-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '12352.1674', 'ambient_counter': 1674, 'ambient_order': 12352, 'ambient_tex': 'C_{772}:C_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 64, 'characteristic': False, 'core_order': 2, 'counter': 50, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '12352.1674.3088.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '3088.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 3088, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '12352.1674', 'aut_centralizer_order': None, 'aut_label': '3088.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '193.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['16.c1.a1', '1544.a1.a1'], 'contains': ['6176.a1.a1'], 'core': '6176.a1.a1', 'coset_action_label': None, 'count': 193, 'diagramx': [1154, -1, 1135, -1, 9068, -1, 3233, -1], 'generators': [6168], 'label': '12352.1674.3088.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '16.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '193.a1.a1', 'old_label': '3088.c1.a1', 'projective_image': '6176.246', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3088.c1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '64.26', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 37056, 'aut_gen_orders': [48, 24, 64, 48], 'aut_gens': [[1, 16], [4273, 3856], [5809, 9008], [3601, 464], [529, 8816]], 'aut_group': None, 'aut_hash': 2013647253270866901, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 296448, 'aut_permdeg': 772, 'aut_perms': [166406514259950795693920368248954736540967271155686351680047657529296457433325983294469233337708988255359577840171958019232667324849424574645511161950002024950291774585827141564053997754016037791146883649120511603332802568632659516471768191744587732678663690253888410247359058388632478489966948148998751361744505810934894037476859732947714264662762285542349246770778834559051996851376961814053726813032559842314927244306083543860087238780697453711739766698654601940904147693343753639029858930554919315756491497159647207054461733993917915091768848871872506770587877811756028983490143172468298489204580538593242448654795749629449582433383486799610789926105163560449874998366417191365941848796582581166577787180434622605580598986900617468622970115984746628885066661160483839886487995783282525482635730315843761954135778842974409956047604552434715287363759351072478657286340271983599952601559324611871960644709758984763130374436076264595793211919845310099865776972031885347342425407501374204108935428633835419152427711644849198809225100317177523639222005935320139381397354352697591385263697942368423779222169918998014112075138548254387879002631507818690503853962362310524641815389492016703056454693022710822049679619024180438554777152640488724921000918749550764760827973341234424813103073597511595325096317359597547475360089019158141150530451217612828119717983016617465339272020976581281482179408746509851638537537158907782914183773236820805334186248573009830174208552375766905349996885754290254867278900588267057175467706652999001965268773129916836914972367505111954603938564597149427656131416554974182560610994089781552083248952967464806172923238826364627993978175383659400246087410320325661665826907566023263981773543272123477598379777721833693278565421808456473376435373232893839178376381683453665500707740874014266484050622509598104149490311062687101151821701468662555057507815894750271998243804, 555696059771055038650973820535451550639619718201129449418117885874270582397844732988663989077429193861697070029911010272853574091442218887278627283268194679185801811129443946350002929470826524240053625233033713894740727200665033566584634919689003857470330577129787895909363104978413596568316711182691750615956435305726237858080556097073112303277313854284879589597930309542892319499051879074762641160963316173677386763449663508011354849248512919493115119594716269065911205738682352109996035389215167659841431662706314123461160018325799392371271265940212467551642559852666051223955277635367047326365607530254683423777708459005534828700984133293712438399274147701651815180576690674933396031130785046759916782898783040552426678540294732344496539524185233550158559229691728349407600730337494260694526731471190339148374433516228794711441923997964755364271064987312079809970430016066192889986195812075123476613338766285344649028860893626504305575791035437329135462320553478327221117391413039781190074071540634051421883313646794447133159678128527117524758760352300287629011877811665075227838164967618111103567913393629487898708122057938514058088978533588667918668620433539691911740158982435132358170949774231839576990481821377724452792749326838274074484697054742572682981610538968514202975552543744626730357967989777557813634343624529472042330798264659523587164899931713672349716830876373083522337432615609358491805698116873008013908952032054332700167863600028898021875707191257254892969072856200607049457200426784336165912906447144429862271768678893687614768493946982262517464103142590421725002902786400627931103476129131190320026328703661277387340284130736673755015491837492351388158563167709054335613283861246386063193327277708346155597379199972636814775075914358110151691357843817339619095850936276808760985960182877389894154921135758479240953143219901186847310910124838031896692248677072563896359331, 529195127319343675920089101384548369427243395741507141733370000174200546698744147189178546633767385933923845802989727446392583072949188953054879388306839582643402359171911896965185810944789436393215151987403523512778258920526117766583931132400810145960444275907962106542843680377878172900198722522379650586235484658969063679234586415065505926960226942361184457604231366353154249429748861043054741436828943766693354718205041994182774676967316845974566666926967665668754075828652922092139023933301257565149548213907087300750171925194429980945191892993185710461931420912719592794926810661451764217166787519906063533978311699229513826597268924621912145287630292678725676865811942284269982977505419137139682143742813164615450843851602106253024399401468440522015034335727168826181995527237769523759542371054581269009564070964537310411078949637890071123124590915795874011531891759380585655576177375269712779572746272434536623050799609080382431047806888686422792394266942480082977352743380837075250948601807556262070273240243404726622517938212943758901668286603953588708370371001140954916448893186497440758389454393956400315600342397097165778313902019260338383160918225043440247455486282812359176034576708065378880575378320417677753015655681366065153361240254052383897933975244748636843476822750719530928384285986315253977962438657700752636184124076333048417124547059529736289142154189960135025163877236174547481552707357420269136021936614586198877442259101938046601460345535765386089026422564626838855071756925610474049795974886589675947356336741351754648068796882760770895081355985788929916095909712960579428683865302797193949804618832994020900157837737471703248193776796970331635990503737540599399092526373723896584281224607627689784475704282585896430531763788602953999838837186818770743263693905366351678272975427750954199756626233852395355253052567377850119181563423194484506976201851825991347588194, 297681368086770200841708220047796983946117155940935419656095402171261134755418135031405641400645493007157499608450367671490541877514775127518973474990691895242985909768827633605972393590847941025931847232250989189417978988651908834052124097396515716594073035208781765602019827346776667673061277899037142494251863616788346831394580700198644652817226967340201517273135845952837431176956607148642449098176490397904268920385667831494737294891478108075255023073522549774048281991032510098223790102419107834314210085034650659970736735540649318085762638612312593331268035646588671167392050265450451969389928616424683508392319361182046514587168164577735620729770878124824067111980022183766417437546520767383977601385374244146007168257578877508531694402084040057125368322260310669301270446058704124027852316384923108542193147652941602681435435042358837831035347182286070387192486325321772024927293934449203861005532471277545310841045692195030536140611131489105993385367027064691863076006968004454820695668257169605806308574736707770048494280074196299613607764834165938641189660111729023621855633241125284994324762401149567442262819753933292582788479400934149842684960090580272390563306847575987949413163535430913995622015821573917273550221692981196639872196038808017031158885483798037625988710732250910243574599521209772929584839922664836413541823630843345989337136382179719541946434511206544502554729604775388180998767533125926159756166543360682816510906094430009647812834561346733046980204957042970686706159084317084366502828658990024867325274627974458461576991402819629501713578656679667929305523839402992539525261681071976542735630275562688228692827659648187409753684180199030712917496572482932862034188941569326997477195642643862710077091578973280562100772603602809699416904124612945330646018846584107346250535409293783858854235517814356673159489114825655288588601144080958840133956356742209992473657], 'aut_phi_ratio': 48.25, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 193, 1, 2], [4, 1, 2, 1], [4, 193, 1, 4], [4, 193, 2, 3], [8, 193, 2, 8], [16, 193, 4, 8], [193, 16, 12, 1], [386, 16, 12, 1], [772, 16, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{386}.C_{96}.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', '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, 1, 1], [2, 193, 2], [4, 1, 2], [4, 193, 10], [8, 193, 16], [16, 193, 32], [193, 16, 12], [386, 16, 12], [772, 16, 24]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '3088.54', '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', '2.1', '2.1', '193.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1674, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3088.54', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 193, 1, 2], [4, 1, 2, 1], [4, 193, 2, 5], [8, 193, 4, 4], [16, 193, 8, 4], [193, 16, 12, 1], [386, 16, 12, 1], [772, 16, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 192, 'exponent': 3088, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 193], 'factors_of_order': [2, 193], 'faithful_reps': [[16, 0, 24]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '6176.246', 'hash': 1674, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 3088, 'inner_gen_orders': [16, 193], 'inner_gens': [[1, 12304], [65, 16]], 'inner_hash': 54, 'inner_nilpotent': False, 'inner_order': 3088, 'inner_split': False, 'inner_tex': 'C_{193}:C_{16}', 'inner_used': [1, 2], 'irrC_degree': 16, 'irrQ_degree': 384, 'irrQ_dim': 384, 'irrR_degree': 32, 'irrep_stats': [[1, 64], [16, 48]], 'label': '12352.1674', 'linC_count': 24, 'linC_degree': 16, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 194, 'linQ_degree_count': 8, 'linQ_dim': 194, 'linQ_dim_count': 8, 'linR_count': 384, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C772:C16', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 31, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 112, 'number_divisions': 21, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 44, 'number_subgroup_classes': 58, 'number_subgroups': 5050, 'old_label': None, 'order': 12352, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 387], [4, 1932], [8, 3088], [16, 6176], [193, 192], [386, 192], [772, 384]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 4, 12], 'outer_gen_pows': [0, 5, 12], 'outer_gens': [[1, 12336], [1, 7312], [3089, 11408]], 'outer_group': '96.165', 'outer_hash': 165, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [767760, 2160, 4032004], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_{12}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 197, 'pgroup': 0, 'primary_abelian_invariants': [4, 16], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 4], [8, 4], [192, 2], [384, 1]], 'representations': {'PC': {'code': '72407528335999452377277843788303020815094084830022278204159326426892925831612632776520499391', 'gens': [1, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -193, 14, 36, 58, 430644, 2531, 11358, 26975, 102, 514757, 6060, 27235, 64706, 124, 595846, 14125, 63524, 75291]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [7189251, 1365920873, 582313729]}, 'Perm': {'d': 197, 'gens': [5290273345984864897683944123082956631220066364900253396726259059766269559200235413715177147698004284025594961028822560923834716647335335646782683178127927701850149621311947644736352530825831559089639200063438532790427221928341897546834098797438127745470625459344068218749828555758253110619759250004403483806498954818604823553161926087080150002071074875160801725560, 9, 7896146472218856502393881215272180448607715302217823038480840402592075442888782334598101578804469491722017008442403110034580758066612966193062153191003165794662509714205876275508506316534606456485998062120044681800379766614251220802300702017288411488349582440302601260771679149486482287636140697371705750316485130991060173970329479343368418131379968662762330588240, 16, 10461532913094680818289408021199360182337152438568301470075869865765861863574813452987192156277912935756095880952558744287790336020715603557128987402820322664978709376847114195610925734074225539086790955889736488207604978092986805271850954882804263039468151199343278442654273620917630499833139965503217978033223387903829743037492645791738257574229789666158530438440, 2592381132973149765779289246531185296419382784883920013901222755849351293628592008151691890954402788253039663753020082609107439334904412326282239603261948437302473854804655963712088645187779112731793528834357671948713737992400807244149705391268119111136677134958116230430763311877262318184849472983323284357135732667011753861924031209391791363940984026715737429240, 520972671180291378873378228799679326959717638111606330328292901165986473129410184711055071299345784828690369857414184422249346580994648235923978424589548271404017967409610673350600057073095348109955036142474517622285973437735974016368518172307132793835132086153054465008777445548393143686583542501166362214033044187300451811977217876794123160958270890813465818123440]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 16], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{772}:C_{16}', 'transitive_degree': 772, 'wreath_data': None, 'wreath_product': False}
