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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1728.12236', 'ambient_counter': 12236, 'ambient_order': 1728, 'ambient_tex': '(C_4\\times \\He_3).\\SD_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 12, 'counter': 32, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.12236.18.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '18.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 18, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '96.18', 'subgroup_hash': 18, 'subgroup_order': 96, 'subgroup_tex': 'C_6:C_{16}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.12236', 'aut_centralizer_order': 16, 'aut_label': '18.c1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '48.35', 'aut_weyl_index': 144, 'centralizer': '108.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1', '9.a1.a1'], 'contains': ['36.c1.a1', '36.g1.a1', '54.c1.a1'], 'core': '144.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [706, -1, 9564, -1, 587, -1, 9534, -1], 'generators': [2, 576, 864, 432, 4, 8], 'label': '1728.12236.18.c1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '18.c1.a1', 'projective_image': '864.2195', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.c1.a1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.16', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 4, 2, 4, 2, 2, 12], 'aut_gens': [[1, 16], [5, 16], [55, 24], [7, 16], [59, 16], [49, 80], [9, 16], [91, 16]], 'aut_group': '192.1520', 'aut_hash': 1520, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 13, 'aut_perms': [1571644081, 1122509520, 490654801, 1575272161, 11290327, 490654800, 1575272191], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 2, 1, 1], [4, 1, 2, 2], [6, 2, 1, 1], [6, 2, 2, 1], [8, 1, 4, 2], [12, 2, 2, 2], [16, 3, 16, 1], [24, 2, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'D_{12}:C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.48', 'autcent_hash': 48, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4:C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 1], [4, 1, 4], [6, 2, 3], [8, 1, 8], [12, 2, 4], [16, 3, 16], [24, 2, 8]], 'center_label': '16.5', 'center_order': 16, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 18, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['48.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 1, 2, 2], [6, 2, 1, 3], [8, 1, 4, 2], [12, 2, 2, 2], [16, 3, 8, 2], [24, 2, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 48, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '12.4', 'hash': 18, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 80], [33, 16]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 16]], 'label': '96.18', 'linC_count': 192, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 4, 'linQ_dim': 10, 'linQ_dim_count': 2, 'linR_count': 8, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6:C16', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 15, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 48, 'number_divisions': 18, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 28, 'number_subgroups': 34, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [8, 8], [12, 8], [16, 48], [24, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4, 2, 2, 4, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[59, 16], [1, 24], [15, 16], [11, 16], [7, 16]], 'outer_group': '32.48', 'outer_hash': 48, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [2799768, 1566, 374407, 1535089, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4:C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 16], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 4], [8, 4]], 'representations': {'PC': {'code': 558650254682015169, 'gens': [1, 5], 'pres': [6, -2, -2, -2, -2, -2, -3, 12, 31, 50, 2404, 88, 2309]}, 'GLZN': {'d': 2, 'p': 21, 'gens': [9409, 120406, 74096, 43861, 135313, 38503]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [557073, 294921, 124284, 688149, 98307, 299959]}, 'Perm': {'d': 21, 'gens': [2574885556409650567, 1, 5367076607632733280, 7946546602070361600, 10508794829940395520, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 16], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6:C_{16}', 'transitive_degree': 96, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [12, 12, 4, 8, 8], 'aut_gens': [[1, 2, 16, 48, 144], [929, 754, 1168, 1200, 144], [1709, 1462, 624, 1184, 1008], [309, 774, 640, 688, 1008], [37, 370, 96, 80, 720], [1453, 402, 704, 1184, 720]], 'aut_group': None, 'aut_hash': 4225599256170198327, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6912, 'aut_permdeg': 288, 'aut_perms': [464963900091594935424607327797031744954452481248853142622202000053441138308792094786404792404911412919063423775939004775809659373534934377617923102658315100717484160777248443840302361436163716924869216453270988507184790752824410588285591531754107476146061129444928566990020627370783586980514839390872922253844057643988294273821719192287698980209229766217250742860190523087713302408302154807798897408413261885341540381703218181050387411990936563661721349081677135245329194019107356287180407328162094905641602626273100417426489039602143709987060110168708909986772112171039008539582885546, 573993276770214604261822194688094286110681732218301215992051200943522804325643464307889069876739250362396452934874140664691836429262405899342365868302406521051133662981422572623495091951865339790131116795902766960917445765454919952578483596722703609062833094535895966683905808209854283998092159390290271483120128503192055281955730278040417919293023230229222975386596363743760900618485139097819855023051704916561442250121913529147480217369447505075936775776386669064724358549289574772825657369462372211232697045378218098756503333159216667730133145180213523802876093474244858012730522718, 39133669117259013966190787678069334823643997507720090879732609265554021390848245378885682536962369613096292148392081414737100893325663932144534627266908128580661630021392109953502172684400923592341246004273815680211390243230206904099382611534891687449021972532524310640462353327331268302482194265820646654664556576554415300651167848438226298634242883045117637000423306849930023741881529557921895740997131406109707198894077635888294346213095430048374264169672692866364846935941148193062334743018743859281479598474776742968031362505808598769512045263302108392735930242353796201983491948, 290968664042432357125821585286422859909014453868817329362396418681798054483764480099564755554135243671470529745765679653923490145705845693639875333614794245707300585400474468999544358235781470996027243851624768209677118619940722584193268700940163951264338269942912638062142055875599212532467653705425200623410054044205157111640885346997777126139297877598413523525535971933603481150655970218736178699678123637934125947328026986544063451949813236865209047063957429537699894853324723463587232283512732422220974004087710081044060943876253529509862677179270485041820888108938354164610056334, 378659006706924660826336355583996499069565419426370678570784840774927119123257333004288428152963615451433650238216805869837205430260566941437665078436077634860995537132321625993794287008802440710393141640716121806532070957255976756794103312872238129897616286181268912383406599040275300843257337341833471640867088364362241073673985660308611301583435459172755099937285775042825779492208277898548137586018820240803930882487310203769381510245589302976871428918024574898485198937330269112693400613742306349917464556622820149600208929646987523253700994947382094369693619055218182852190508357], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 72, 2, 1], [3, 2, 1, 1], [3, 24, 1, 1], [4, 2, 1, 1], [4, 18, 1, 1], [4, 72, 2, 1], [6, 2, 1, 1], [6, 18, 1, 2], [6, 24, 1, 1], [6, 144, 2, 1], [8, 18, 2, 2], [12, 4, 1, 1], [12, 36, 1, 1], [12, 48, 1, 1], [12, 72, 2, 2], [16, 54, 4, 2], [24, 36, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2:S_3.C_2^4.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '1728.46113', 'autcentquo_hash': 46113, 'autcentquo_nilpotent': False, 'autcentquo_order': 1728, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SU(3,2):C_2^3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [2, 72, 2], [3, 2, 1], [3, 24, 1], [4, 2, 1], [4, 18, 1], [4, 72, 2], [6, 2, 1], [6, 18, 2], [6, 24, 1], [6, 144, 2], [8, 18, 4], [12, 4, 1], [12, 36, 1], [12, 48, 1], [12, 72, 4], [16, 54, 8], [24, 36, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '864.2195', 'commutator_count': 1, 'commutator_label': '216.25', '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', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 12236, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 72, 1, 2], [3, 2, 1, 1], [3, 24, 1, 1], [4, 2, 1, 1], [4, 18, 1, 1], [4, 72, 2, 1], [6, 2, 1, 1], [6, 18, 1, 2], [6, 24, 1, 1], [6, 144, 1, 2], [8, 18, 2, 2], [12, 4, 1, 1], [12, 36, 1, 1], [12, 48, 1, 1], [12, 72, 4, 1], [16, 54, 8, 1], [24, 36, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 48, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 4]], 'familial': False, 'frattini_label': '12.2', 'frattini_quotient': '144.182', 'hash': 12236, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 8, 3, 3, 6], 'inner_gens': [[1, 1302, 80, 624, 1584], [1309, 2, 128, 1184, 720], [705, 1234, 16, 624, 144], [1153, 642, 1168, 48, 144], [289, 1154, 16, 48, 144]], 'inner_hash': 2195, 'inner_nilpotent': False, 'inner_order': 864, 'inner_split': True, 'inner_tex': '(C_2\\times \\He_3).\\SD_{16}', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 48, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 14], [6, 8], [8, 4], [12, 6], [16, 1]], 'label': '1728.12236', 'linC_count': 64, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 4, 'linQ_dim': 14, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': '(C4*He3).SD16', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 26, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 41, 'number_divisions': 26, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 125, 'number_subgroup_classes': 153, 'number_subgroups': 3513, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 163], [3, 26], [4, 164], [6, 350], [8, 72], [12, 376], [16, 432], [24, 144]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [432, 0], 'outer_gens': [[433, 2, 16, 48, 144], [1, 434, 16, 48, 144]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 43, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 2], [6, 2], [8, 6], [12, 5], [16, 1], [24, 2]], 'representations': {'PC': {'code': 1527782808690746636509188454173786280571102744841675489645123823725605161967561503382553, 'gens': [1, 2, 5, 6, 7], 'pres': [9, -2, -2, -2, -2, -3, 3, -2, -2, -3, 23437, 46, 23654, 74, 31395, 3918, 3604, 2893, 922, 211, 33701, 31982, 16439, 8456, 2147, 99798, 22695, 186, 103687, 51856, 214, 93320, 46673]}, 'Perm': {'d': 43, 'gens': [103051769102521726361107864058747403149389484297274, 140586915037552063253734078109681513361997375312780, 174652063397549908043361945803818962866068440137126, 4497682605676, 208918857227046410725795775456009378596421386413676, 5901332335801, 1546306431198056630058652294838848897230869262336000, 3053381643260330370887589682026615822877776261120000, 4492680797991609120023789961411117769433838833664000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_4\\times \\He_3).\\SD_{16}', 'transitive_degree': 144, 'wreath_data': None, 'wreath_product': False}
