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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '960.4617', 'ambient_counter': 4617, 'ambient_order': 960, 'ambient_tex': 'C_2\\times C_{12}:C_{40}', 'central': False, 'central_factor': False, 'centralizer_order': 480, 'characteristic': True, 'core_order': 48, 'counter': 95, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '960.4617.20.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '20.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '20.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{20}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '48.44', 'subgroup_hash': 44, 'subgroup_order': 48, 'subgroup_tex': 'C_2^2\\times C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.4617', 'aut_centralizer_order': 384, 'aut_label': '20.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '64.202', 'aut_weyl_index': 384, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '10.a1.a1'], 'contains': ['40.c1.a1', '40.c1.b1', '40.c1.c1', '40.c1.d1', '40.d1.a1', '40.g1.a1', '40.g1.b1', '60.b1.a1'], 'core': '20.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4413, 6708, 3184, 8413, 4303, 2044, 2852, 7208], 'generators': [1, 480, 8, 240, 640], 'label': '960.4617.20.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '20.b1.a1', 'normal_contained_in': ['4.b1.a1', '10.a1.a1'], 'normal_contains': ['40.c1.d1', '40.c1.c1', '40.c1.b1', '40.c1.a1', '40.d1.a1', '40.g1.a1', '40.g1.b1', '60.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '20.b1.a1', 'projective_image': '120.24', 'quotient_action_image': '2.1', 'quotient_action_kernel': '10.2', 'quotient_action_kernel_order': 10, 'quotient_fusion': None, 'short_label': '20.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '48.44', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [3, 2, 4], [1, 2, 20], [2, 3, 4], [1, 2, 6], [1, 2, 7], [1, 26, 29], [25, 2, 30], [1, 2, 28]], 'aut_group': '384.20089', 'aut_hash': 20089, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 10, 'aut_perms': [85945, 848929, 166680, 1961646, 1270440, 1538646, 2682390, 848928], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [3, 1, 2, 1], [4, 1, 8, 1], [6, 1, 2, 1], [6, 1, 12, 1], [12, 1, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '384.20089', 'autcent_hash': 20089, 'autcent_nilpotent': False, 'autcent_order': 384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4:S_4', '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, 7], [3, 1, 2], [4, 1, 8], [6, 1, 14], [12, 1, 16]], 'center_label': '48.44', 'center_order': 48, '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', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 44, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 1], [4, 1, 2, 4], [6, 1, 2, 7], [12, 1, 4, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 91, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '24.15', 'hash': 44, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48]], 'label': '48.44', 'linC_count': 5824, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 96, 'linQ_dim': 5, 'linQ_dim_count': 96, 'linR_count': 96, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C12', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 48, 'number_divisions': 24, 'number_normal_subgroups': 54, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 54, 'number_subgroups': 54, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 8], [6, 14], [12, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[3, 2, 4], [1, 2, 20], [2, 3, 4], [1, 2, 6], [1, 2, 7], [1, 26, 29], [25, 2, 30], [1, 2, 28]], 'outer_group': '384.20089', 'outer_hash': 20089, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 10, 'outer_perms': [85945, 848929, 166680, 1961646, 1270440, 1538646, 2682390, 848928], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 4]], 'representations': {'PC': {'code': 26739073, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -3, 42, 58]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [705946296092, 141602720900, 706461794101]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [10319436418, 9789111396, 6526074264, 4548416880, 6762482606]}, 'Perm': {'d': 11, 'gens': [2400, 3628800, 40320, 4, 744]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 12], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{12}', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '160.190', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [12, 2, 4, 12, 4, 4, 12, 12], 'aut_gens': [[1, 2, 16], [489, 82, 784], [481, 482, 464], [1, 330, 376], [9, 327, 121], [9, 495, 121], [1, 243, 953], [1, 170, 697], [489, 563, 120]], 'aut_group': None, 'aut_hash': 8705921932039392806, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24576, 'aut_permdeg': 100, 'aut_perms': [37433194337509727086677096417602409332873590680868142621317684259648306524069009685322355492908159320903128370352976196130433034992560069994883516510890010498, 46762583017232808364387424932981613081551126871918113207671502269764024849830403718544965232440953513896532982216268175159381959658155278094363681478734931810, 41136683953199927572255973611498828143764695841224350793808848752935261067278233087303206286358500580661067295578422399999875974703527698781534491661408881120, 35511060721225348463154405970025248916142037872386014738481352644012934991968546052420194230896604341090549387966847766311949591172363159079442859390909324341, 7506480432156414137049233574229366832973563217582061419984606519342930438662648563552363041744405693208336549379946069782265152214799643838533904109244666146, 29963498303919535773514095027393596125249264250915131025430511278348597369674135380749490965746030478041516673396015073996059197845474914057269494644823376167, 12201292945509774881952449482200691179153952848418804059910576470623247527237386260639446819870072026244908620284952744992337735022799503482208052701441307160, 15908931885806252475083040165068161801863447366220132769502292267649926699145161331308709720976197405763094939263137786003242067870149274371074400638020993138], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 4, 1], [3, 2, 1, 1], [4, 1, 2, 2], [4, 1, 4, 1], [4, 2, 4, 2], [5, 1, 4, 1], [6, 2, 1, 3], [6, 2, 4, 1], [8, 6, 16, 1], [10, 1, 4, 3], [10, 1, 16, 1], [12, 2, 2, 2], [12, 2, 4, 1], [12, 2, 8, 2], [15, 2, 4, 1], [20, 1, 8, 2], [20, 1, 16, 1], [20, 2, 16, 2], [30, 2, 4, 3], [30, 2, 16, 1], [40, 6, 64, 1], [60, 2, 8, 2], [60, 2, 16, 1], [60, 2, 32, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^2\\times C_4\\times C_2^6.C_2^3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 7419489452941580063, 'autcent_nilpotent': True, 'autcent_order': 2048, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5.C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 2, 1], [4, 1, 8], [4, 2, 8], [5, 1, 4], [6, 2, 7], [8, 6, 16], [10, 1, 28], [12, 2, 24], [15, 2, 4], [20, 1, 32], [20, 2, 32], [30, 2, 28], [40, 6, 64], [60, 2, 96]], 'center_label': '80.45', 'center_order': 80, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4617, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1], ['96.11', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [4, 1, 2, 4], [4, 2, 1, 4], [4, 2, 2, 2], [5, 1, 4, 1], [6, 2, 1, 7], [8, 6, 4, 4], [10, 1, 4, 7], [12, 2, 2, 8], [12, 2, 4, 2], [15, 2, 4, 1], [20, 1, 8, 4], [20, 2, 4, 4], [20, 2, 8, 2], [30, 2, 4, 7], [40, 6, 16, 4], [60, 2, 8, 8], [60, 2, 16, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 10416, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '120.45', 'hash': 4617, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 6], 'inner_gens': [[1, 2, 16], [1, 2, 176], [1, 802, 16]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': False, 'inner_tex': 'D_6', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 160], [2, 200]], 'label': '960.4617', 'linC_count': 380928, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 64, 'linQ_dim': 10, 'linQ_dim_count': 32, 'linR_count': 512, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*C12:C40', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 42, 'number_characteristic_subgroups': 58, 'number_conjugacy_classes': 360, 'number_divisions': 80, 'number_normal_subgroups': 206, 'number_subgroup_autclasses': 140, 'number_subgroup_classes': 276, 'number_subgroups': 432, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 24], [5, 4], [6, 14], [8, 96], [10, 28], [12, 48], [15, 8], [20, 96], [30, 56], [40, 384], [60, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4, 4, 4, 4, 4, 4, 4], 'outer_gen_pows': [0, 640, 0, 0, 0, 0, 0], 'outer_gens': [[1, 406, 473], [9, 87, 593], [9, 10, 368], [9, 7, 473], [1, 402, 953], [1, 734, 849], [489, 735, 696]], 'outer_group': None, 'outer_hash': 3853072307116489657, 'outer_nilpotent': True, 'outer_order': 2048, 'outer_permdeg': 256, 'outer_perms': [831413025724334233688267775293694053388641183810004038463381998612189566868136473715572798592589292715778122081871160640338831039477533716480953719068937788180039651181876821737493044921963074857414097983856291235790360941217002912279994332901233299182443481224669067588040454181276640194796977035686902522620928255240661851548823245447819062714442117202478476384688470615831833525132953378620684710748522873902415898097621320954527873690432083063715790973142946633346809599656313159861872357406376280646539, 507514505523228419563697823017452735682928700897615277383061600683852161719262296495682899818630539325994138095718178229366384883339994476266533085270856122008036368384006050730050937266063694993299109737256388777461770431963774110309009862140422418620449123952036524100982717673535185247631103007369049843704032409760145904740811016612548339869542772801057729128556189435870585397144427500258988862697160309045456026945185258625790987770033108213128325272809209792949744701658318609967252714289965779673103, 652819741752258236724400176084121056525537989209738542649637815048769817960608616681620226529855340097868805716526346275113226315150005812500464999925993927569799568775560042700546521257857259057416785182377988217325913829847699598220210427596811607645123752678050383817811351660897979735289246174014866804096452066298521598351939228498124291260835650680838723604071784175610561835922933707084674676465347156919036846834497688950120487869259786583895224609830688527703024887359829120173969395168811899478080, 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756228366202849548449747208691501787300461547674857139130869183679338367588559666940452834352266507198828526784706941343916550466746230562018462230018382087933877382775903164255629289405608353136832165313158756389996894814839329584266823584931278097658082899407059845660028435685504734588841993964370678696867492920211295117124426077086038917736546507596954491245317629397129286345512296523533800293466493996757224058529287277024832205087410404293847709649713930902177345502485858135039567914458466932846342, 696490368193091158767698570379604136810381571784633358986353695213148534537890533761296982308568285279660018336821001284915904675369357588077023904843369214779979604634106415148251148569936996542820938803087033200745392216196354327762932662004328405751332895524288009102811524798739366563069511539494865179744812208062530115504965998295377425342490072366376100601882686490530816522331130650646817503112509245209505864474019796154062836644020632803764230131353429875030933379290393959828926182001980259746423, 376396182985388191250641986030687741748410952639746561846105053639383236185984623935014298414847453236885521591398444553933467027681300514930694583046867020703230360831194833811925908641324650878770373798282724648681254919973605897316041320738202933579548188860612555955724298526726619841506434845878225300533869735666256249934252146892296928010257235940199089236439715470075867470492131869519504494908978423635650541657456596150794345270611492577435630632486417169328902090482344493626249960823191748952637], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^7.C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 8, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 22], [8, 18], [16, 14], [32, 2]], 'representations': {'PC': {'code': 13556308985205377147893857295520890887, 'gens': [1, 2, 5], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -5, 41, 66, 3532, 116, 8461, 141, 19726, 222]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [7187425, 18177151, 6612431, 289675, 12470437, 3765583, 12362371, 482087]}, 'Perm': {'d': 22, 'gens': [2432904711270572160, 6402374784580320, 22318721399520, 33, 1612044000, 44460928512000, 1078852320, 53523844179886080000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 40], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{12}:C_{40}', 'transitive_degree': 960, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '20.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1], [11], [17]], 'aut_group': '8.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 6, 'aut_perms': [120, 9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', '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], [5, 1, 4], [10, 1, 4], [20, 1, 8]], 'center_label': '20.2', 'center_order': 20, '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', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 8]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '10.2', 'hash': 2, 'hyperelementary': 10, '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': 8, 'irrQ_dim': 8, 'irrR_degree': 2, 'irrep_stats': [[1, 20]], 'label': '20.2', 'linC_count': 8, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 2, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 4, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C20', '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': 20, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 4], [10, 4], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[11], [17]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [4, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1]], 'representations': {'PC': {'code': 51395, 'gens': [1], 'pres': [3, -2, -2, -5, 6, 16]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58438750843733785]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504, 252]}, 'Perm': {'d': 9, 'gens': [131040, 96, 41040]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [20], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
