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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '158400.i', 'ambient_counter': 9, 'ambient_order': 158400, 'ambient_tex': '\\GL(2,11):D_6', 'central': False, 'central_factor': False, 'centralizer_order': 60, 'characteristic': True, 'core_order': 39600, 'counter': 9, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '158400.i.4.B', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': False, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '39600.e', 'subgroup_hash': 1398855719071205022, 'subgroup_order': 39600, 'subgroup_tex': '\\SL(2,11):C_{30}', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '158400.i', 'aut_centralizer_order': None, 'aut_label': '4.B', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2640.B', 'complements': ['39600.I'], 'conjugacy_class_count': 1, 'contained_in': ['2.C', '2.D', '2.E'], 'contains': ['8.a1', '12.A', '20.B', '44.B', '48.D', '220.X'], 'core': '4.B', 'coset_action_label': None, 'count': 1, 'diagramx': [4984, 2540, 8260, 5826], 'generators': [41772741070013040, 11619123519371993, 12229776919405614, 37595466963011736, 36891648239530184, 2526212370389246], 'label': '158400.i.4.B', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.B', 'normal_contained_in': ['2.C', '2.D', '2.E'], 'normal_contains': ['8.a1', '12.A', '20.B', '2640.B'], 'normalizer': '1.a1', 'old_label': '4.b1', 'projective_image': None, 'quotient_action_image': '4.2', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '4.B', 'subgroup_fusion': None, 'weyl_group': '2640.a'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '30.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 660, 'aut_gen_orders': [20, 22, 20, 12, 4], 'aut_gens': [[12531822321003912, 15394321519121429, 20258619725292814, 40860042266609549, 11619123519371993, 33418192856010432], [20886370535006520, 44800057445045130, 29206002569800029, 6114901428503014, 13251566580699389, 25063644642007824], [12531822321003912, 40559846859009190, 31165639787311143, 22216482884836568, 38925579311070212, 33418192856010432], [20886370535006520, 33550852467829997, 8847669042709828, 6114901428503014, 13251566580699389, 25063644642007824], [37595466963011736, 20677489079083759, 3125974002126509, 20278855777548780, 18646723749506926, 8354548214002608], [12531822321003912, 43284222713555421, 37887156225256559, 22216482884836568, 11619123519371993, 33418192856010432]], 'aut_group': None, 'aut_hash': 371462752101153955, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21120, 'aut_permdeg': 116, 'aut_perms': [25924768341761906241593633937383167295097942136100488783871806374662972066833310859172583129590718914942620648924438343750701244749157988134425836165020161634508515826492520818342243199162091, 31916197147843944626143254999144284283240233391769499114509428992993249451926527916152579908862902121131692111070053889122851429782796043998117447567215742216440369077234941194675333403924599, 29106384650864185486895571568263409791649306806527658905350321046881946974225233596536783257564461143205612915135434524925830650112917447578445875548265374548289608567896113732126781152833277, 33262120550530042997358082263049425334344413254701732142958323622034094865519203669435212074314255789382621693999150969692044972455648818346167512066157401980258299606796141854569158788721007, 25889606932450126396000251504721977447236945236360279775808420152188708135142763452006764433400384105579142633920746440126702700370810447231773218601967022447001852446237232415912397438063593], 'aut_phi_ratio': 2.2, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 110, 1, 1], [3, 1, 2, 1], [3, 110, 1, 1], [3, 110, 2, 1], [4, 1, 2, 1], [4, 110, 1, 1], [5, 1, 4, 1], [5, 132, 1, 2], [5, 132, 4, 2], [6, 1, 2, 1], [6, 110, 1, 1], [6, 110, 2, 3], [6, 110, 4, 1], [10, 1, 4, 1], [10, 110, 4, 1], [10, 132, 1, 2], [10, 132, 4, 2], [11, 60, 2, 1], [12, 1, 4, 1], [12, 110, 1, 2], [12, 110, 2, 4], [12, 110, 4, 1], [15, 1, 8, 1], [15, 110, 4, 1], [15, 110, 8, 1], [15, 132, 2, 2], [15, 132, 8, 2], [20, 1, 8, 1], [20, 110, 4, 1], [20, 132, 2, 2], [20, 132, 8, 2], [22, 60, 2, 1], [30, 1, 8, 1], [30, 110, 4, 1], [30, 110, 8, 3], [30, 110, 16, 1], [30, 132, 2, 2], [30, 132, 8, 2], [33, 60, 4, 1], [44, 60, 4, 1], [55, 60, 8, 1], [60, 1, 16, 1], [60, 110, 4, 2], [60, 110, 8, 4], [60, 110, 16, 1], [60, 132, 4, 2], [60, 132, 16, 2], [66, 60, 4, 1], [110, 60, 8, 1], [132, 60, 8, 1], [165, 60, 16, 1], [220, 60, 16, 1], [330, 60, 16, 1], [660, 60, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times C_4\\times \\PSL(2,11).C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 660, 'autcentquo_group': '1320.133', 'autcentquo_hash': 133, 'autcentquo_nilpotent': False, 'autcentquo_order': 1320, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PGL(2,11)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 110, 1], [3, 1, 2], [3, 110, 3], [4, 1, 2], [4, 110, 1], [5, 1, 4], [5, 132, 10], [6, 1, 2], [6, 110, 11], [10, 1, 4], [10, 110, 4], [10, 132, 10], [11, 60, 2], [12, 1, 4], [12, 110, 14], [15, 1, 8], [15, 110, 12], [15, 132, 20], [20, 1, 8], [20, 110, 4], [20, 132, 20], [22, 60, 2], [30, 1, 8], [30, 110, 44], [30, 132, 20], [33, 60, 4], [44, 60, 4], [55, 60, 8], [60, 1, 16], [60, 110, 56], [60, 132, 40], [66, 60, 4], [110, 60, 8], [132, 60, 8], [165, 60, 16], [220, 60, 16], [330, 60, 16], [660, 60, 32]], 'center_label': '60.4', 'center_order': 60, 'central_product': True, 'central_quotient': '660.13', 'commutator_count': 1, 'commutator_label': '1320.13', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1', '660.13'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2640.l', 1], ['3.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 110, 1, 1], [3, 1, 2, 1], [3, 110, 1, 1], [3, 110, 2, 1], [4, 1, 2, 1], [4, 110, 1, 1], [5, 1, 4, 1], [5, 132, 2, 1], [5, 132, 4, 2], [6, 1, 2, 1], [6, 110, 1, 1], [6, 110, 2, 5], [10, 1, 4, 1], [10, 110, 4, 1], [10, 132, 2, 1], [10, 132, 4, 2], [11, 60, 2, 1], [12, 1, 4, 1], [12, 110, 2, 3], [12, 110, 4, 2], [15, 1, 8, 1], [15, 110, 4, 1], [15, 110, 8, 1], [15, 132, 4, 1], [15, 132, 8, 2], [20, 1, 8, 1], [20, 110, 4, 1], [20, 132, 4, 1], [20, 132, 8, 2], [22, 60, 2, 1], [30, 1, 8, 1], [30, 110, 4, 1], [30, 110, 8, 5], [30, 132, 4, 1], [30, 132, 8, 2], [33, 60, 4, 1], [44, 60, 4, 1], [55, 60, 8, 1], [60, 1, 16, 1], [60, 110, 8, 3], [60, 110, 16, 2], [60, 132, 8, 1], [60, 132, 16, 2], [66, 60, 4, 1], [110, 60, 8, 1], [132, 60, 8, 1], [165, 60, 16, 1], [220, 60, 16, 1], [330, 60, 16, 1], [660, 60, 32, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 36576, 'exponent': 660, 'exponents_of_order': [4, 2, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[6, 0, 32], [10, 0, 48], [12, 0, 32]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': None, 'hash': 1398855719071205022, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [1, 3, 2, 1, 1, 1], 'inner_gens': [[12531822321003912, 15394321519121429, 20258619725292814, 40860042266609549, 11619123519371993, 33418192856010432], [12531822321003912, 15394321519121429, 34082170458900506, 40860042266609549, 11619123519371993, 33418192856010432], [12531822321003912, 24000369396632122, 20258619725292814, 40860042266609549, 11619123519371993, 33418192856010432], [12531822321003912, 15394321519121429, 20258619725292814, 40860042266609549, 11619123519371993, 33418192856010432], [12531822321003912, 15394321519121429, 20258619725292814, 40860042266609549, 11619123519371993, 33418192856010432], [12531822321003912, 15394321519121429, 20258619725292814, 40860042266609549, 11619123519371993, 33418192856010432]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 660, 'inner_split': True, 'inner_tex': '\\PSL(2,11)', 'inner_used': [2, 3], 'irrC_degree': 6, 'irrQ_degree': 160, 'irrQ_dim': 160, 'irrR_degree': 12, 'irrep_stats': [[1, 30], [5, 60], [6, 60], [10, 150], [11, 30], [12, 120]], 'label': '39600.e', 'linC_count': 32, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 26, 'linQ_degree_count': 4, 'linQ_dim': 26, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SL(2,11):C30', '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, 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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 80, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 450, 'number_divisions': 72, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 206, 'number_subgroup_classes': 268, 'number_subgroups': 14368, 'old_label': None, 'order': 39600, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 111], [3, 332], [4, 112], [5, 1324], [6, 1212], [10, 1764], [11, 120], [12, 1544], [15, 3968], [20, 3088], [22, 120], [30, 7488], [33, 240], [44, 240], [55, 480], [60, 11456], [66, 240], [110, 480], [132, 480], [165, 960], [220, 960], [330, 960], [660, 1920]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [4177274107001304, 4177274107001304, 11930796867425820, 4177274107001304], 'outer_gens': [[12531822321003912, 11687950943229949, 36705855301915694, 22216482884836568, 11619123519371993, 33418192856010432], [12531822321003912, 15394321519121429, 30282944696259841, 40860042266609549, 38925579311070212, 33418192856010432], [12531822321003912, 16368056914121971, 7007106000247367, 40860042266609549, 11619123519371993, 33418192856010432], [20886370535006520, 6789690027253329, 22069658221081063, 19366156759931367, 37293136249742816, 25063644642007824]], 'outer_group': '32.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 5040, 120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 56, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [4, 2], [8, 2], [10, 6], [11, 2], [20, 7], [22, 2], [24, 3], [40, 10], [44, 2], [48, 8], [80, 7], [88, 2], [96, 8], [160, 4], [192, 3]], 'representations': {'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 15394321519121429, 20258619725292814, 40860042266609549, 11619123519371993, 33418192856010432]}, 'Perm': {'d': 56, 'gens': [13637212782977098589958011183434947383955502402031874518649449534758525587, 26547518693203102626819063621763006582981654261699169284309253485940223507, 11760, 39504368882042211577451826726636679884674630647924386526137691653098878084, 52436367040888822821003753652370908349761551934181747193400622930001195843, 20163]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [30], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SL(2,11):C_{30}', 'transitive_degree': 720, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [12, 20, 2, 12, 4, 12, 12], 'aut_gens': [[12531822321003912, 4352264157421504, 32568456490485961, 40860042266609549, 11619123519371993, 33418192856010432, 14949399392055802], [37595466963011736, 22172445105364590, 6231944007346175, 20278855777548780, 31897979080935279, 8354548214002608, 19059573810010056], [12531822321003912, 12203504442998908, 37732792264734881, 22216482884836568, 11619123519371993, 33418192856010432, 28895927227292736], [12531822321003912, 36971069053669371, 21373902538270612, 40860042266609549, 38925579311070212, 33418192856010432, 14949399392055802], [37595466963011736, 2989113762266784, 43678666633028089, 21911298838890817, 18646723749506926, 8354548214002608, 7355025179958710], [37595466963011736, 26093524962596618, 13591539101859796, 20278855777548780, 31897979080935279, 8354548214002608, 43155129000900322], [37595466963011736, 11473174719574658, 37569799079402544, 20278855777548780, 31897979080935279, 8354548214002608, 19059573810010056], [37595466963011736, 39474015381000881, 19015914176850362, 20278855777548780, 18646723749506926, 8354548214002608, 45710095517991618]], 'aut_group': None, 'aut_hash': 1854200450572961862, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 506880, 'aut_permdeg': 148, 'aut_perms': [2174609647141187731657745954091550616931294154774243153366941789680492741496064956714927812159497502338725460986864964452972501740394460153062086019521102186641340594572671440581903895457945425900007802850353754533158450412723448341524586786330067259912662321, 2029080722664901929978991646517995522495452927687779870486348652791172900171694857726609710381068819921671769472024364870747486344060163529509226442624251976897137269992850760131114795367634041290094914385024429230074490204423993141727062960924805582417726091, 869702774003151890482120629321676972952224110060681423813314256566264532229746832018255635381967131815125967747757118523660625188338222788876594512142818411681284122489481830878474652178879611602469439421113098125122140907298025851523760706044503746392231697, 86043002803816214950013379338954559625046657390024904395339409833854513491629254897685401504129762472497387216099308545019529505440264044565838950253963428729954633803481837259505426260876896726896601582846555238416017703806194353762792614890489090693670685, 1617297571724020083983435132968383537622293295374022829860357790152387703623057060993096673091404257226069836650058840675283116449867903585664581817140430037438878947027963134117973545843455950942409884237757433699624032332285241725761487478061160279514183095, 1853838773146062390127458675915707917039787155176957624328287564678143042313795956534232618173454363803843150907590995834470437830005787437746921776850703646467828557320748672732346987530378778616901172907728527643215114722055831897716227601250734081397969154, 1964372981202244685081095360484665890515167118914463934453266780992441328514591600149353497862630725453886797743040925759101728256302306091383909804189011248456619203885178510066130614289157960918032010040409419918083872323584897538373113303933091908270055211], 'aut_phi_ratio': 13.2, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 2, 1], [2, 110, 1, 1], [2, 132, 1, 1], [2, 396, 2, 1], [3, 2, 1, 1], [3, 110, 1, 1], [3, 220, 1, 1], [4, 2, 1, 1], [4, 110, 1, 1], [4, 132, 1, 1], [4, 330, 2, 1], [5, 1, 4, 1], [5, 132, 1, 2], [5, 132, 4, 2], [6, 2, 1, 1], [6, 110, 1, 1], [6, 220, 1, 3], [6, 220, 2, 1], [6, 264, 1, 1], [6, 660, 2, 1], [8, 110, 2, 1], [8, 220, 1, 1], [8, 660, 2, 1], [10, 1, 4, 1], [10, 6, 8, 1], [10, 110, 4, 1], [10, 132, 1, 2], [10, 132, 2, 2], [10, 132, 4, 3], [10, 132, 8, 2], [10, 396, 8, 1], [10, 792, 2, 4], [10, 792, 8, 4], [11, 120, 1, 1], [12, 2, 2, 1], [12, 110, 1, 2], [12, 220, 1, 4], [12, 220, 2, 1], [12, 264, 1, 1], [12, 660, 2, 1], [15, 2, 4, 1], [15, 110, 4, 1], [15, 220, 4, 1], [15, 264, 1, 2], [15, 264, 4, 2], [20, 2, 4, 1], [20, 110, 4, 1], [20, 132, 4, 1], [20, 264, 1, 4], [20, 264, 4, 4], [20, 330, 8, 1], [22, 120, 1, 1], [22, 720, 2, 1], [24, 110, 2, 2], [24, 220, 1, 2], [24, 220, 2, 6], [24, 660, 2, 2], [30, 2, 4, 1], [30, 110, 4, 1], [30, 220, 4, 3], [30, 220, 8, 1], [30, 264, 1, 2], [30, 264, 2, 2], [30, 264, 4, 3], [30, 264, 8, 2], [30, 660, 8, 1], [33, 240, 1, 1], [40, 110, 8, 1], [40, 220, 4, 1], [40, 660, 8, 1], [44, 240, 1, 1], [55, 120, 4, 1], [60, 2, 8, 1], [60, 110, 4, 2], [60, 220, 4, 4], [60, 220, 8, 1], [60, 264, 2, 4], [60, 264, 4, 1], [60, 264, 8, 4], [60, 660, 8, 1], [66, 240, 1, 1], [110, 120, 4, 1], [110, 720, 8, 1], [120, 110, 8, 2], [120, 220, 4, 2], [120, 220, 8, 6], [120, 660, 8, 2], [132, 240, 2, 1], [165, 240, 4, 1], [220, 240, 4, 1], [330, 240, 4, 1], [660, 240, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times C_6).C_2^5.\\PSL(2,11).C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.45', 'autcent_hash': 45, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 660, 'autcentquo_group': None, 'autcentquo_hash': 2842205202387851101, 'autcentquo_nilpotent': False, 'autcentquo_order': 15840, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times \\PSL(2,11).C_2\\times S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 2], [2, 110, 1], [2, 132, 1], [2, 396, 2], [3, 2, 1], [3, 110, 1], [3, 220, 1], [4, 2, 1], [4, 110, 1], [4, 132, 1], [4, 330, 2], [5, 1, 4], [5, 132, 10], [6, 2, 1], [6, 110, 1], [6, 220, 5], [6, 264, 1], [6, 660, 2], [8, 110, 2], [8, 220, 1], [8, 660, 2], [10, 1, 4], [10, 6, 8], [10, 110, 4], [10, 132, 34], [10, 396, 8], [10, 792, 40], [11, 120, 1], [12, 2, 2], [12, 110, 2], [12, 220, 6], [12, 264, 1], [12, 660, 2], [15, 2, 4], [15, 110, 4], [15, 220, 4], [15, 264, 10], [20, 2, 4], [20, 110, 4], [20, 132, 4], [20, 264, 20], [20, 330, 8], [22, 120, 1], [22, 720, 2], [24, 110, 4], [24, 220, 14], [24, 660, 4], [30, 2, 4], [30, 110, 4], [30, 220, 20], [30, 264, 34], [30, 660, 8], [33, 240, 1], [40, 110, 8], [40, 220, 4], [40, 660, 8], [44, 240, 1], [55, 120, 4], [60, 2, 8], [60, 110, 8], [60, 220, 24], [60, 264, 44], [60, 660, 8], [66, 240, 1], [110, 120, 4], [110, 720, 8], [120, 110, 16], [120, 220, 56], [120, 660, 16], [132, 240, 2], [165, 240, 4], [220, 240, 4], [330, 240, 4], [660, 240, 8]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': None, 'commutator_count': 1, 'commutator_label': '3960.a', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '5.1', '660.13'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['31680.j', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 2], [2, 110, 1, 1], [2, 132, 1, 1], [2, 396, 1, 2], [3, 2, 1, 1], [3, 110, 1, 1], [3, 220, 1, 1], [4, 2, 1, 1], [4, 110, 1, 1], [4, 132, 1, 1], [4, 330, 1, 2], [5, 1, 4, 1], [5, 132, 2, 1], [5, 132, 4, 2], [6, 2, 1, 1], [6, 110, 1, 1], [6, 220, 1, 5], [6, 264, 1, 1], [6, 660, 1, 2], [8, 110, 2, 1], [8, 220, 1, 1], [8, 660, 1, 2], [10, 1, 4, 1], [10, 6, 4, 2], [10, 110, 4, 1], [10, 132, 2, 1], [10, 132, 4, 8], [10, 396, 4, 2], [10, 792, 2, 4], [10, 792, 4, 8], [11, 120, 1, 1], [12, 2, 2, 1], [12, 110, 2, 1], [12, 220, 1, 2], [12, 220, 2, 2], [12, 264, 1, 1], [12, 660, 1, 2], [15, 2, 4, 1], [15, 110, 4, 1], [15, 220, 4, 1], [15, 264, 2, 1], [15, 264, 4, 2], [20, 2, 4, 1], [20, 110, 4, 1], [20, 132, 4, 1], [20, 264, 2, 2], [20, 264, 4, 4], [20, 330, 4, 2], [22, 120, 1, 1], [22, 720, 1, 2], [24, 110, 4, 1], [24, 220, 2, 3], [24, 220, 4, 2], [24, 660, 2, 2], [30, 2, 4, 1], [30, 110, 4, 1], [30, 220, 4, 5], [30, 264, 2, 1], [30, 264, 4, 8], [30, 660, 4, 2], [33, 240, 1, 1], [40, 110, 8, 1], [40, 220, 4, 1], [40, 660, 4, 2], [44, 240, 1, 1], [55, 120, 4, 1], [60, 2, 8, 1], [60, 110, 8, 1], [60, 220, 4, 2], [60, 220, 8, 2], [60, 264, 4, 3], [60, 264, 8, 4], [60, 660, 4, 2], [66, 240, 1, 1], [110, 120, 4, 1], [110, 720, 4, 2], [120, 110, 16, 1], [120, 220, 8, 3], [120, 220, 16, 2], [120, 660, 8, 2], [132, 240, 2, 1], [165, 240, 4, 1], [220, 240, 4, 1], [330, 240, 4, 1], [660, 240, 8, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 2251105920, 'exponent': 1320, 'exponents_of_order': [6, 2, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[20, 0, 48], [24, 0, 40]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': None, 'hash': 3349893979029791745, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 660, 'inner_gen_orders': [1, 12, 6, 3, 2, 1, 2], 'inner_gens': [[12531822321003912, 4352264157421504, 32568456490485961, 40860042266609549, 11619123519371993, 33418192856010432, 14949399392055802], [12531822321003912, 4352264157421504, 23211670150963183, 40860042266609549, 11619123519371993, 33418192856010432, 20921491951666157], [12531822321003912, 15276570035176916, 32568456490485961, 40860042266609549, 11619123519371993, 33418192856010432, 10532514518949524], [12531822321003912, 4352264157421504, 32568456490485961, 40860042266609549, 11619123519371993, 33418192856010432, 25411413845255961], [12531822321003912, 4352264157421504, 32568456490485961, 40860042266609549, 11619123519371993, 33418192856010432, 35560754788803230], [12531822321003912, 4352264157421504, 32568456490485961, 40860042266609549, 11619123519371993, 33418192856010432, 14949399392055802], [12531822321003912, 41588194236696405, 43615646349369483, 22216482884836568, 38925579311070212, 33418192856010432, 14949399392055802]], 'inner_hash': 2842205202387851101, 'inner_nilpotent': False, 'inner_order': 15840, 'inner_split': True, 'inner_tex': 'C_2\\times S_3.\\PSL(2,11).C_2', 'inner_used': [2, 3, 5, 7], 'irrC_degree': 20, 'irrQ_degree': 192, 'irrQ_dim': 192, 'irrR_degree': None, 'irrep_stats': [[1, 40], [2, 20], [10, 100], [11, 40], [12, 80], [20, 140], [22, 20], [24, 115]], 'label': '158400.i', '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': False, 'metacyclic': False, 'monomial': False, 'name': 'GL(2,11):D6', 'ngens': 7, '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, 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': 154, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 555, 'number_divisions': 156, 'number_normal_subgroups': 60, 'number_subgroup_autclasses': 1274, 'number_subgroup_classes': 2050, 'number_subgroups': 329628, 'old_label': None, 'order': 158400, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1047], [3, 332], [4, 904], [5, 1324], [6, 2796], [8, 1760], [10, 39828], [11, 120], [12, 3128], [15, 3968], [20, 8896], [22, 1560], [24, 6160], [30, 19104], [33, 240], [40, 7040], [44, 240], [55, 480], [60, 23072], [66, 240], [110, 6240], [120, 24640], [132, 480], [165, 960], [220, 960], [330, 960], [660, 1920]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [11619123519371993, 4177274107001304, 4177274107001304, 4177274107001304], 'outer_gens': [[12531822321003912, 4352264157421504, 32568456490485961, 40860042266609549, 11619123519371993, 33418192856010432, 16571680830183658], [12531822321003912, 8956223301562360, 43615646349369483, 22216482884836568, 11619123519371993, 33418192856010432, 14949399392055802], [12531822321003912, 41980667810948290, 32568456490485961, 40860042266609549, 11619123519371993, 33418192856010432, 14949399392055802], [37595466963011736, 17022685441436090, 11284039172477472, 21911298838890817, 31897979080935279, 8354548214002608, 20059019052703338]], 'outer_group': '32.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [120, 362880, 5040, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 56, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 8], [8, 4], [10, 12], [11, 8], [20, 10], [22, 4], [24, 9], [40, 15], [44, 8], [48, 21], [80, 14], [88, 4], [96, 14], [160, 3], [192, 6], [320, 4]], 'representations': {'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 4352264157421504, 32568456490485961, 40860042266609549, 11619123519371993, 33418192856010432, 14949399392055802]}, 'Perm': {'d': 56, 'gens': [13872413664758609238761520250906940982964589392073775845329459812743658277, 26778438218240057666001789685445759377670691656380194903351721892646109492, 52, 26778438218240057666001789685445759377670691656380194903351721892646103732, 39739487401935409737657616190442916597272313448680933977957219103010909373, 52650107829080096216594757004972599207107093805829219177894332693625728813, 65346591841521841460996043099383663455515021988271356814458919693741508592]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\GL(2,11):D_6', 'transitive_degree': 720, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': 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, 3]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
