-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '165888.eb', 'ambient_counter': 106, 'ambient_order': 165888, 'ambient_tex': 'C_4\\times A_4^3:S_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 165888, 'counter': 1, 'cyclic': False, 'direct': None, 'hall': 6, 'label': '165888.eb.1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '1.a1', 'outer_equivalence': True, 'perfect': False, 'proper': False, 'quotient': '1.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_1', 'simple': False, 'solvable': True, 'special_labels': ['R', 'L0', 'D0', 'C0'], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '165888.eb', 'subgroup_hash': 813647509787346299, 'subgroup_order': 165888, 'subgroup_tex': 'C_4\\times A_4^3:S_4', 'supersolvable': False, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '165888.eb', 'aut_centralizer_order': None, 'aut_label': '1.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '41472.A', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': '1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [26, 2969, 6, 373], 'generators': [791394514560, 10867651201560, 95880, 241003462, 19006560, 270655227360, 7, 387132496560, 1093680, 8033837137200, 15307625906040, 355659096960, 355477656960, 1960645680, 474904782142], 'label': '165888.eb.1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1', 'normal_contained_in': [], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '1.a1', 'projective_image': '41472.j', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1.a1', 'subgroup_fusion': None, 'weyl_group': '41472.j'}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 72, 'aut_gen_orders': [8, 6, 12, 6], 'aut_gens': [[1408754672767, 1500846370697], [6722052392887, 4577362968257], [15042388965967, 13868106749417], [1361106810720, 7245948302777], [13078881573840, 10063361070742]], 'aut_group': '331776.gh', 'aut_hash': 9100794241782841809, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 331776, 'aut_permdeg': 866, 'aut_perms': [894623464274330486614947381646352639647764329118308747206470554401106890601983122727600098887147012796700633117464186139857186281403609786444739375261679992588303150745802768281300658052052189527959652024487146434966918887719563227518416286705226818902943405683950900931429844728615395256338455179981607623485618334008322505732913645885098043797815448472677147925812096761782502274244631663845443968398365435460791911213820215870478815493181211330113814134220111194145979837486868198985503826111193059060795184816156735917375392945079335505966876176903582118720667569515018127325987408846034682613807433621863686272414402210488591000942328298671680893910187014643437131945981318704603993012277262852270182785611971413608040981811911644749440280086917704423144415840394334036545992298028797705665069310327603054259636686622495430167852000340188887683811530441609987619753012771956126973064552897123239771200870558459753028216483959878323867329327135699151802729398091154009305980720771311209472034669604527845504415674079488868502495367671200932413409244461210164212987802814622911665806837890712496383764639767676410549532922625368046248523931613476365816532881926639357366683810434253601757285340208924169364403693138625292703597504510711727629044956686307102110001480223641910032186204749781134826601578931491881611224343708612424256014101855444351362962946732163176124841435617582450170037742542799322794253717334363115203516574455548158221693390360609951069674249686803057522187095687131577391891803104028760104396299130135193799899343195690730213236110452419469863912988568134684434849294711714976624179738547910809649783214906908950748040415531407080843230895910415087429050327034574265321043889254378151736615364981966990218181456046274589434180306134418405614435165313815750042606962008657491757543666726263412673150401439510585767152965922901554330206923206834042028076179889445716363709931335421391243366315825173670669113788611818765847383090358188194360789032944108561205631435766218051598820555798280546597711181655408188590663602065944968126235628417562733138182077558305993309982904708298490670228340285273756066977635838272703444069688103603485691800549, 2165035176079147889460575736485324813023616625160848671481443949939127024034532634243925137020772870584151456416640139598981692752367398192726551207909330026241737799188128956868660051543102279667777718711629879944960490003464734290639149435074099505594316980247566679754307488786998421004575570389241760849860593073150389747717867441424016572031474707727908616512531727065408208492430397796974957276023579790373326988394019187532469454365535553848005363285526992854987301985502861222968488993556099216432274657174529217760220477297438093946972765122316599838966878804747143420268159875005117260379459702842473190960822435687971630201549703092299259083633538587148891583305100609784103161945860454142872013901863243896299125701908378435051230491182168493206312756471633340142040489403468022370193940235293855093177494739804734898019476960554517201077070929130498502303667696378762032265855743694596841153701017157412986607202824276636470132756573120208596401056143165046908914617993883273712108401160483032755480092174431987304042962208913272729064362776766369588488346009048159728560938327229761152901273227599683829299940463285191073189295662006281673865618805968328434163530176846850884544899132187062157986590530347155849743481844233265628248557408008393074950901633730702156195867057885732307463137484870228695988593139881218412541710379489963408200074526629559026651885218519976213422197662224113398912144845380844308663908927273005549277665979824362951955478719945627827439843588051049779899009869277997639688079522858841715952596204965093302758809965848364216766886202347568993618365527975080585114125094911739121818015469746517519310009270025943738004752557090047925820009398766120900958501500597693039176529963223046373784053652335437638886894652040354766091979694788072193808051697783653001225648181561104098776246831391711570563124951014753939399910599755164959816840447295584381168792675530769847737509976856060222077112978221076138377085918991885572589482434952014796576508482981103122614529026959422358825897043035407325628403070584440620093328329757838222654229814083882463880426810624944810528953736091111842969911142425413310941741338155615967690957502, 2704918606667017043888198672413381860275082292487199010006196247332898192992522699944419417121355362130845546532857754414896364261529419058724401807051301113271648699121303269545828904273422617002308006199967152199611338331256836068808234464088670370812678335659816246024985237420687844116521710756975012461746726168198856489109896749221165979946786124294493780987057994581993731628435737488289484891929698168767999474330516503951324235480551862818660798508394345806388993128303482129445159582955837319063283092411699282873113277694167286996712530361852846433047677861486434813703923163741561545123061399679519078549971052971538156943408912819895623964128956474767686095521502281830461616690693644563343361897566263641268676185479210525107496811485485525577515304020390518887265386903857464415077923210908260124797792700763846283342442488614026396442080690008879512425527626765904237699519063090943905679322751173211301815532525016642536858969544835627581719997184424585121579672352897925547406867364867103801393498747081883062158707684023839939960983666872182912078859680088212299576567179770156944346560938525044760006430563725890883469445030935542253485652898961181409857057185430508905658549609529850822453504089241122607767010266819476915082245971971957785572809458727872995561494491185179284195781388140736321670249360175911964879956840198672695831194059496030163702654319250690069720599901136541817251070385225056036943744429396600234034746403233983030209709234853173333416345433713033192588078483007511674340489912403222176493578020473250655643656441282271525139510644710590016080910645316822360720278183087346765074527570779435094712210691809167344064588281157488134346997087949865385827796522446419730018048402926522212556147845590454600012346786760577727453525349406044816387838690261868832350731867253534238923030121187483687825586632676766203966516401948270023717221581563327882480068696794970017010651541309590756000967709543529855488939753208449841591488603140490471835074211259185826100611303832724595186742414205091550923444154647692659514079552268845829492194734590543220933093826867586544392487807192196863651879902653240526688983658852348111005667424, 4355735944114798232015222551731374669340153342679062947068981075806577716422863492512223461242397115104016739221267666323190859474033265026817044553284831702893537149056977752656708581728144224739645566781948669397712848834293355939873287159080519020718142347300103572967656172413251995780193408435450961554031745103340611291560455258095552617999859818095462234725665633230868968742720129567604653108003544616849537596643149743398716341627527213735868904327126492548618852700141800549692880242920429305744539224590429231467453850850728409795788319229019196185130402055461070827292584134553288844622386369543406345079307609133518760928224416034047724543098021189278414584260620711174450130066743761592828123634271768484735847881630834040861214414238898086070057098996166572881051185505928814295263118201343958862755905004613036477328905320619615112031903087663410627408797012219239821478009836388212345978487957861314074138634093660328996056855554232268624142415736353734037150369971285462722036207918789006091227582933552693885335019642771409818237376500065815034117735413977639424204738614798126624465507349512967514703437683605816164350566376146110795163249248891755792525671755023912995398346290854042328738721685270119557793679538518121340379628909527275069429007332452114571654782765721091390444242261702879495037793320171699803308464712098955451769719370767234762468386955595830679391872051916134768656105883494553775496068763656644432953053685988742977370051775738579979361416753814415831898987365365402038393006983462277107529284227950776933188930541190415092162603207010296848250768156443596368109495535866032577352414678382169918505031797895294034563701248531443706107633693560050824927642471524409071790861389458181473816638031608997216388659839658930102150813411110117045249157432357887601395051672225685750213149324620960922752525534064550987919314516882363806388699243559158023898449283132013635490794276842841273687322877597339573424452186900860018297051465276142816446412268418951633146652681144881124177613970477127357623678909007249909748484857466850712300993511716141880899546959967005206526765506188805159272000118503450096794094965961552131765071203], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 27, 1, 4], [2, 108, 1, 2], [2, 324, 1, 2], [2, 432, 2, 1], [3, 24, 1, 1], [3, 192, 1, 1], [3, 512, 1, 1], [3, 1152, 1, 1], [4, 1, 2, 1], [4, 9, 2, 1], [4, 27, 2, 2], [4, 108, 1, 2], [4, 108, 2, 2], [4, 216, 1, 2], [4, 216, 2, 1], [4, 324, 1, 2], [4, 324, 2, 2], [4, 432, 2, 5], [4, 648, 1, 2], [4, 648, 2, 1], [4, 1296, 2, 6], [6, 24, 1, 1], [6, 144, 1, 2], [6, 192, 1, 1], [6, 216, 1, 2], [6, 512, 1, 1], [6, 576, 1, 2], [6, 864, 1, 2], [6, 1152, 1, 1], [6, 3456, 1, 2], [6, 3456, 2, 1], [8, 432, 2, 2], [8, 1296, 2, 2], [9, 4608, 2, 1], [12, 24, 2, 1], [12, 144, 2, 1], [12, 192, 2, 1], [12, 216, 2, 1], [12, 512, 2, 1], [12, 576, 2, 1], [12, 864, 1, 2], [12, 864, 2, 2], [12, 1152, 2, 1], [12, 1728, 1, 2], [12, 1728, 2, 1], [12, 1728, 4, 2], [12, 3456, 2, 4], [18, 4608, 2, 1], [24, 1728, 4, 2], [36, 4608, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times S_4^3).D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 72, 'autcentquo_group': '82944.d', 'autcentquo_hash': 6931629392945487332, 'autcentquo_nilpotent': False, 'autcentquo_order': 82944, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'A_4^3.(C_2\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [2, 27, 4], [2, 108, 2], [2, 324, 2], [2, 432, 2], [3, 24, 1], [3, 192, 1], [3, 512, 1], [3, 1152, 1], [4, 1, 2], [4, 9, 2], [4, 27, 4], [4, 108, 6], [4, 216, 4], [4, 324, 6], [4, 432, 10], [4, 648, 4], [4, 1296, 12], [6, 24, 1], [6, 144, 2], [6, 192, 1], [6, 216, 2], [6, 512, 1], [6, 576, 2], [6, 864, 2], [6, 1152, 1], [6, 3456, 4], [8, 432, 4], [8, 1296, 4], [9, 4608, 2], [12, 24, 2], [12, 144, 2], [12, 192, 2], [12, 216, 2], [12, 512, 2], [12, 576, 2], [12, 864, 6], [12, 1152, 2], [12, 1728, 12], [12, 3456, 8], [18, 4608, 2], [24, 1728, 8], [36, 4608, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '41472.j', 'commutator_count': 1, 'commutator_label': '20736.r', 'complements_known': False, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 15, 'conjugacy_classes_known': True, 'counter': 106, 'cyclic': False, 'derived_length': 5, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['41472.j', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 27, 1, 4], [2, 108, 1, 2], [2, 324, 1, 2], [2, 432, 1, 2], [3, 24, 1, 1], [3, 192, 1, 1], [3, 512, 1, 1], [3, 1152, 1, 1], [4, 1, 2, 1], [4, 9, 2, 1], [4, 27, 2, 2], [4, 108, 1, 2], [4, 108, 2, 2], [4, 216, 1, 2], [4, 216, 2, 1], [4, 324, 1, 2], [4, 324, 2, 2], [4, 432, 1, 4], [4, 432, 2, 3], [4, 648, 1, 2], [4, 648, 2, 1], [4, 1296, 1, 6], [4, 1296, 2, 3], [6, 24, 1, 1], [6, 144, 1, 2], [6, 192, 1, 1], [6, 216, 1, 2], [6, 512, 1, 1], [6, 576, 1, 2], [6, 864, 1, 2], [6, 1152, 1, 1], [6, 3456, 1, 4], [8, 432, 1, 2], [8, 432, 2, 1], [8, 1296, 1, 2], [8, 1296, 2, 1], [9, 4608, 1, 2], [12, 24, 2, 1], [12, 144, 2, 1], [12, 192, 2, 1], [12, 216, 2, 1], [12, 512, 2, 1], [12, 576, 2, 1], [12, 864, 1, 2], [12, 864, 2, 2], [12, 1152, 2, 1], [12, 1728, 1, 2], [12, 1728, 2, 3], [12, 1728, 4, 1], [12, 3456, 1, 2], [12, 3456, 2, 3], [18, 4608, 1, 2], [24, 1728, 2, 2], [24, 1728, 4, 1], [36, 4608, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 72, 'exponents_of_order': [11, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[9, 0, 8], [18, 0, 2], [27, 0, 12], [36, 0, 8], [54, 0, 12], [81, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '82944.o', 'hash': 813647509787346299, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 72, 'inner_gen_orders': [4, 9], 'inner_gens': [[1408754672767, 11976219900737], [2790750792367, 1500846370697]], 'inner_hash': 4961977302514827477, 'inner_nilpotent': False, 'inner_order': 41472, 'inner_split': True, 'inner_tex': 'A_4^3:S_4', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 8], [2, 4], [3, 8], [6, 16], [8, 12], [9, 16], [12, 8], [18, 4], [27, 24], [36, 16], [54, 24], [81, 8]], 'label': '165888.eb', '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': 'C4*A4^3:S4', 'ngens': 2, '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, 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': 90, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 148, 'number_divisions': 105, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 14195, 'number_subgroup_classes': 15920, 'number_subgroups': 4765433, 'old_label': None, 'order': 165888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 1855], [3, 1880], [4, 26048], [6, 19304], [8, 6912], [9, 9216], [12, 59200], [18, 9216], [24, 13824], [36, 18432]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1408754672760, 1500846370702], [1408754672760, 1500846370697], [5426624811007, 13360009622537]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 10, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 6], [8, 6], [9, 8], [12, 8], [16, 3], [18, 6], [24, 3], [27, 12], [36, 9], [54, 14], [72, 4], [81, 4], [108, 6], [162, 2], [216, 1]], 'representations': {'PC': {'code': '63160286675392184216458933786033274469142055384192784219940781148552140595833172523061819287019647846324092945916504621013713792491538246959302826962517560635103073208132070856591173453567519266123256076710060887444807762286904196069195934557619298556762002292852193748099419610173854588843062428489038936285217570040608198414526352744989904488919368345429898770353279361845139389883338179564355663756535989573852126094595620038419825609557296707747502038063033817801139349034075392336', 'gens': [1, 2, 5, 6, 8, 10, 12, 13, 14, 15], 'pres': [15, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 803160, 4866781, 76, 1877942, 122, 4763043, 1085808, 10558804, 308719, 1457134, 982624, 154864, 4598645, 1119980, 1481255, 388040, 83045, 260, 13366086, 1300341, 1179396, 1549851, 906, 17959687, 3841942, 2796517, 1893652, 277987, 79282, 13417, 352, 5145128, 3803783, 4445318, 411533, 42188, 178283, 23716809, 8179224, 691239, 2547054, 540069, 29784, 20349, 82014, 28929, 444, 7413130, 7945, 1021720, 233695, 47590, 23845, 25855, 18940, 15863051, 7931546, 1866281, 2099576, 42206, 1721, 1010907, 4296282, 273837, 252807, 42237, 14172, 4827, 3672, 1179388, 4898923, 22738, 544393, 15238, 22813, 7708, 1423, 22161614, 16912829, 2381444, 3207659, 923474, 465839, 220829, 4169, 38609]}, 'Perm': {'d': 16, 'gens': [1408754672767, 1500846370697]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 288, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times A_4^3:S_4', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 72, 'aut_gen_orders': [8, 6, 12, 6], 'aut_gens': [[1408754672767, 1500846370697], [6722052392887, 4577362968257], [15042388965967, 13868106749417], [1361106810720, 7245948302777], [13078881573840, 10063361070742]], 'aut_group': '331776.gh', 'aut_hash': 9100794241782841809, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 331776, 'aut_permdeg': 866, 'aut_perms': [894623464274330486614947381646352639647764329118308747206470554401106890601983122727600098887147012796700633117464186139857186281403609786444739375261679992588303150745802768281300658052052189527959652024487146434966918887719563227518416286705226818902943405683950900931429844728615395256338455179981607623485618334008322505732913645885098043797815448472677147925812096761782502274244631663845443968398365435460791911213820215870478815493181211330113814134220111194145979837486868198985503826111193059060795184816156735917375392945079335505966876176903582118720667569515018127325987408846034682613807433621863686272414402210488591000942328298671680893910187014643437131945981318704603993012277262852270182785611971413608040981811911644749440280086917704423144415840394334036545992298028797705665069310327603054259636686622495430167852000340188887683811530441609987619753012771956126973064552897123239771200870558459753028216483959878323867329327135699151802729398091154009305980720771311209472034669604527845504415674079488868502495367671200932413409244461210164212987802814622911665806837890712496383764639767676410549532922625368046248523931613476365816532881926639357366683810434253601757285340208924169364403693138625292703597504510711727629044956686307102110001480223641910032186204749781134826601578931491881611224343708612424256014101855444351362962946732163176124841435617582450170037742542799322794253717334363115203516574455548158221693390360609951069674249686803057522187095687131577391891803104028760104396299130135193799899343195690730213236110452419469863912988568134684434849294711714976624179738547910809649783214906908950748040415531407080843230895910415087429050327034574265321043889254378151736615364981966990218181456046274589434180306134418405614435165313815750042606962008657491757543666726263412673150401439510585767152965922901554330206923206834042028076179889445716363709931335421391243366315825173670669113788611818765847383090358188194360789032944108561205631435766218051598820555798280546597711181655408188590663602065944968126235628417562733138182077558305993309982904708298490670228340285273756066977635838272703444069688103603485691800549, 2165035176079147889460575736485324813023616625160848671481443949939127024034532634243925137020772870584151456416640139598981692752367398192726551207909330026241737799188128956868660051543102279667777718711629879944960490003464734290639149435074099505594316980247566679754307488786998421004575570389241760849860593073150389747717867441424016572031474707727908616512531727065408208492430397796974957276023579790373326988394019187532469454365535553848005363285526992854987301985502861222968488993556099216432274657174529217760220477297438093946972765122316599838966878804747143420268159875005117260379459702842473190960822435687971630201549703092299259083633538587148891583305100609784103161945860454142872013901863243896299125701908378435051230491182168493206312756471633340142040489403468022370193940235293855093177494739804734898019476960554517201077070929130498502303667696378762032265855743694596841153701017157412986607202824276636470132756573120208596401056143165046908914617993883273712108401160483032755480092174431987304042962208913272729064362776766369588488346009048159728560938327229761152901273227599683829299940463285191073189295662006281673865618805968328434163530176846850884544899132187062157986590530347155849743481844233265628248557408008393074950901633730702156195867057885732307463137484870228695988593139881218412541710379489963408200074526629559026651885218519976213422197662224113398912144845380844308663908927273005549277665979824362951955478719945627827439843588051049779899009869277997639688079522858841715952596204965093302758809965848364216766886202347568993618365527975080585114125094911739121818015469746517519310009270025943738004752557090047925820009398766120900958501500597693039176529963223046373784053652335437638886894652040354766091979694788072193808051697783653001225648181561104098776246831391711570563124951014753939399910599755164959816840447295584381168792675530769847737509976856060222077112978221076138377085918991885572589482434952014796576508482981103122614529026959422358825897043035407325628403070584440620093328329757838222654229814083882463880426810624944810528953736091111842969911142425413310941741338155615967690957502, 2704918606667017043888198672413381860275082292487199010006196247332898192992522699944419417121355362130845546532857754414896364261529419058724401807051301113271648699121303269545828904273422617002308006199967152199611338331256836068808234464088670370812678335659816246024985237420687844116521710756975012461746726168198856489109896749221165979946786124294493780987057994581993731628435737488289484891929698168767999474330516503951324235480551862818660798508394345806388993128303482129445159582955837319063283092411699282873113277694167286996712530361852846433047677861486434813703923163741561545123061399679519078549971052971538156943408912819895623964128956474767686095521502281830461616690693644563343361897566263641268676185479210525107496811485485525577515304020390518887265386903857464415077923210908260124797792700763846283342442488614026396442080690008879512425527626765904237699519063090943905679322751173211301815532525016642536858969544835627581719997184424585121579672352897925547406867364867103801393498747081883062158707684023839939960983666872182912078859680088212299576567179770156944346560938525044760006430563725890883469445030935542253485652898961181409857057185430508905658549609529850822453504089241122607767010266819476915082245971971957785572809458727872995561494491185179284195781388140736321670249360175911964879956840198672695831194059496030163702654319250690069720599901136541817251070385225056036943744429396600234034746403233983030209709234853173333416345433713033192588078483007511674340489912403222176493578020473250655643656441282271525139510644710590016080910645316822360720278183087346765074527570779435094712210691809167344064588281157488134346997087949865385827796522446419730018048402926522212556147845590454600012346786760577727453525349406044816387838690261868832350731867253534238923030121187483687825586632676766203966516401948270023717221581563327882480068696794970017010651541309590756000967709543529855488939753208449841591488603140490471835074211259185826100611303832724595186742414205091550923444154647692659514079552268845829492194734590543220933093826867586544392487807192196863651879902653240526688983658852348111005667424, 4355735944114798232015222551731374669340153342679062947068981075806577716422863492512223461242397115104016739221267666323190859474033265026817044553284831702893537149056977752656708581728144224739645566781948669397712848834293355939873287159080519020718142347300103572967656172413251995780193408435450961554031745103340611291560455258095552617999859818095462234725665633230868968742720129567604653108003544616849537596643149743398716341627527213735868904327126492548618852700141800549692880242920429305744539224590429231467453850850728409795788319229019196185130402055461070827292584134553288844622386369543406345079307609133518760928224416034047724543098021189278414584260620711174450130066743761592828123634271768484735847881630834040861214414238898086070057098996166572881051185505928814295263118201343958862755905004613036477328905320619615112031903087663410627408797012219239821478009836388212345978487957861314074138634093660328996056855554232268624142415736353734037150369971285462722036207918789006091227582933552693885335019642771409818237376500065815034117735413977639424204738614798126624465507349512967514703437683605816164350566376146110795163249248891755792525671755023912995398346290854042328738721685270119557793679538518121340379628909527275069429007332452114571654782765721091390444242261702879495037793320171699803308464712098955451769719370767234762468386955595830679391872051916134768656105883494553775496068763656644432953053685988742977370051775738579979361416753814415831898987365365402038393006983462277107529284227950776933188930541190415092162603207010296848250768156443596368109495535866032577352414678382169918505031797895294034563701248531443706107633693560050824927642471524409071790861389458181473816638031608997216388659839658930102150813411110117045249157432357887601395051672225685750213149324620960922752525534064550987919314516882363806388699243559158023898449283132013635490794276842841273687322877597339573424452186900860018297051465276142816446412268418951633146652681144881124177613970477127357623678909007249909748484857466850712300993511716141880899546959967005206526765506188805159272000118503450096794094965961552131765071203], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 27, 1, 4], [2, 108, 1, 2], [2, 324, 1, 2], [2, 432, 2, 1], [3, 24, 1, 1], [3, 192, 1, 1], [3, 512, 1, 1], [3, 1152, 1, 1], [4, 1, 2, 1], [4, 9, 2, 1], [4, 27, 2, 2], [4, 108, 1, 2], [4, 108, 2, 2], [4, 216, 1, 2], [4, 216, 2, 1], [4, 324, 1, 2], [4, 324, 2, 2], [4, 432, 2, 5], [4, 648, 1, 2], [4, 648, 2, 1], [4, 1296, 2, 6], [6, 24, 1, 1], [6, 144, 1, 2], [6, 192, 1, 1], [6, 216, 1, 2], [6, 512, 1, 1], [6, 576, 1, 2], [6, 864, 1, 2], [6, 1152, 1, 1], [6, 3456, 1, 2], [6, 3456, 2, 1], [8, 432, 2, 2], [8, 1296, 2, 2], [9, 4608, 2, 1], [12, 24, 2, 1], [12, 144, 2, 1], [12, 192, 2, 1], [12, 216, 2, 1], [12, 512, 2, 1], [12, 576, 2, 1], [12, 864, 1, 2], [12, 864, 2, 2], [12, 1152, 2, 1], [12, 1728, 1, 2], [12, 1728, 2, 1], [12, 1728, 4, 2], [12, 3456, 2, 4], [18, 4608, 2, 1], [24, 1728, 4, 2], [36, 4608, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times S_4^3).D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 72, 'autcentquo_group': '82944.d', 'autcentquo_hash': 6931629392945487332, 'autcentquo_nilpotent': False, 'autcentquo_order': 82944, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'A_4^3.(C_2\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [2, 27, 4], [2, 108, 2], [2, 324, 2], [2, 432, 2], [3, 24, 1], [3, 192, 1], [3, 512, 1], [3, 1152, 1], [4, 1, 2], [4, 9, 2], [4, 27, 4], [4, 108, 6], [4, 216, 4], [4, 324, 6], [4, 432, 10], [4, 648, 4], [4, 1296, 12], [6, 24, 1], [6, 144, 2], [6, 192, 1], [6, 216, 2], [6, 512, 1], [6, 576, 2], [6, 864, 2], [6, 1152, 1], [6, 3456, 4], [8, 432, 4], [8, 1296, 4], [9, 4608, 2], [12, 24, 2], [12, 144, 2], [12, 192, 2], [12, 216, 2], [12, 512, 2], [12, 576, 2], [12, 864, 6], [12, 1152, 2], [12, 1728, 12], [12, 3456, 8], [18, 4608, 2], [24, 1728, 8], [36, 4608, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '41472.j', 'commutator_count': 1, 'commutator_label': '20736.r', 'complements_known': False, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 15, 'conjugacy_classes_known': True, 'counter': 106, 'cyclic': False, 'derived_length': 5, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['41472.j', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 27, 1, 4], [2, 108, 1, 2], [2, 324, 1, 2], [2, 432, 1, 2], [3, 24, 1, 1], [3, 192, 1, 1], [3, 512, 1, 1], [3, 1152, 1, 1], [4, 1, 2, 1], [4, 9, 2, 1], [4, 27, 2, 2], [4, 108, 1, 2], [4, 108, 2, 2], [4, 216, 1, 2], [4, 216, 2, 1], [4, 324, 1, 2], [4, 324, 2, 2], [4, 432, 1, 4], [4, 432, 2, 3], [4, 648, 1, 2], [4, 648, 2, 1], [4, 1296, 1, 6], [4, 1296, 2, 3], [6, 24, 1, 1], [6, 144, 1, 2], [6, 192, 1, 1], [6, 216, 1, 2], [6, 512, 1, 1], [6, 576, 1, 2], [6, 864, 1, 2], [6, 1152, 1, 1], [6, 3456, 1, 4], [8, 432, 1, 2], [8, 432, 2, 1], [8, 1296, 1, 2], [8, 1296, 2, 1], [9, 4608, 1, 2], [12, 24, 2, 1], [12, 144, 2, 1], [12, 192, 2, 1], [12, 216, 2, 1], [12, 512, 2, 1], [12, 576, 2, 1], [12, 864, 1, 2], [12, 864, 2, 2], [12, 1152, 2, 1], [12, 1728, 1, 2], [12, 1728, 2, 3], [12, 1728, 4, 1], [12, 3456, 1, 2], [12, 3456, 2, 3], [18, 4608, 1, 2], [24, 1728, 2, 2], [24, 1728, 4, 1], [36, 4608, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 72, 'exponents_of_order': [11, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[9, 0, 8], [18, 0, 2], [27, 0, 12], [36, 0, 8], [54, 0, 12], [81, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '82944.o', 'hash': 813647509787346299, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 72, 'inner_gen_orders': [4, 9], 'inner_gens': [[1408754672767, 11976219900737], [2790750792367, 1500846370697]], 'inner_hash': 4961977302514827477, 'inner_nilpotent': False, 'inner_order': 41472, 'inner_split': True, 'inner_tex': 'A_4^3:S_4', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 8], [2, 4], [3, 8], [6, 16], [8, 12], [9, 16], [12, 8], [18, 4], [27, 24], [36, 16], [54, 24], [81, 8]], 'label': '165888.eb', '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': 'C4*A4^3:S4', 'ngens': 2, '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, 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': 90, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 148, 'number_divisions': 105, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 14195, 'number_subgroup_classes': 15920, 'number_subgroups': 4765433, 'old_label': None, 'order': 165888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 1855], [3, 1880], [4, 26048], [6, 19304], [8, 6912], [9, 9216], [12, 59200], [18, 9216], [24, 13824], [36, 18432]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1408754672760, 1500846370702], [1408754672760, 1500846370697], [5426624811007, 13360009622537]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 10, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 6], [8, 6], [9, 8], [12, 8], [16, 3], [18, 6], [24, 3], [27, 12], [36, 9], [54, 14], [72, 4], [81, 4], [108, 6], [162, 2], [216, 1]], 'representations': {'PC': {'code': '63160286675392184216458933786033274469142055384192784219940781148552140595833172523061819287019647846324092945916504621013713792491538246959302826962517560635103073208132070856591173453567519266123256076710060887444807762286904196069195934557619298556762002292852193748099419610173854588843062428489038936285217570040608198414526352744989904488919368345429898770353279361845139389883338179564355663756535989573852126094595620038419825609557296707747502038063033817801139349034075392336', 'gens': [1, 2, 5, 6, 8, 10, 12, 13, 14, 15], 'pres': [15, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 803160, 4866781, 76, 1877942, 122, 4763043, 1085808, 10558804, 308719, 1457134, 982624, 154864, 4598645, 1119980, 1481255, 388040, 83045, 260, 13366086, 1300341, 1179396, 1549851, 906, 17959687, 3841942, 2796517, 1893652, 277987, 79282, 13417, 352, 5145128, 3803783, 4445318, 411533, 42188, 178283, 23716809, 8179224, 691239, 2547054, 540069, 29784, 20349, 82014, 28929, 444, 7413130, 7945, 1021720, 233695, 47590, 23845, 25855, 18940, 15863051, 7931546, 1866281, 2099576, 42206, 1721, 1010907, 4296282, 273837, 252807, 42237, 14172, 4827, 3672, 1179388, 4898923, 22738, 544393, 15238, 22813, 7708, 1423, 22161614, 16912829, 2381444, 3207659, 923474, 465839, 220829, 4169, 38609]}, 'Perm': {'d': 16, 'gens': [1408754672767, 1500846370697]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 288, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times A_4^3:S_4', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 1]], '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': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, 'wreath_data': None, 'wreath_product': False}
