652126711913072569718497421547172383083840103384006344978235102157040058336966981688483237, 45220971679821825038892228939748489186443268248488768024602646390501451442362452081207723807059577660108, 13548814129488995593687348922586061482053134371466431979643162679244086569377961746867070179875324249805, 60676832691453999795100506521613435786659966781277130140826414222186852902883340189517347132457600829073, 9791473905517437841599544141750357176146316099590674586233783140945514020460472820150895035809170560], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 36, 1, 1], [3, 2, 1, 1], [3, 24, 1, 1], [4, 18, 1, 1], [4, 36, 1, 1], [6, 18, 1, 1], [6, 72, 1, 1], [8, 54, 1, 2], [12, 36, 1, 3]], 'aut_supersolvable': False, 'aut_tex': '\\He_3:\\SD_{16}', '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': '432.520', 'autcentquo_hash': 520, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\He_3:\\SD_{16}', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 36, 1], [3, 2, 1], [3, 24, 1], [4, 18, 1], [4, 36, 1], [6, 18, 1], [6, 72, 1], [8, 54, 2], [12, 36, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '432.520', 'commutator_count': 1, 'commutator_label': '108.15', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 520, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 36, 1, 1], [3, 2, 1, 1], [3, 24, 1, 1], [4, 18, 1, 1], [4, 36, 1, 1], [6, 18, 1, 1], [6, 72, 1, 1], [8, 54, 2, 1], [12, 36, 1, 1], [12, 36, 2, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 144, 'exponent': 24, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 2], [6, 1, 2], [12, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '144.182', 'hash': 520, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [4, 2, 3, 3, 12, 6, 3], 'inner_gens': [[20189134, 782163, 26377746, 16225603, 41053625, 7191393, 1305911], [25834994, 32004859, 20206118, 22023521, 26337246, 5741406, 29656017], [38901070, 2447529, 22023521, 5509503, 4548587, 18041215, 1305911], [30677867, 15936914, 35125891, 20206118, 27993288, 1546886, 1305911], [2492512, 8374639, 20206118, 28136091, 16262215, 26089305, 1305911], [21768073, 18902489, 12908038, 39808163, 9961417, 20438021, 1305911], [20189134, 18902489, 22023521, 20206118, 16262215, 20438021, 1305911]], 'inner_hash': 520, 'inner_nilpotent': False, 'inner_order': 432, 'inner_split': True, 'inner_tex': '\\He_3:\\SD_{16}', 'inner_used': [1, 2, 3], 'irrC_degree': 6, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 4], [2, 3], [6, 4], [8, 2], [12, 1]], 'label': '432.520', 'linC_count': 4, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 2, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'He3:SD16', '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': 14, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 14, 'number_divisions': 12, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 40, 'number_subgroups': 511, 'old_label': None, 'order': 432, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 45], [3, 26], [4, 54], [6, 90], [8, 108], [12, 108]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1], [6, 2], [8, 2], [12, 2]], 'representations': {'PC': {'code': 315743925718442161789025476127419366510060295, 'gens': [1, 2, 5, 6, 7], 'pres': [7, -2, -2, -2, -2, -3, 3, -3, 85, 36, 254, 58, 4491, 438, 165, 4037, 7404, 355, 530, 537, 14118, 7069]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [124895238226433067, 41238302028140584]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [20189134, 32004859, 22023521, 20206118, 16262215, 20438021, 1305911]}, 'Perm': {'d': 27, 'gens': [15836702746677418303130425, 15593415493553930535327482, 594568345998959105472690, 108062212610944756771170, 7550102686338507196661760000, 2516538717432998222691201987, 806634631153204606767248884]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\He_3:\\SD_{16}', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
