-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1584.67', 'ambient_counter': 67, 'ambient_order': 1584, 'ambient_tex': 'D_{792}', 'central': False, 'central_factor': False, 'centralizer_order': 792, 'characteristic': True, 'core_order': 99, 'counter': 20, 'cyclic': True, 'direct': False, 'hall': 33, 'label': '1584.67.16.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '16.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '16.7', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 7, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_8', 'simple': False, 'solvable': True, 'special_labels': ['L3', 'C4'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '99.1', 'subgroup_hash': 1, 'subgroup_order': 99, 'subgroup_tex': 'C_{99}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1584.67', 'aut_centralizer_order': 3168, 'aut_label': '16.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '60.13', 'aut_weyl_index': 3168, 'centralizer': '2.b1.a1', 'complements': ['99.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '8.b1.a1', '8.b1.b1'], 'contains': ['48.a1.a1', '176.a1.a1'], 'core': '16.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [301, 801, 9689, 536, 347, 689, 822, 689], 'generators': [1408, 1056, 144], 'label': '1584.67.16.a1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '16.a1.a1', 'normal_contained_in': ['8.a1.a1'], 'normal_contains': ['48.a1.a1', '176.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '16.a1.a1', 'projective_image': '1584.67', 'quotient_action_image': '2.1', 'quotient_action_kernel': '8.1', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '16.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '99.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 30, 'aut_gen_orders': [2, 30], 'aut_gens': [[1], [89], [79]], 'aut_group': '60.13', 'aut_hash': 13, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 60, 'aut_permdeg': 12, 'aut_perms': [362880, 40285473], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1], [11, 1, 10, 1], [33, 1, 20, 1], [99, 1, 60, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{30}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 30, 'autcent_group': '60.13', 'autcent_hash': 13, 'autcent_nilpotent': True, 'autcent_order': 60, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{30}', '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, 2], [9, 1, 6], [11, 1, 10], [33, 1, 20], [99, 1, 60]], 'center_label': '99.1', 'center_order': 99, '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', '11.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1], [11, 1, 10, 1], [33, 1, 20, 1], [99, 1, 60, 1]], 'element_repr_type': 'PC', 'elementary': 33, 'eulerian_function': 1, 'exponent': 99, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [3, 11], 'faithful_reps': [[1, 0, 60]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '33.1', 'hash': 1, 'hyperelementary': 33, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 60, 'irrQ_dim': 60, 'irrR_degree': 2, 'irrep_stats': [[1, 99]], 'label': '99.1', 'linC_count': 60, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 30, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C99', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 99, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 99, 'order_factorization_type': 22, 'order_stats': [[1, 1], [3, 2], [9, 6], [11, 10], [33, 20], [99, 60]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 30], 'outer_gen_pows': [0, 0], 'outer_gens': [[89], [79]], 'outer_group': '60.13', 'outer_hash': 13, 'outer_nilpotent': True, 'outer_order': 60, 'outer_permdeg': 12, 'outer_perms': [362880, 40285473], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{30}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [9, 11], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [6, 1], [10, 1], [20, 1], [60, 1]], 'representations': {'PC': {'code': 3523119, 'gens': [1], 'pres': [3, -3, -3, -11, 9, 22]}, 'GLFp': {'d': 2, 'p': 89, 'gens': [48560782]}, 'Perm': {'d': 20, 'gens': [1013709170073600000, 36288000, 243332058851481600]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [99], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{99}', 'transitive_degree': 99, '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': 2, 'aut_exponent': 3960, 'aut_gen_orders': [18, 120, 180, 30, 30, 18], 'aut_gens': [[1, 2], [1257, 746], [679, 1570], [53, 314], [793, 562], [325, 1270], [731, 878]], 'aut_group': None, 'aut_hash': 2750653646408301733, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 190080, 'aut_permdeg': 792, 'aut_perms': [3349976607832693470664652740372905687235704876775373214741611000225789227807070238969093142638140445740451214632580357901964683364646689435479280987066922995607089786843175623766314318278892123156709151061629991558242370590047137637462869339917968623127795048718943398587060959528638123680479812676623035357045434174807948576549261142795774664551820761360604958394794983859338659722776552669197934371701501193549730188967662106133613751221794928994220520961177981296455971511249119105713142270653696833064521693037481008207141287221485203250385081937214519424888925586482380912856829094919540404749169778445148503781039416652627990521366965275297135706816403627923175072295381862269249152189792818576477020898912544500548620461265630610910074100911705360894562501511364137721283385677917855120796649422094942712617456513230548169887548391266984277151297138879201406460125118065852208601609230321043706152031051239155241211454544428455227613823137113347422128193300266055814455036328659945405465611901291221322768188723679440395460528709318932885683796090303241190535263532017868704252709856892823206518752871209802321864636559827760806066157403320190541651117842353251399519314876325739803284005161849705553947467177414280270433396988733027953811233976081187445948828552947252064663458204311018477238387074161993378759633698475090474088211122085980948187722208496157667562356451765666982335080463701569956713003790491406412961639764723773847608529893640576848382629060040063500949865872560177391857118011681552608402291956156855600502422070615610858211218551932569897694404074972705329158653668251431738512701080232737053457901521228749273981333384936491252323917894564730453951903953858664412302946196578691804012760761800895814899602856742094061281649944943431321087277977865588579132530459568456514336422652928467412096154556493023349436116246956433881662763975588698448010326074094805354745966100690439755656671572421207630327027168080759205168187276, 3671998187383577390016839505771017290397153562025455789258905641413168738799759369770606792182635021263335130461293817522158646345068237519099908484659830042260311646217013006362191897984574259317492685539769725007686591991466320594424252102553454338178638243004457036992464396038074336858894707473095340440272767329991931580188760326383462388122600255651339229005907694411878892737173899903694279603794489473983915508959232280250081058981078431233461419216711554063476521456157064521307072698441028602032909391875590778512012771948906455829525881238059292296275743470927855636416217332024883506442365164188469850736598592464936823833650937981971797462749118303910244298171421146416310138866581628865164606882445167544684379779011952491483404855874152526233738500645119431370731890426900872483874553265931990083152245074415691498270156495656400328312579730234794981633028823468154706154273282163583251310983176383342512549406068627645249670664798033206364077477645112821174871924125327225718274428479774977577720416167418566394304933840039444614633619636004444897596546923049567537367313100002102869388232143902469147393777984744334050015782163756892832564954831403393337740306075275990266673304897874825099501492334579942501769637950813812691029094832982011806392818584297257558193430234891762051972027374048409887341213672861096421565385189484685946650003736263814637682483270419010470006223301894572590040885846872154293541243648018086518593944521337412051772845434905020184677714724689238191892747396295956081769533835169533595945369208446851818191490238615925129670606181394804438102427517413765846383473736048678293967350244620651269009881112289286229002968217350908340353308891878623593486154962885515555160285190372774674742350473255381149805009142784468119342897051295250011085897130054023548953995296028565207836003696607792532414130353907983812116567760343218177395290777214516408525499471212483281677891306381911091080034145082092039978744421, 3456233266496093283982664717867451962655088935572602383284012928438298102897079797788677870100776495641481092566595413572117953674286879694581510902456596914921228758100218452476941731887168796795211857575289165285617256784214348646877053667946036086873154776281517372782943297968542326067987776811795031505401948955221049094806958877484023111222916209379090900441279485207702354682468186692749831794652676828522748039973980044748006981348134402188567931088897882986692770432255049734006684569152666513191677427719765724894304987723540466776274341874175442945887039386994264041920148293863486901159290600943505717196008446501787234841133196404245759240221577334770449942051538760688425839997919767384553228787261876227745524771827591576695238479597131950781322931525533654239707029314943002305677275991502765111469899316140483908526897959617588456443537048638936484054003328370488638599912080831890735243398331562508467168670831748947274080562278688362696811781096268120657090807953519897290797176283513932830925283772164885610196070556544205518797668588139415974703435046686449856567560019571498829686340898117741462190866989084188723152289306600665640014144363425381322031028590989421771302616492394127409138970031593711691262225490814201863125429358741575333809932070159229245566891480692389507369020022642961217951171593154278646897699548386811881720439315174451888897437660290071184248703480654570136167524616968554748146025465460963704426777574355423161579689652031408254148312971746552747620918995550414682280931778404396830569519558343808250543290528059614706662690265994834970838851712632274795492832017922169526348024424080125590375820140484072036249855440287519440685000347494254178639536934901367790737715136393180782554001716574608007776111952036725328944860939590757259930991196263455399735844733752616457522264220882739737700942371491378067637516851438329013520327820499575154488914551606715409339561655405789057449626842822907732027695413, 1296946517411143757884399385532046674940589824979153178738633273530444793756579619441728973567673629876653520404180610647043782512989056219743249763032119403374293636159040691679589664571532218824995799515808783230053922293491891705236689440699854937273883121975000206293320766276516769572131739779731657412274237223430876812284488545187073767543276596059474935369347594697271495698779545275879703979567101520654558535026803724652891956824700021262881250506622437583478388574754320610965308539715805478749505657359949292245980548801352235987188601654063002021130041690867409104761115809768691922563525858625201316515177380136232746139387072959533232271239893759884815017129887207377257631818346371021204721728873258912539962550100451040426938966789655975302805479293763303877199687860242832085966019839123326140412896721189217816294748605994679218917588648718588856582179470388206015311582371389540946808507519443508197095313780146936702963095843823744600298122017700868364706466378171428159937253866196248095372117983187930920341813544772260756838045700946936636172797639654228915569542414819110078345169883313448721543614148077547435280781775279890650764348636238943493305254590825963885485265358901672931439357263055591603370070894197034153897009739562564022983121906088179766923139616066247586377112111563840135886831497284221624683054424200861719361102760948003981880093021779517059397686017462620056544588647961277191396160118298500797578352905574809269584696221927064473346602849956437984678540448241287490311962214587067654571585047025383800499592441064317579662723653958470332765905276114939629468212218877588291620764650388268035295594477330250761286733285871544881987808305289374237261309912847191202002875188679538062462104882657678895091035071541760740945125273359146750915789821759112067505606894438955698404443201307558239067130368286330576464356576538863906759369598204839285704691786804986846727556874539439219888009095747737958987994649, 1651648379892540628853818727966395646793113566229299500699698836580140455289482175778838991512913708752078212434508653836161673774939922207230395303846525572183414779796063615631532322393795201549595116782003860579866081212734205107521839907527160116472276067527582852479078283400890565992296261213082916899203486358603856184442459667924715577772141220193230940339434798433931611662498183588684923759041179882763118225905714388306778054027723872752752900962902100845713086438183743795195219259180034016010307368286193259664011232287899682282921790329133594033626656702624772389136258084606437313684517055657690361931482497329760620771478973648941248828175444686805389058700114256495120014716801358813334743524935386530343180953987129636343489757042881506269812106037986517644164633103203142044360093232060911299984611921603105319905943678097880857017612943559262178328050270459995650478755183076855698857692998425289196702636202002527620491247988889975165520501673071625305603446354890628844392164274337691576988939732773181237610240886038861085921585547541452910314538218755802135147286301071500437744722669444578895963080571399688638278853638149655083988897890331251315310662103536040025672817214886400404799021863510394243573017641002200675881129706019801323043665087366820283735861235300931223913328165522062733758241890140748199405439554904250730837743194104523444738934049118462289221637494247989670567247570714734313993620992171040495438082825754242669433267187364745406271922510137168655254738233886243089442505394380006118313642691717795048343166945934682728208991591744785266615958533164493798711182744488144906907661034462315118698142009471486177986350623390929304626392830679829094540639060329725650185515314100152563930967152334060759735990181619075777748877375655006210242406397083351586959947203902765203210453866308257753733578204742823256317305014116713906975142420573159818748626375066213389793931985006245524405283488846473453917613251, 1126933378458298769917802778493604929226761366292384424149052423571937408358175150842303400826224127920795200751095957408372395876481379383160026265007243814533261307112250759443987080578482280445243112284826303804452080035981714106466695299158800194710918516452361545361984822958999798992606249897356446192693369013786487722998801473639099538879236892153895114442462545037374630590222517601329868674929830747069449309274574037083113588032084457723211232061052079815669684665404474307971149530729527229700138698032894745790310388718476060676552815869732768636660114786303937493758191933136585316067300100458239346113074169912210227819212374928811368267449667326462396098929201326101970918684933342832634009283506572939535317570505998896648492843559414005454173416206142577712358381035030768117198990071386736371412780810447016507604666779029611182648786548450609526230041545592025264910880790146207863319981228491198340299776526011769935446896736697632190697971552743294002917081639975872179758153698935237735686598009595250518499081501180780351550566281734113866853617402528033336339468738069832217469676134278504114967470780760948752684612094232031232948744519972685030167151052583980356698068538545063727882045745822876867917270895300928135020578324674625105461995185299110366908182527781776443791723287119370117326702508529818830834829682186196685116319225783839145646860290419667040170185630857959321707550957212200711013406629367874384379931845838773746407549307646861320612471059852472212318167132502308951582810456773701231363493463524209437479820017453601754183633533312571000429560710323660277166036052948251888850466329326996717766002366267601797183115152179091855910732202111743631118479475204165546031413142203440213117748970237091334324423888313776939324997850173032828226262900509885548459992697399835779120685278115780730791920277231108134101562444704352538707990992469744822763685625180684014598361478034907048541167208384925406872131223], 'aut_phi_ratio': 396.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 396, 2, 1], [3, 2, 1, 1], [4, 2, 1, 1], [6, 2, 1, 1], [8, 2, 2, 1], [9, 2, 3, 1], [11, 2, 5, 1], [12, 2, 2, 1], [18, 2, 3, 1], [22, 2, 5, 1], [24, 2, 4, 1], [33, 2, 10, 1], [36, 2, 6, 1], [44, 2, 10, 1], [66, 2, 10, 1], [72, 2, 12, 1], [88, 2, 20, 1], [99, 2, 30, 1], [132, 2, 20, 1], [198, 2, 30, 1], [264, 2, 40, 1], [396, 2, 60, 1], [792, 2, 120, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{99}.C_{60}.C_2^5', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1980, 'autcentquo_group': None, 'autcentquo_hash': 7697546556299087385, 'autcentquo_nilpotent': False, 'autcentquo_order': 47520, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{99}.C_{30}.C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 396, 2], [3, 2, 1], [4, 2, 1], [6, 2, 1], [8, 2, 2], [9, 2, 3], [11, 2, 5], [12, 2, 2], [18, 2, 3], [22, 2, 5], [24, 2, 4], [33, 2, 10], [36, 2, 6], [44, 2, 10], [66, 2, 10], [72, 2, 12], [88, 2, 20], [99, 2, 30], [132, 2, 20], [198, 2, 30], [264, 2, 40], [396, 2, 60], [792, 2, 120]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '792.25', 'commutator_count': 1, 'commutator_label': '396.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 67, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 396, 1, 2], [3, 2, 1, 1], [4, 2, 1, 1], [6, 2, 1, 1], [8, 2, 2, 1], [9, 2, 3, 1], [11, 2, 5, 1], [12, 2, 2, 1], [18, 2, 3, 1], [22, 2, 5, 1], [24, 2, 4, 1], [33, 2, 10, 1], [36, 2, 6, 1], [44, 2, 10, 1], [66, 2, 10, 1], [72, 2, 12, 1], [88, 2, 20, 1], [99, 2, 30, 1], [132, 2, 20, 1], [198, 2, 30, 1], [264, 2, 40, 1], [396, 2, 60, 1], [792, 2, 120, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 792, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[2, 1, 120]], 'familial': True, 'frattini_label': '12.2', 'frattini_quotient': '132.9', 'hash': 67, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 396, 'inner_gen_orders': [2, 396], 'inner_gens': [[1, 1582], [5, 2]], 'inner_hash': 25, 'inner_nilpotent': False, 'inner_order': 792, 'inner_split': False, 'inner_tex': 'D_{396}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 240, 'irrQ_dim': 240, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 395]], 'label': '1584.67', 'linC_count': 120, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 4, 'linQ_dim': 20, 'linQ_dim_count': 4, 'linR_count': 120, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D792', '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, 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': 25, 'number_characteristic_subgroups': 25, 'number_conjugacy_classes': 399, 'number_divisions': 26, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 66, 'number_subgroups': 2364, 'old_label': None, 'order': 1584, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 793], [3, 2], [4, 2], [6, 2], [8, 4], [9, 6], [11, 10], [12, 4], [18, 6], [22, 10], [24, 8], [33, 20], [36, 12], [44, 20], [66, 20], [72, 24], [88, 40], [99, 60], [132, 40], [198, 60], [264, 80], [396, 120], [792, 240]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 2, 2, 30], 'outer_gen_pows': [1408, 1408, 0, 330], 'outer_gens': [[1409, 1010], [1409, 398], [1, 218], [551, 1202]], 'outer_group': '240.208', 'outer_hash': 208, 'outer_nilpotent': True, 'outer_order': 240, 'outer_permdeg': 16, 'outer_perms': [40279680, 362880, 1307674368000, 6227026593], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{30}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 2], [6, 2], [8, 1], [10, 2], [12, 1], [20, 3], [24, 1], [40, 2], [60, 2], [80, 1], [120, 1], [240, 1]], 'representations': {'PC': {'code': 1202941996938997147046784599807845591982060000119, 'gens': [1, 2], 'pres': [7, -2, -2, -2, -2, -3, -3, -11, 22149, 36, 33182, 58, 44131, 80, 54884, 137, 64517, 166, 70566]}, 'Perm': {'d': 28, 'gens': [868708690480067649339017647, 7891827264921600, 13916445459840000, 21069517179110400, 12145280938112229701713920000, 4364893, 23454917710790789398855680000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{792}', 'transitive_degree': 792, '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': 2, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 8], 'aut_gens': [[1, 2], [1, 10], [1, 14], [3, 2]], 'aut_group': '32.43', 'aut_hash': 43, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [10824, 18366, 7679], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 2, 1], [4, 2, 1, 1], [8, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_8:C_2', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 2], [4, 2, 1], [8, 2, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '8.3', 'commutator_count': 1, 'commutator_label': '4.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 2], [4, 2, 1, 1], [8, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 8, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 2]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '4.2', 'hash': 7, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4], 'inner_gens': [[1, 14], [5, 2]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 3]], 'label': '16.7', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D8', 'ngens': 2, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 7, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 11, 'number_subgroups': 19, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 9], [4, 2], [8, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 2], 'outer_gens': [[1, 10], [3, 2]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1]], 'representations': {'PC': {'code': 2499614, 'gens': [1, 2], 'pres': [4, -2, 2, -2, -2, 113, 21, 146, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20182740, 20401702]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [2059, 2042]}, 'Perm': {'d': 8, 'gens': [23616, 143, 16577, 5167]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_8', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}