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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '4608.v', 'ambient_counter': 22, 'ambient_order': 4608, 'ambient_tex': 'D_{24}.C_{96}', 'central': False, 'central_factor': False, 'centralizer_order': 192, 'characteristic': False, 'core_order': 384, 'counter': 35, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4608.v.6._.J', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '6.g1.b1', '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': 6, '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': '768.6725', 'subgroup_hash': 6725, 'subgroup_order': 768, 'subgroup_tex': '\\OD_{128}:C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4608.v', 'aut_centralizer_order': 4, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '512.7529762', 'aut_weyl_index': 24, 'centralizer': '24._.O', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2._.B', '3._.B'], 'contains': ['12._.G', '12._.P', '12._.U', '12._.BA', '12._.BC', '18._.C'], 'core': '12._.G', 'coset_action_label': None, 'count': 3, 'diagramx': [430, -1, 772, -1, 1473, -1, 1545, -1], 'generators': [1380299136, 179726450, 1337164788, 783607297, 805174465, 37442, 35945290, 330696668, 776418264], 'label': '4608.v.6._.J', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2._.B', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3._.B', 'old_label': '6.g1.b1', 'projective_image': '24.6', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6._.J', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '384.5547', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 48, 'aut_gen_orders': [8, 16, 4, 8, 2], 'aut_gens': [[37442, 35945290, 582313729], [7188864, 215671740, 805174465], [37442, 647015220, 7152001], [7188864, 1250896092, 74305], [4187521, 812363554, 582313729], [4187521, 553557466, 805174465]], 'aut_group': '1536.408624996', 'aut_hash': 737399360701677454, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 40, 'aut_perms': [325838670943426261918469296229327890454717330939, 270923988142122130139059073519113415242384626291, 480166140746563606592002533329982691105801020608, 783487430196486756147624897331733598899917703805, 20920239780829431635361255089544895107252001509], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [3, 1, 2, 1], [4, 1, 2, 1], [4, 2, 3, 1], [6, 1, 2, 1], [6, 2, 6, 1], [8, 1, 4, 1], [8, 2, 6, 1], [12, 1, 4, 1], [12, 2, 6, 1], [16, 1, 8, 1], [16, 2, 12, 1], [24, 1, 8, 1], [24, 2, 12, 1], [32, 1, 16, 1], [32, 2, 24, 1], [48, 1, 16, 1], [48, 2, 24, 1], [64, 1, 32, 1], [64, 2, 48, 1], [96, 1, 32, 1], [96, 2, 48, 1], [192, 1, 64, 1], [192, 2, 96, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times C_{16}\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '256.55608', 'autcent_hash': 55608, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\times C_{16}', '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, 1], [2, 2, 3], [3, 1, 2], [4, 1, 2], [4, 2, 3], [6, 1, 2], [6, 2, 6], [8, 1, 4], [8, 2, 6], [12, 1, 4], [12, 2, 6], [16, 1, 8], [16, 2, 12], [24, 1, 8], [24, 2, 12], [32, 1, 16], [32, 2, 24], [48, 1, 16], [48, 2, 24], [64, 1, 32], [64, 2, 48], [96, 1, 32], [96, 2, 48], [192, 1, 64], [192, 2, 96]], 'center_label': '192.2', 'center_order': 192, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 6725, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['256.6725', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [3, 1, 2, 1], [4, 1, 2, 1], [4, 2, 1, 3], [6, 1, 2, 1], [6, 2, 2, 3], [8, 1, 4, 1], [8, 2, 2, 3], [12, 1, 4, 1], [12, 2, 2, 3], [16, 1, 8, 1], [16, 2, 4, 3], [24, 1, 8, 1], [24, 2, 4, 3], [32, 1, 16, 1], [32, 2, 8, 3], [48, 1, 16, 1], [48, 2, 8, 3], [64, 1, 32, 1], [64, 2, 16, 3], [96, 1, 32, 1], [96, 2, 16, 3], [192, 1, 64, 1], [192, 2, 32, 3]], 'element_repr_type': 'GLFp', 'elementary': 2, 'eulerian_function': 93184, 'exponent': 192, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 64]], 'familial': False, 'frattini_label': '32.1', 'frattini_quotient': '24.15', 'hash': 6725, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 1, 2], 'inner_gens': [[37442, 35945290, 805174465], [37442, 35945290, 582313729], [7188864, 35945290, 582313729]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 3], 'irrC_degree': 2, 'irrQ_degree': 128, 'irrQ_dim': 128, 'irrR_degree': 4, 'irrep_stats': [[1, 384], [2, 96]], 'label': '768.6725', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'OD128:C6', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 26, 'number_conjugacy_classes': 480, 'number_divisions': 50, 'number_normal_subgroups': 122, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 128, 'number_subgroups': 134, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 8], [6, 14], [8, 16], [12, 16], [16, 32], [24, 32], [32, 64], [48, 64], [64, 128], [96, 128], [192, 256]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 48, 'outer_gen_orders': [2, 2, 2, 48], 'outer_gen_pows': [7189058, 7189058, 7189058, 7189058], 'outer_gens': [[37442, 122213986, 805174465], [1380298945, 35945290, 4193504], [37442, 244427972, 582313729], [1380298945, 913010366, 7152001]], 'outer_group': '384.14574', 'outer_hash': 14574, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 192, 'outer_perms': [74193784810592653619580128927085726225848957825593168032519189078935098101496302827068770905797618738739921555785732093934079873986494508521154709093619253686940595978650282675365174514530968651418798277854769991699817714451282432609501708363752734142734090649541162242954269213584429131649001600045489612302059804527813653603544120178079967483796283042977, 4903665568502735207020628056819927700806173351020351263654240982239242051805799007839994332223077131254648721400272574989819653530757788582640862098720324248076745916479010095639861022640259985353537021066029949484957604548473094093273707338724471345837129344830946968540746587012276794482572874493512651924611397613669505141401026769852880, 3765648501821508456899681948466005364038697859949157279871157226054482411635718006104082157734394602633884569955422651541413678712145696307314671362912051833475460245818723915573850474881159482290189988569526417128138809942940441945004155756824654698283120071887389898629241636955087749418242731802459222815018922661585050929298220521326905234821260365444, 128482673503410564905371123381663084823940214159283676749261505043381736322478910156103870230608205973934340087187289764262512685997407459867465124849882683406652148912994979571203050617882427769307634140371102083177487067854154149700692870459078576675392700760967718378840442469516377515852070928880919081110479335229762346352928101464445313485134485971872], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{48}:C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 131, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 32, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 8], [8, 8], [16, 8], [32, 4], [64, 1], [128, 1]], 'representations': {'PC': {'code': '7395354281119244219896848118446081', 'gens': [1, 2, 3], 'pres': [9, -2, -2, -2, -2, -2, -2, -2, -2, -3, 6949, 74, 102, 130, 158, 186, 214]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [37442, 35945290, 582313729]}, 'Perm': {'d': 131, 'gens': [619063061798386334790069036752671682201287559923027721256289271909227441621677101847129342270651767433334432117735693981738969999008387907043028965763710461519657049717917284364319232785519517533311302893568000000000000000, 221577240763167038119109839447201024829703303304193331009606347304918802524636103405360277559762058094718602523668276989723117657113784854954523992182881899941855253176704585555254949232345931352526416031979508942727825760, 208543654031972675811019022052163236968537591767456424791448791213859880889247801384876088711812476338122170711722096821169103178616669061431205200685998718401275063917074384508084046539833363971250585483794291097600000000, 201975753734440983168030992648345208264212376388114871673391689480898138382292855727386580549799782664444032009028475721194954356535389660928998977348840843979867145661227551577927113563778364907203537514660624295657472000, 97703926718454645262521481622263669305231509288075143274768690420952684674725941701719143922097299003221079930603232925676017985822782413399392578373059946913543597937743311740647828241809656186635859466309924032688179200, 45568013210461477802436320934552585957398418966841536608637824613634519950608643500909480558047922198413920646381047001919078571147955484131645072042534240569788816898293180645392877172846446348689281672731066812729456320, 19500056623652699785979096803241248616917686955914956470906338149014306958708988487073399602816145714391524121343899707830797054580243243248140351426483485096903470317130119225339378328918829664757907481433996085127825760, 6467241133426074922228152049989658139556312540995027677021616250812091641311811409166050027660830708985040706124699047409306243427358608599418179156332255942688689445158231826643474837546795637905688944437945541028258664, 4]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 96], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\OD_{128}:C_6', 'transitive_degree': 384, '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': '384.5547', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 48, 'aut_gen_orders': [16, 24, 24, 16, 8, 48, 48, 8], 'aut_gens': [[37442, 395398321, 35945290], [6882766, 115025093, 913010366], [481149, 395398321, 158159276], [287377, 992089873, 833930728], [37442, 992089873, 71890580], [343540, 1301219333, 1250896092], [4039104, 395398321, 862686960], [3187202, 395398321, 1049602468], [5622862, 1272463101, 589502756]], 'aut_group': None, 'aut_hash': 2675128990628703738, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 58, 'aut_perms': [1209836495979212660927431096201921996793240735553015761880256400263328622580693, 460404172012635887940176008295548619537204047757590778225083279146513706895115, 174652649647282533900418686626609112483306293175539093391243115546255472235550, 1597639767924934956743718311830340876443831605492568598712822068395741430687228, 1765178233114510148290555356882976171987611967425069352141110052884293477525307, 1074154375750739599809226010366626104094520919021880789240330648703660255467695, 2331923544139373031937616687423484820025328957153845510685169469375855377795222, 460087216947899050909888636323091739061296835940936754624644615139315293135174], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 4, 1], [4, 12, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 3], [6, 2, 4, 1], [6, 12, 4, 1], [8, 1, 4, 1], [8, 2, 2, 3], [8, 12, 4, 1], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 2], [12, 2, 8, 2], [12, 2, 16, 1], [12, 12, 4, 1], [16, 1, 8, 1], [16, 2, 4, 1], [16, 2, 8, 1], [16, 12, 8, 1], [24, 1, 8, 1], [24, 2, 4, 7], [24, 2, 8, 4], [24, 12, 8, 1], [32, 1, 16, 1], [32, 2, 8, 1], [32, 2, 16, 1], [32, 12, 16, 1], [48, 1, 16, 1], [48, 2, 8, 3], [48, 2, 16, 4], [48, 2, 32, 1], [48, 12, 16, 1], [64, 1, 32, 1], [64, 2, 16, 1], [64, 2, 32, 1], [64, 12, 32, 1], [96, 1, 32, 1], [96, 2, 16, 3], [96, 2, 32, 4], [96, 2, 64, 1], [96, 12, 32, 1], [192, 1, 64, 1], [192, 2, 32, 3], [192, 2, 64, 4], [192, 2, 128, 1], [192, 12, 64, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:((C_4\\times C_8).C_2^6.C_2)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '256.55608', 'autcent_hash': 55608, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\times C_{16}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '48.38', 'autcentquo_hash': 38, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 12, 2], [3, 1, 2], [3, 2, 3], [4, 1, 2], [4, 2, 5], [4, 12, 2], [6, 1, 2], [6, 2, 11], [6, 12, 4], [8, 1, 4], [8, 2, 6], [8, 12, 4], [12, 1, 4], [12, 2, 46], [12, 12, 4], [16, 1, 8], [16, 2, 12], [16, 12, 8], [24, 1, 8], [24, 2, 60], [24, 12, 8], [32, 1, 16], [32, 2, 24], [32, 12, 16], [48, 1, 16], [48, 2, 120], [48, 12, 16], [64, 1, 32], [64, 2, 48], [64, 12, 32], [96, 1, 32], [96, 2, 240], [96, 12, 32], [192, 1, 64], [192, 2, 480], [192, 12, 64]], 'center_label': '192.2', 'center_order': 192, 'central_product': True, 'central_quotient': '24.6', 'commutator_count': 1, 'commutator_label': '12.2', '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', '2.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 22, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1536.10843710', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 2, 2], [4, 12, 1, 2], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 5], [6, 12, 2, 2], [8, 1, 4, 1], [8, 2, 2, 3], [8, 12, 2, 2], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 10], [12, 12, 2, 2], [16, 1, 8, 1], [16, 2, 4, 1], [16, 2, 8, 1], [16, 12, 4, 2], [24, 1, 8, 1], [24, 2, 4, 7], [24, 2, 8, 4], [24, 12, 4, 2], [32, 1, 16, 1], [32, 2, 8, 1], [32, 2, 16, 1], [32, 12, 8, 2], [48, 1, 16, 1], [48, 2, 8, 3], [48, 2, 16, 6], [48, 12, 8, 2], [64, 1, 32, 1], [64, 2, 16, 1], [64, 2, 32, 1], [64, 12, 16, 2], [96, 1, 32, 1], [96, 2, 16, 3], [96, 2, 32, 6], [96, 12, 16, 2], [192, 1, 64, 1], [192, 2, 32, 3], [192, 2, 64, 6], [192, 12, 32, 2]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 2236416, 'exponent': 192, 'exponents_of_order': [9, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 256]], 'familial': False, 'frattini_label': '64.50', 'frattini_quotient': '72.48', 'hash': 1280522339947909588, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 12, 1], 'inner_gens': [[37442, 1337164657, 35945290], [1850291, 395398321, 35945290], [37442, 395398321, 35945290]], 'inner_hash': 6, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': False, 'inner_tex': 'D_{12}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 128, 'irrQ_dim': 128, 'irrR_degree': None, 'irrep_stats': [[1, 384], [2, 1056]], 'label': '4608.v', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D24.C96', 'ngens': 3, 'nilpotency_class': -1, 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