-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '27000.b', 'ambient_counter': 2, 'ambient_order': 27000, 'ambient_tex': 'C_{15}^2:(C_6\\times F_5)', 'central': False, 'central_factor': False, 'centralizer_order': 5, 'characteristic': True, 'core_order': 1350, 'counter': 46, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '27000.b.20.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '20.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '20.3', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'F_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '1350.55', 'subgroup_hash': 55, 'subgroup_order': 1350, 'subgroup_tex': 'C_{15}^2:C_6', 'supersolvable': False, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '27000.b', 'aut_centralizer_order': None, 'aut_label': '20.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '5400.a1', 'complements': ['1350.e1', '1350.d1', '1350.h1', '1350.g1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1', '10.a1'], 'contains': ['40.a1', '60.a1', '60.b1', '500.a1'], 'core': '20.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [707, 717, 1325, 532], 'generators': [60, 5760, 19200, 18000, 4, 6504], 'label': '27000.b.20.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '20.a1', 'normal_contained_in': ['4.a1'], 'normal_contains': ['40.a1', '60.a1'], 'normalizer': '1.a1', 'old_label': '20.a1', 'projective_image': '27000.b', 'quotient_action_image': '20.3', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '20.a1', 'subgroup_fusion': None, 'weyl_group': '5400.r'}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [12, 24, 24, 10], 'aut_gens': [[1, 6, 90], [239, 750, 918], [731, 186, 1296], [569, 1254, 144], [803, 390, 954]], 'aut_group': None, 'aut_hash': 3552034962649702167, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 64800, 'aut_permdeg': 159, 'aut_perms': [2779396467288410159565464921002005117768873592713251285510204965542090334080928819131907235659568166080983478443796582161924817351887656693835421936305465275443435925832877330194565379956107246414273586147870338662309304515808712378752268439643188412334280336624295582944358248453596, 476214588898476795702816111418800638729267952184618872725675376430739011492696858779792682455528777791039443035408534850879964249562443426896877548532625234612708646709579182803040099344158310860009710064051171320675915531281783190948511073460653165159903690057210421319286340960859, 2625558598344821134053418280025668442537218373584005705507356554534884179008740504275693064254699020024166292245050487271482542648527958600438790521615548924663356827622792987877876779549432423603793787466983294416313035083424470146357444459080477617119396059795615491657817034417922, 1717816903755929108866128065759566144615417698011643478948453464519787240645676833803447171422936217456327331864270358350375144093259256878248155576060810014580604220991495326676797827463987375883149399338124307211471331998246132721241061284230933952817222949548187636608628775354186], 'aut_phi_ratio': 180.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 75, 2, 1], [3, 150, 2, 1], [5, 3, 8, 1], [6, 225, 2, 1], [10, 27, 8, 1], [15, 6, 8, 1], [15, 6, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{15}^2.(C_{24}\\times S_3).C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': None, 'autcentquo_hash': 3552034962649702167, 'autcentquo_nilpotent': False, 'autcentquo_order': 64800, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{15}^2.(C_{24}\\times S_3).C_2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 1], [3, 6, 1], [3, 75, 2], [3, 150, 2], [5, 3, 8], [6, 225, 2], [10, 27, 8], [15, 6, 32]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1350.55', 'commutator_count': 1, 'commutator_label': '225.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 55, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 75, 2, 1], [3, 150, 2, 1], [5, 3, 4, 2], [6, 225, 2, 1], [10, 27, 4, 2], [15, 6, 4, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 30, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 24]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '450.23', 'hash': 55, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [6, 15, 15], 'inner_gens': [[1, 1020, 1314], [427, 6, 90], [217, 6, 90]], 'inner_hash': 55, 'inner_nilpotent': False, 'inner_order': 1350, 'inner_split': True, 'inner_tex': 'C_{15}^2:C_6', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 6], [2, 3], [3, 16], [6, 33]], 'label': '1350.55', 'linC_count': 24, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 4, 'linQ_dim': 18, 'linQ_dim_count': 4, 'linR_count': 20, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C15^2:C6', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 11, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 58, 'number_divisions': 19, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 56, 'number_subgroups': 900, 'old_label': None, 'order': 1350, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 9], [3, 458], [5, 24], [6, 450], [10, 216], [15, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 24], 'outer_gen_pows': [0, 0], 'outer_gens': [[5, 1020, 936], [1, 816, 666]], 'outer_group': '48.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [363001, 4446963], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{24}:C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [6, 1], [12, 4], [24, 8]], 'representations': {'PC': {'code': 1642896269639909458121783892599715307278711, 'gens': [1, 3, 5], 'pres': [6, 2, 3, 3, 5, 3, 5, 12, 18362, 11564, 68, 6483, 13833, 39424, 14050, 118, 41477, 335]}, 'Perm': {'d': 24, 'gens': [12289, 28202720198792099401913, 51932, 98499, 55290747866795289273600, 4559297579257342521600]}}, 'schur_multiplier': [15], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^2:C_6', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [12, 12, 15, 24, 4], 'aut_gens': [[1, 12, 120, 1800], [10249, 3204, 4224, 16728], [23185, 7332, 26664, 14232], [10801, 18612, 120, 1800], [1225, 12228, 5064, 2328], [14549, 1236, 6720, 7320]], 'aut_group': None, 'aut_hash': 1525260094185992089, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1296000, 'aut_permdeg': 284, 'aut_perms': [38732379701156988517253430827897135047041564754096829949613818993619614235583464715381602228542487151608891486195576387482308889695651830723922997990000202419687245377365102127784679536977354342843504662184027460086775723004862765550900139382266781669874650502988867590014473202316537652660003832014307424529288487672510823453185886432724171032747662616681895236331128373024183260829727155180952556242889019012335959771992480419982724745000885137760518359536776021811371743067307000039426815954046689437416983647884350366365704388993961899337093914360831994118552802220449008, 52406346103905871530992818771475320875115206086934635447922106811442990415499416061501656126177278074930086539926159968302062925285814974479770363036001643005067957327028546639054747051470195287004788181135455025231504132026479052927532800197788498904663161894066936861366245491966843830083631527181088616822767318745312551343684691978855721472129314034323425501402396680578075655806164177975470977174259477230539738420931542279740598164546624181331914974852710670543042089511551990757677611883854753514233790769185497457721574183476511217750980749570142058267952918293285557, 97386090563404931797387742471368054191490165187291762069564329091449242070866209529348790925502402052700149290180319914964878350847577500684896627024890833196450230974130453515915423600630574109198253965401584441565035756173192844167775659932055574373585931521814794780627493983137039274274507487702493865314651591480605190431892256275353785851645103476147243697248680812277398399564300480114055628790156573986272362324573281935104608745802983991566860765006648229557697743170442630012892097230869577767540887711126688831297574858701505384658138342671163011729640630557229073, 30042392820482583797575311516997456893856576969652934873733041045795455037765343834382013335133776050602729092781142946868873570647611906768741824730761393624301098899945430453405267188120612694439439611639203160322869984464489375128603680557277923203953712532422083522226222414150753746879011583050553501666129219691170687608709086848584468581247623768541764798533993423729706860331473573027308316728134322310927946398804263207541406337582715792517749838597699113927492392471641007012619489320828618989188371646665311602749405564583042095139997702366773911818877946261174638, 93792734908860622515349497299208821181864921029335241747715937114312349486719913064827678410251921328606321259400149033786024999687768866293071122518949643691458690218595140042134104479454174517716997616843077007685967689355580012426472720635911099844589766149883411600961729532896372718407992323590240608552171981258653407013998642343080066209350516369239153166408391775714283994189894764097335346337619426041681690645962338957449220068182936989240785650709598154391920075444130736920576172018159142259399772398563502327356734908534961755983311012385886434809549769274117813], 'aut_phi_ratio': 180.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 125, 1, 1], [2, 1125, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 75, 2, 1], [3, 150, 2, 1], [4, 125, 1, 2], [4, 1125, 1, 2], [5, 4, 1, 1], [5, 12, 2, 1], [5, 12, 8, 1], [6, 225, 2, 1], [6, 250, 1, 1], [6, 375, 2, 1], [6, 750, 1, 1], [6, 750, 2, 1], [6, 1125, 2, 1], [10, 36, 1, 1], [10, 108, 2, 1], [10, 108, 8, 1], [12, 250, 1, 2], [12, 375, 2, 2], [12, 750, 1, 2], [12, 750, 2, 2], [12, 1125, 2, 2], [15, 8, 1, 1], [15, 24, 1, 1], [15, 24, 2, 1], [15, 24, 6, 1], [15, 24, 8, 1], [15, 24, 24, 1], [15, 300, 2, 1], [15, 600, 2, 1], [30, 900, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_5\\times C_{15}).C_{15}.C_{12}^2.C_2^3', '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': 120, 'autcentquo_group': None, 'autcentquo_hash': 1525260094185992089, 'autcentquo_nilpotent': False, 'autcentquo_order': 1296000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(C_5\\times C_{15}^2).(C_6\\times C_{12}).C_2^4', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 125, 1], [2, 1125, 1], [3, 2, 1], [3, 6, 1], [3, 75, 2], [3, 150, 2], [4, 125, 2], [4, 1125, 2], [5, 4, 1], [5, 12, 10], [6, 225, 2], [6, 250, 1], [6, 375, 2], [6, 750, 3], [6, 1125, 2], [10, 36, 1], [10, 108, 10], [12, 250, 2], [12, 375, 4], [12, 750, 6], [12, 1125, 4], [15, 8, 1], [15, 24, 41], [15, 300, 2], [15, 600, 2], [30, 900, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '27000.b', 'commutator_count': 1, 'commutator_label': '1125.14', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 125, 1, 1], [2, 1125, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 75, 2, 1], [3, 150, 2, 1], [4, 125, 2, 1], [4, 1125, 2, 1], [5, 4, 1, 1], [5, 12, 1, 10], [6, 225, 2, 1], [6, 250, 1, 1], [6, 375, 2, 1], [6, 750, 1, 1], [6, 750, 2, 1], [6, 1125, 2, 1], [10, 36, 1, 1], [10, 108, 1, 10], [12, 250, 2, 1], [12, 375, 4, 1], [12, 750, 2, 1], [12, 750, 4, 1], [12, 1125, 4, 1], [15, 8, 1, 1], [15, 24, 1, 41], [15, 300, 2, 1], [15, 600, 2, 1], [30, 900, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 96, 'exponent': 60, 'exponents_of_order': [3, 3, 3], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[24, 1, 24]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '9000.h', 'hash': 5150692036991545229, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [12, 10, 15, 15], 'inner_gens': [[1, 2388, 2256, 14688], [26545, 12, 1320, 19800], [26785, 612, 120, 1800], [16033, 9012, 120, 1800]], 'inner_hash': 5150692036991545229, 'inner_nilpotent': False, 'inner_order': 27000, 'inner_split': True, 'inner_tex': 'C_{15}^2:(C_6\\times F_5)', 'inner_used': [1, 2], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 24], [2, 12], [4, 6], [6, 4], [8, 3], [12, 20], [24, 41]], 'label': '27000.b', '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': False, 'monomial': True, 'name': 'C15^2:(C6*F5)', '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, 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': 43, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 110, 'number_divisions': 88, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 376, 'number_subgroup_classes': 1208, 'number_subgroups': 120912, 'old_label': None, 'order': 27000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1259], [3, 458], [4, 2500], [5, 124], [6, 5950], [10, 1116], [12, 11000], [15, 2792], [30, 1800]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 24], 'outer_gen_pows': [0, 0], 'outer_gens': [[5, 21996, 1320, 13152], [1, 16980, 16344, 2232]], 'outer_group': '48.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [1134121, 16899843], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{24}:C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 7], [6, 2], [8, 4], [12, 21], [16, 1], [24, 41]], 'representations': {'PC': {'code': '831613064970188750733038989289677827215555169542525661843810451465981051565728994947897529712818355838062963645317728604787575', 'gens': [1, 4, 6, 8], 'pres': [9, 2, 2, 3, 2, 5, 3, 5, 3, 5, 18, 46, 85971, 33924, 127785, 102, 992524, 487093, 191182, 121829, 706334, 180815, 5972, 212, 411270, 714435, 177306, 1057543, 57904, 388393, 118834, 286, 1372472, 44729, 139508]}, 'Perm': {'d': 24, 'gens': [55293320953936707760513, 28263632005523734496725]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^2:(C_6\\times F_5)', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 5], 'aut_gens': [[1, 4], [1, 12], [5, 4]], 'aut_group': '20.3', 'aut_hash': 3, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20, 'aut_permdeg': 5, 'aut_perms': [18, 34], 'aut_phi_ratio': 2.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 5, 1, 2], [5, 4, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_5', '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': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [4, 5, 2], [5, 4, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '20.3', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 5, 2, 1], [5, 4, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '20.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 5], 'inner_gens': [[1, 12], [13, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'F_5', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [4, 1]], 'label': '20.3', 'linC_count': 1, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'F5', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 14, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 5], [4, 10], [5, 4]], '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': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1]], 'representations': {'PC': {'code': 2551363, 'gens': [1, 3], 'pres': [3, -2, -2, -5, 6, 110, 77]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [23561683, 28172809]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33, 10], 'family': 'AGL'}, {'d': 1, 'q': 5, 'gens': [33, 10], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 127]}, 'Perm': {'d': 5, 'gens': [10, 23, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'F_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
