-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '111168.a', 'ambient_counter': 1, 'ambient_order': 111168, 'ambient_tex': 'C_3\\times F_{193}', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 9264, 'counter': 15, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '111168.a.12.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '12.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '12.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 12, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{12}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '9264.a', 'subgroup_hash': 1125378713036621189, 'subgroup_order': 9264, 'subgroup_tex': 'C_{193}:C_{48}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '111168.a', 'aut_centralizer_order': None, 'aut_label': '12.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '37056.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '6.b1.a1'], 'contains': ['24.b1.a1', '36.a1.a1', '2316.b1.a1'], 'core': '12.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3307, 3415, 5088, 4147, 5768, 6949, 5138, 4546], 'generators': [48, 96, 576, 24, 64, 12], 'label': '111168.a.12.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '12.b1.a1', 'normal_contained_in': ['4.a1.a1', '6.b1.a1'], 'normal_contains': ['24.b1.a1', '36.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '12.b1.a1', 'projective_image': '111168.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.b1.a1', 'subgroup_fusion': None, 'weyl_group': '37056.a'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '48.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 37056, 'aut_gen_orders': [192, 193], 'aut_gens': [[1, 48], [1, 1824], [8353, 48]], 'aut_group': '37056.a', 'aut_hash': 1432757702296672986, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 37056, 'aut_permdeg': 386, 'aut_perms': [18202997354635245150892255989052078862324293648724958368126716940691920535959005634576873633047165930115742948639497607149985008814470247760376591946876302043482086331984736299260723882303331222012124126668067597993528912551487304778908917793727800626446706704642776929760328898919486890280078021076301248764630172854178180343032964914127215787636518076893756718271613820021372630746316944403872287675554468696658066630892186589325169524798818591881767240803212286618030335091524078007103961401261727194980422963759873880072906058994470357226626997875948153249004424444070384257297602137989352370612497703276955194921479276381596273162965074755650924213036370490537912654941587751197045256832910131567401486434912696254678409037649603377525620032869241652978840399576053072935556803922527095181160032778344648377680199856578281572490, 18205302384957157755055156569623903666905842658916497356723082077194361344460742914480021779077602715804199459350032043104352455191694538660489271763241534236130128208476706713824938118840277928538692783689743207103698697502098302572498900153412975643693820487732867610066082883960014320323442833471836938680498062038840916064923174243816064609022929688919702156920680198399069852315958836255361762655123981445409114129418794919848962586117174152863875399384247549973822539528879919056510693149446851409087058156994651945320042202866321430179933694809841143005126725269074653088447892479450789863477601269222413337034328518963699801093637421300770137242197936003071405370334918065333121430068518785038304213142307472859982701713953930245728726977182402072344516500986567339698032943998766041911862376065094947750608288693605901833330], 'aut_phi_ratio': 12.0625, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 193, 1, 1], [3, 193, 1, 2], [4, 193, 1, 2], [6, 193, 1, 2], [8, 193, 1, 4], [12, 193, 1, 4], [16, 193, 1, 8], [24, 193, 1, 8], [48, 193, 1, 16], [193, 48, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_{193}', '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': 37056, 'autcentquo_group': '37056.a', 'autcentquo_hash': 1432757702296672986, 'autcentquo_nilpotent': False, 'autcentquo_order': 37056, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{193}', 'cc_stats': [[1, 1, 1], [2, 193, 1], [3, 193, 2], [4, 193, 2], [6, 193, 2], [8, 193, 4], [12, 193, 4], [16, 193, 8], [24, 193, 8], [48, 193, 16], [193, 48, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '9264.a', 'commutator_count': 1, 'commutator_label': '193.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '193.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 193, 1, 1], [3, 193, 2, 1], [4, 193, 2, 1], [6, 193, 2, 1], [8, 193, 4, 1], [12, 193, 4, 1], [16, 193, 8, 1], [24, 193, 8, 1], [48, 193, 16, 1], [193, 48, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1536, 'exponent': 9264, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 193], 'factors_of_order': [2, 3, 193], 'faithful_reps': [[48, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '9264.a', 'hash': 1125378713036621189, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 9264, 'inner_gen_orders': [48, 193], 'inner_gens': [[1, 6288], [3025, 48]], 'inner_hash': 1125378713036621189, 'inner_nilpotent': False, 'inner_order': 9264, 'inner_split': False, 'inner_tex': 'C_{193}:C_{48}', 'inner_used': [1, 2], 'irrC_degree': 48, 'irrQ_degree': 192, 'irrQ_dim': 192, 'irrR_degree': 48, 'irrep_stats': [[1, 48], [48, 4]], 'label': '9264.a', 'linC_count': 4, 'linC_degree': 48, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 192, 'linQ_degree_count': 1, 'linQ_dim': 192, 'linQ_dim_count': 1, 'linR_count': 4, 'linR_degree': 48, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C193:C48', 'ngens': 6, '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': 49, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 52, 'number_divisions': 11, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 20, 'number_subgroups': 1748, 'old_label': None, 'order': 9264, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 193], [3, 386], [4, 386], [6, 386], [8, 772], [12, 772], [16, 1544], [24, 1544], [48, 3088], [193, 192]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [37], 'outer_gens': [[1, 5424]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 193, 'pgroup': 0, 'primary_abelian_invariants': [16, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 2], [8, 2], [16, 1], [192, 1]], 'representations': {'PC': {'code': '304948676297405235447272883343606527584290600013267327', 'gens': [1, 6], 'pres': [6, -2, -2, -2, -2, -3, -193, 12, 31, 50, 69, 226373, 152939, 27233, 23567, 11693]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [7189251, 1028035235]}, 'Perm': {'d': 193, 'gens': [3736912403715067823373930328090042301118071099263380248296033042098831181667560352326718897338544085825149263737056408415709006732108258862743226885098880392104451438623432876896433823628124241668884733810694388112728520756678761417685406680052919578851231628761582855361434817280889393251820274265418636323217076383099469416640523834979635188698848768964, 360543977123933955894873607406092231048530816186021997934492360295585960503047095542725922589101604379606877057976328002973281753774605562734925235144287932489359869292327859851952029156223951256417137599141956174390567982235158481051420838745994207613288558596922386468168760559887165664607179609365078563217122702898368286236902907803644770069537114840035]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [48], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{193}:C_{48}', 'transitive_degree': 193, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '576.182', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 37056, 'aut_gen_orders': [64, 192, 96], 'aut_gens': [[1, 192], [97153, 39552], [105793, 64320], [47809, 18816]], 'aut_group': None, 'aut_hash': 9034218873832238734, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 222336, 'aut_permdeg': 579, 'aut_perms': [415332439196822892389352437816130401390738383370221364844878850607550778333990999776184605568421278475280244453511953039962181494122189821781883263316959201531902154435173394089394881934386442975626803816682013366274710639327603048148709203263335850956934513705677432612445360081946015157071655246361766139655035408612165506562077446916418628964643217990337766352996222601170027446653298548723903021005585090748174612486989215916127204701984643312424419565310829136987071596031819888738883484494631912286631467416080596207115760718910261030352200258429464228694285393933605670082097777061678748728778910376089906175830219538826740427819069543445641473111518135030865041383761505802995633042264157695385149095812910305730873962133804210048811778408408912044895041794289016373876993393003850343067915221826771638027998018289443816891858117689277909315562197355454440756908444550351612076844757970360026059982795028072217662618178674174347631582979577052959925128972855693498719179982748932220795907650991275021926868086525363090353655986424650695238183662501765618102081991333595375807675522468968658278057534484857352976226502006309087252976077697563757811289195230490163803952647899670731559865265643180967467213966712908123993183585039763847724928805148033274967628307145129222343322411324931038858051345174327003752114021781110525226462752885580313, 479014167272169279755996534749016869201360125409707869346777139173855325732973306407581827360637861416796628583967537605174750440755172951647611148360420886066818966625224493036121140593991361702574439173487931495542811324288522316193184742900760726240752065536218699257748196284431579881855064224518805697295732303159061258681671081176040188032548833476269580387262908306563250660702611019277068476953424205735136714001191742818540650758360743051137638423493431011048231585738140104365197692678118733943399624276121231952910352335809352534900979894500301522060798687180784338268714734310069283898407485255332153678510128023079965874650141745439653517794871949674493658599361325135322238631828400809628963234626171640799212597628544539653848153537551703888160925703847542959186641445837725502276602925122671623118778042891665514630761156855503193641295628483318460990688232589593362604657098181334407559208546438440116238131525874637673754413479433224759030646838233940605423973996483539483765545005350674521063200626975803892704173618464958000037145440102744057197009069087597840592403845425088701173293366696779715011794067656736491752480204104369899761052146486746011901145681350543544281549199692102699926114539513794074642925875249292216791419433139874592421827462737631936905234500848852439091584890275128310285682039693192033515132543417787127, 277760953236342837117292557174462821114287163390486698520044107397102630341828911334724634063986710842920444830955313299925385350983244268886021022315981237795549780737884741923651084913239180727343406296955777942433557918598624112300413804496960863368252179927543364492282896784532251429949042415925461844126162445734642961450965380443036076222027786381896161962090515095762416255165795351898158842639957857049238957473589440156894832176991940550709600986239377877538544743956803923387193193865066176949238129341122274150540946851597339194782674124160123517928207485441918334711385663320613461637191579989079478741038608798295078518503987684983548650849968716598703795924874995717040585144950493270494927740533175639582291434685719726656706800993299794242981898778028252854083548476628038358490539279012898734102971616777329387637325227448025943318834668524333064387760248758936069676844022436243570342772220747138128162070159407304816491385443347970526774849970968076693118499504812073473738620026244292676174046524243626716974721534065429730115396995305226243592090361756712031130218463359952789100999205852776151465902423211012566157221428957791334117857340426667880379703257574413329883898048665482750203385023283066989113542296839356503238609620710525212771820102003142672456708042283414397442459359716168084828115094291803017438146292553014690], 'aut_phi_ratio': 6.03125, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 193, 1, 1], [3, 1, 2, 1], [3, 193, 3, 2], [4, 193, 1, 2], [6, 193, 2, 1], [6, 193, 3, 2], [8, 193, 1, 4], [12, 193, 2, 2], [12, 193, 3, 4], [16, 193, 1, 8], [24, 193, 2, 4], [24, 193, 3, 8], [32, 193, 1, 16], [48, 193, 2, 8], [48, 193, 3, 16], [64, 193, 1, 32], [96, 193, 2, 16], [96, 193, 3, 32], [192, 193, 2, 32], [192, 193, 3, 64], [193, 192, 1, 1], [579, 192, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{579}.C_{96}.C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 37056, 'autcentquo_group': '37056.a', 'autcentquo_hash': 1432757702296672986, 'autcentquo_nilpotent': False, 'autcentquo_order': 37056, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{193}', 'cc_stats': [[1, 1, 1], [2, 193, 1], [3, 1, 2], [3, 193, 6], [4, 193, 2], [6, 193, 8], [8, 193, 4], [12, 193, 16], [16, 193, 8], [24, 193, 32], [32, 193, 16], [48, 193, 64], [64, 193, 32], [96, 193, 128], [192, 193, 256], [193, 192, 1], [579, 192, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '37056.a', 'commutator_count': 1, 'commutator_label': '193.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '193.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['37056.a', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 193, 1, 1], [3, 1, 2, 1], [3, 193, 2, 3], [4, 193, 2, 1], [6, 193, 2, 4], [8, 193, 4, 1], [12, 193, 4, 4], [16, 193, 8, 1], [24, 193, 8, 4], [32, 193, 16, 1], [48, 193, 16, 4], [64, 193, 32, 1], [96, 193, 32, 4], [192, 193, 64, 4], [193, 192, 1, 1], [579, 192, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24576, 'exponent': 37056, 'exponents_of_order': [6, 2, 1], 'factors_of_aut_order': [2, 3, 193], 'factors_of_order': [2, 3, 193], 'faithful_reps': [[192, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '111168.a', 'hash': 2544430592072231885, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 37056, 'inner_gen_orders': [192, 193], 'inner_gens': [[1, 28416], [82945, 192]], 'inner_hash': 1432757702296672986, 'inner_nilpotent': False, 'inner_order': 37056, 'inner_split': True, 'inner_tex': 'F_{193}', 'inner_used': [1, 2], 'irrC_degree': 192, 'irrQ_degree': 384, 'irrQ_dim': 384, 'irrR_degree': None, 'irrep_stats': [[1, 576], [192, 3]], 'label': '111168.a', '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': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*F193', 'ngens': 9, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 258, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 579, 'number_divisions': 37, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 84, 'number_subgroups': 7764, 'old_label': None, 'order': 111168, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 193], [3, 1160], [4, 386], [6, 1544], [8, 772], [12, 3088], [16, 1544], [24, 6176], [32, 3088], [48, 12352], [64, 6176], [96, 24704], [192, 49408], [193, 192], [579, 384]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[37057, 110976], [74113, 192]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 196, 'pgroup': 0, 'primary_abelian_invariants': [64, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 5], [8, 5], [16, 5], [32, 5], [64, 4], [192, 1], [384, 1]], 'representations': {'PC': {'code': '43023836931081284013122713998333971413484062145605317493274726077956947008935476624168211114647392625691859060173537147616629038650528127', 'gens': [1, 8], 'pres': [9, -2, -2, -2, -2, -2, -2, -3, -3, -193, 18, 46, 74, 102, 130, 158, 2045959, 3324688, 1175065, 654946, 42379, 36772, 59893, 286, 6905096, 2216177, 1714634, 1084787, 142928, 123983, 61298]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [201293672, 7189251, 776418240]}, 'Perm': {'d': 196, 'gens': [26998242104082005447251696691331464774121971917348421755781267835361429941668435221333290080630207260549917888723056059763539993175776658074908879762949012617169070415501257720842862810632964336955359428101490488135920181880491972015344559396903845870899558118113549760267322033639158277229321994017930485074559887727912882386576752937183479668551985879937835304, 2631778338730623662950964945824670653980764756662906445448946890283312750222914027144913759101874195455749646321665281014480436912763364625294578572203894275170402936839769371092982389742011367078542571256093075126368574007362860032302134699363967331838336966533207438392507094484694472919274785125127394266993739467967407942441269210233369459195139463953392461104, 53651049977427163004782649094143068173087154853604693633795859769783469643879806996027569149175074794409538134264301490310872601961150471788286697107340058590630568151646039445332768032894779979465847323816958134956968491642077503940495849602428272197081574774182305031506649821005649262198902553255728399362769286478552688840974860927203438716481864507921529723]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 192], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times F_{193}', 'transitive_degree': 579, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '12.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2], 'aut_gens': [[1], [5], [7]], 'aut_group': '4.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [1, 6], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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], [2, 1, 1], [3, 1, 2], [4, 1, 2], [6, 1, 2], [12, 1, 4]], 'center_label': '12.2', 'center_order': 12, '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': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 12, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 12]], 'label': '12.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C12', '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': 12, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [12, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5], [7]], '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': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [4, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 3865, 'gens': [1], 'pres': [3, -2, -2, -3, 6, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20970031]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [619]}, 'Perm': {'d': 7, 'gens': [2400, 4, 744]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [12], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
