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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '148224.c', 'ambient_counter': 3, 'ambient_order': 148224, 'ambient_tex': 'C_4\\times F_{193}', 'central': False, 'central_factor': True, 'centralizer_order': 4, 'characteristic': False, 'core_order': 37056, 'counter': 10, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '148224.c.4.d1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.d1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '37056.a', 'subgroup_hash': 1432757702296672986, 'subgroup_order': 37056, 'subgroup_tex': 'F_{193}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '148224.c', 'aut_centralizer_order': None, 'aut_label': '4.d1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '37056.a1.a1', 'complements': ['37056.a1.a1', '37056.c1.a1', '37056.d1.a1', '37056.d1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['8.e1.a1', '12.d1.b1', '772.d1.b1'], 'core': '4.d1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [9194, 8888, 6802, 6762, 9752, 9783, 7217, 1987], 'generators': [39960, 768, 138336, 102960, 128268, 387, 64, 40710], 'label': '148224.c.4.d1.b1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.d1.b1', 'normal_contained_in': ['2.b1.a1'], 'normal_contains': ['8.e1.a1', '12.d1.b1'], 'normalizer': '1.a1.a1', 'old_label': '4.d1.b1', 'projective_image': '148224.c', 'quotient_action_image': '1.1', 'quotient_action_kernel': '4.1', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '4.d1.b1', 'subgroup_fusion': None, 'weyl_group': '37056.a'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '192.2', '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, 192], [1, 7296], [2305, 192]], 'aut_group': '37056.a', 'aut_hash': 1432757702296672986, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 37056, 'aut_permdeg': 386, 'aut_perms': [1015897978262432392259055287089892981816792561978363223255333473475737856427285178449728078832142098810257134463116375065859472669066769340462826473923700815816715297528878323552999848973820897553920692175306802551549615553935426493665741152855867759193797974943152777879123825483610458818655871960149344676974904001412987096454266886527422601514403216482556085712676425550775173521768169803986095320747984982324618401376108442689862182154832751432733363788817004754919138001108277216924113238932625935930578015329534455333672367823351711880176143254939555543871144198961323807859841384368081681520371006834220735429730934025874388385432045162196351235394259601946456131293921558655701599547522278769237430083677403190278339414972458324518920176937993912971401333958477856889504283727431524308853053880657412913081727344836977245813, 16768477725994331310980273365242639776193574816821378450319305405109541234703080131912948273361190499032900891057925862686825902663187341373176250143145920509693319181735861360192000158599885394975719688140979264916958272922099800851015704838925006074549781473230662032304143072939904748254933999342343963853114702557930041511480361899931009648248000083229507040500328311999966549092138936884991481320703290461932012421505361215740868613024914584620982499431217446978052749012756042802404310532426824718196334586209599568268342169869593920091946702741022378448155227775822663137333584532590908935300963894025050350084855136313004320186087226430499654261923729419354324885966616838970896677588920588087451031160209444336069724487797654548898045644243685856350161949414131919676395199632269214422084528131956500195865489662220468721433], 'aut_phi_ratio': 3.015625, '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], [32, 193, 1, 16], [48, 193, 1, 16], [64, 193, 1, 32], [96, 193, 1, 32], [192, 193, 1, 64], [193, 192, 1, 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], [32, 193, 16], [48, 193, 16], [64, 193, 32], [96, 193, 32], [192, 193, 64], [193, 192, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '37056.a', 'commutator_count': 1, 'commutator_label': '193.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '193.1'], 'composition_length': 8, '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], [32, 193, 16, 1], [48, 193, 16, 1], [64, 193, 32, 1], [96, 193, 32, 1], [192, 193, 64, 1], [193, 192, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24576, 'exponent': 37056, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 193], 'factors_of_order': [2, 3, 193], 'faithful_reps': [[192, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '37056.a', 'hash': 1432757702296672986, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 37056, 'inner_gen_orders': [192, 193], 'inner_gens': [[1, 7296], [29953, 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': 192, 'irrQ_dim': 192, 'irrR_degree': 192, 'irrep_stats': [[1, 192], [192, 1]], 'label': '37056.a', 'linC_count': 1, 'linC_degree': 192, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 192, 'linQ_degree_count': 1, 'linQ_dim': 192, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 192, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'F193', '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': 193, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 193, 'number_divisions': 15, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 28, 'number_subgroups': 2524, 'old_label': None, 'order': 37056, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 193], [3, 386], [4, 386], [6, 386], [8, 772], [12, 772], [16, 1544], [24, 1544], [32, 3088], [48, 3088], [64, 6176], [96, 6176], [192, 12352], [193, 192]], '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': 193, 'pgroup': 0, 'primary_abelian_invariants': [64, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 2], [8, 2], [16, 2], [32, 2], [64, 1], [192, 1]], 'representations': {'PC': {'code': '45515334396045693417352492702089104139933519998455127445076196666909319532055151772031', 'gens': [1, 8], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -193, 16, 41, 66, 91, 116, 141, 466951, 571407, 482327, 211999, 99879, 41903, 20791]}, 'Lie': [{'d': 1, 'q': 193, 'family': 'AGL'}], 'GLFp': {'d': 2, 'p': 193, 'gens': [1222139866, 7189251]}, 'Perm': {'d': 193, 'gens': [3736912403715067823373930137238782926940486211704502117646706774297507816909184480220827687385088022109878516812058348746439524608309512279021534356236286352563830963703243089734601131223999278981612140241588394397731041228552900348797553170702951233696315378294945193488148967793869908071010690095770856334416356968327716174690930322639391880050203758277, 360543977123933955894873607183984202701225184627736786742378976763316385113450809182742747783566636004265785086266179301763773247492174059616538819301043441731929110106194375521425437253761115869162642447344255394411380716857064995071679049023774827770143547077105222236630337046250074446772508791881697130914185821465492952000214588675897225165107542491565]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [192], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'F_{193}', 'transitive_degree': 193, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '768.497', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 37056, 'aut_gen_orders': [96, 192, 96, 24], 'aut_gens': [[1, 192], [33793, 26688], [71233, 24384], [84289, 92736], [142081, 113472]], 'aut_group': None, 'aut_hash': 2013647253270866901, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 296448, 'aut_permdeg': 965, 'aut_perms': [228286409572473312765910102677468167363698581525830012964612443572959946963616331122121005394657651513276552529195904084632239595986701968432975049456707244885659582408911704234207063847510849341767875446100801298229528319245414477107670594478094914180726576053699179832946581997114206131715501018941492694538598561580881153781384302538980182012541114087185789329776531333843885988021108009256600274186653863873767623903676955425430648141880622168861700774811678511685373378066140191502233131189087020364602051318081593980383102290724337179487092042248415855801143228096033986959380789295905144272779591365038300110375754885563458992100948404909020083055302633629296771213137189565805712161991905195150295205901718737811824846412658376616251560628044301604890734988674831572584749637738054611783345736989507322450803172204527496543227746190302621301935223932175209896088933635283539267257485233213010282956976772911264717384690454916762079675944999906240955366201440942485607928498462319128461622048896658472630926024689151841875163337566649625099893595494222110954670965623852346313357224612359644193642363684216971936737624120264461938987965922622604559119869193074024157847945985504812760890072899045289592570103587085125229988283791349373466808678257665046032644181238203566828350756121750349642403320609284225310750789228993911241507011135338572610778579416881956087237358318506701192011497792248182923765985210167579089509735497604310165524767097317194698208744643714712465486908201808392660052735954836628779486203172614543846670781025633720294170146428885561068315657967516082502093436201000583197787575281114002605834688220647751078074776662297135734440087064281924272749727157897556884938096239225023755434252293314294251678330096296458888828533743715208642806612149901710096112478767477929030831691924601324254254410565513678491839655691066350694919261073154776373114044330177084475260477938355393034806673605086846811539409454722023078512426461726818129300561404051703658924865566169846989864846741727306522736339686060604887887232642467931538069180516965716038345744266005385710643601995762806993447861442109486902037541507937175130094411101309813430043700142853573726588802698452448624775007930866793652290656983314275859026286893247816852725236948465618799268143869566002522200473986073429821703777822898718037653566311988007348230594366358168185301838420146598294845100730861243644739607340962102971497268508595497291435459191824101429211139189216, 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'aut_phi_ratio': 6.03125, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 193, 1, 2], [3, 193, 1, 2], [4, 1, 2, 1], [4, 193, 1, 4], [4, 193, 2, 3], [6, 193, 1, 6], [8, 193, 1, 8], [8, 193, 2, 4], [12, 193, 1, 8], [12, 193, 2, 8], [16, 193, 1, 16], [16, 193, 2, 8], [24, 193, 1, 16], [24, 193, 2, 8], [32, 193, 2, 32], [48, 193, 1, 32], [48, 193, 2, 16], [64, 193, 4, 32], [96, 193, 2, 64], [192, 193, 4, 64], [193, 192, 1, 1], [386, 192, 1, 1], [772, 192, 2, 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], [3, 193, 2], [4, 1, 2], [4, 193, 10], [6, 193, 6], [8, 193, 16], [12, 193, 24], [16, 193, 32], [24, 193, 32], [32, 193, 64], [48, 193, 64], [64, 193, 128], [96, 193, 128], [192, 193, 256], [193, 192, 1], [386, 192, 1], [772, 192, 2]], 'center_label': '4.1', 'center_order': 4, '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', '2.1', '2.1', '3.1', '193.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['37056.a', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 193, 1, 2], [3, 193, 2, 1], [4, 1, 2, 1], [4, 193, 2, 5], [6, 193, 2, 3], [8, 193, 4, 4], [12, 193, 4, 6], [16, 193, 8, 4], [24, 193, 8, 4], [32, 193, 16, 4], [48, 193, 16, 4], [64, 193, 32, 4], [96, 193, 32, 4], [192, 193, 64, 4], [193, 192, 1, 1], [386, 192, 1, 1], [772, 192, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24576, 'exponent': 37056, 'exponents_of_order': [8, 1, 1], 'factors_of_aut_order': [2, 3, 193], 'factors_of_order': [2, 3, 193], 'faithful_reps': [[192, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '74112.a', 'hash': 6270743351508985570, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 37056, 'inner_gen_orders': [192, 193], 'inner_gens': [[1, 59328], [89089, 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, 768], [192, 4]], 'label': '148224.c', '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': 'C4*F193', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 339, 'number_characteristic_subgroups': 57, 'number_conjugacy_classes': 772, 'number_divisions': 55, 'number_normal_subgroups': 89, 'number_subgroup_autclasses': 136, 'number_subgroup_classes': 172, 'number_subgroups': 16108, 'old_label': None, 'order': 148224, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 387], [3, 386], [4, 1932], [6, 1158], [8, 3088], [12, 4632], [16, 6176], [24, 6176], [32, 12352], [48, 12352], [64, 24704], [96, 24704], [192, 49408], [193, 192], [386, 192], [772, 384]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[74113, 74304], [111169, 73920]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 197, 'pgroup': 0, 'primary_abelian_invariants': [4, 64, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 10], [8, 8], [16, 8], [32, 8], [64, 4], [192, 2], [384, 1]], 'representations': {'PC': {'code': '801195960766777251515249611722655493968456296447262404514148888468398602649995147657883464142499164087162493182682857037244345719493094025152212215326094907297793723060129046189656768703', 'gens': [1, 8], 'pres': [10, -2, -2, -2, -2, -2, -2, -3, -2, -2, -193, 20, 51, 82, 113, 144, 175, 4746247, 4032017, 80667, 846757, 680687, 52377, 72307, 237, 10679048, 2401938, 181468, 237638, 697728, 117778, 162608, 268, 8908809, 5337619, 403229, 528039, 624049, 261659, 129669]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [201293672, 7189251, 582313729]}, 'Perm': {'d': 197, 'gens': [5317955674209268645617152432553202640807188273834003711618161725943022945440574621977274207290503481956318310156875032987594827835403112310375356268422576077255405134659525978319744091339490880602816400890776795146458791719821335790313622344675857048609930274880074232444557450505461259678804038149458769174394294388911436873653218025885951232033883911332077006849, 7923213596605446593501618377023026514549442866072610958678929025489575939776169496613620735833633265400625273063165537595125526108947677480702249540735735036707315769552099444902911966765282312463635945853599008019869106885647068960011669112792309447862047523339509247428887675331507893702854913292800428304753771729027821695261920178158893523542883342375499098329, 518381186806547031996934387974868152041144612571180781617816510329213385192551789597049989487735309542392702316162555187895988311544961117734829585099699128224619423286120351280871434821429904522867156799112191144521626172643798708163320998082693389226821248226982832081086117306903716717054767698306435879371644347918205751975465202325222763303300783114009329408000]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 192], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times F_{193}', 'transitive_degree': 772, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
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{'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}
