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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1024.dii', 'ambient_counter': 2245, 'ambient_order': 1024, 'ambient_tex': 'C_4:D_8.D_8', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 128, 'counter': 10, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1024.dii.4.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.f1.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': 4, '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': '256.4591', 'subgroup_hash': 4591, 'subgroup_order': 256, 'subgroup_tex': 'C_2\\times D_8:C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1024.dii', 'aut_centralizer_order': 8, 'aut_label': '4.f1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '512.10483304', 'aut_weyl_index': 16, 'centralizer': '128.n1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.c1.a1', '8.j1.a1', '8.l1.a1', '8.r1.a1', '8.r1.b1'], 'core': '8.c1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [3686, -1, 3459, -1, 3680, -1, 3463, -1], 'generators': [16, 942, 744], 'label': '1024.dii.4.f1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '4.f1.a1', 'projective_image': '512.1866', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.f1.a1', 'subgroup_fusion': None, 'weyl_group': '64.128'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 2, 4, 2, 2, 1, 2, 8], 'aut_gens': [[1015839, 909669, 360847, 820097], [1015839, 909669, 688785, 820097], [1015839, 172731, 360847, 820097], [1015839, 909669, 623247, 295041], [1015839, 909669, 360591, 820097], [1015839, 909669, 885647, 819585], [1016351, 909669, 360847, 820097], [1015839, 909669, 360847, 820097], [1015839, 909669, 360975, 820097], [1015839, 910309, 360847, 819585]], 'aut_group': None, 'aut_hash': 28497905344746472, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 2048, 'aut_permdeg': 80, 'aut_perms': [1800560732148413132708404140174791845127315925955176095487847776327353922989490176869464625849984505321960757670308607, 1811885013326619430539861661335506499671452115598780494794540501385250355940577653905784130319597353898583379392681647, 69802727921481516442288714566379043188082055593961718061552090826226048136302739582267537829400857363971724995028376912, 42959827142474288499842649459278512549009484438267753655682555151441312664096724294903031243571848521900398031208060464, 68013201229546526683064147263530166286241439231613067664848102374694596583149882575142529386215652262485384860620952896, 2600169981593980723667011705669101621225747749977110816233621184180121013959814217770038040084275099513164913497013, 0, 28643612703279969512002472413983547345400488247945453552815831579371662325110245511434983847858871329124418715492681536, 29028843128460161322000874120225002423241005101819399268920260093981232670894875879670523199531656577762122824444058176], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 1, 2], [2, 8, 2, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 4, 2], [4, 8, 2, 1], [8, 2, 2, 4], [8, 4, 1, 4], [8, 4, 16, 1], [8, 8, 4, 1], [16, 8, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2.C_2^6.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.206', 'autcent_hash': 206, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_4^2:C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '32.46', 'autcentquo_hash': 46, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 8, 2], [4, 1, 4], [4, 2, 10], [4, 8, 2], [8, 2, 8], [8, 4, 20], [8, 8, 4], [16, 8, 8]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '32.9', 'commutator_count': 1, 'commutator_label': '8.1', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4591, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['128.68', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 8, 1, 2], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 4], [4, 8, 1, 2], [8, 2, 2, 4], [8, 4, 1, 4], [8, 4, 4, 4], [8, 8, 2, 2], [16, 8, 4, 2]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 2688, 'exponent': 16, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '32.3', 'frattini_quotient': '8.5', 'hash': 4591, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [1, 2, 8, 2], 'inner_gens': [[1015839, 909669, 360847, 820097], [1015839, 909669, 360975, 820097], [1015839, 910309, 360847, 819585], [1015839, 909669, 361359, 820097]], 'inner_hash': 9, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'D_4:C_4', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [4, 8]], 'label': '256.4591', 'linC_count': 128, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 16, 'linQ_dim': 9, 'linQ_dim_count': 16, 'linR_count': 16, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D8:C8', 'ngens': 3, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 55, 'number_conjugacy_classes': 64, 'number_divisions': 34, 'number_normal_subgroups': 79, 'number_subgroup_autclasses': 142, 'number_subgroup_classes': 199, 'number_subgroups': 439, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 23], [4, 40], [8, 128], [16, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2, 2], 'outer_gen_pows': [32769, 32769, 32769, 820097, 32769, 32769], 'outer_gens': [[1015839, 909669, 688785, 820097], [1015839, 172731, 360847, 820097], [1015839, 909669, 623247, 295041], [1015839, 909669, 360591, 820097], [1015839, 909669, 885647, 819585], [1016351, 909669, 360847, 820097]], 'outer_group': '64.267', 'outer_hash': 267, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 12, 'outer_perms': [39916800, 362880, 5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 14], [8, 4]], 'representations': {'PC': {'code': 2485932669228251987805585516, 'gens': [1, 2, 3, 6], 'pres': [8, -2, 2, 2, -2, -2, 2, -2, -2, 2842, 66, 1419, 91, 2892, 1941, 141, 166]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [33025, 303403, 32897, 1015839, 294913, 557057, 33281, 909669]}, 'Perm': {'d': 18, 'gens': [382308966971551, 366713871151825, 1, 798430735152288, 819354500754768, 1204687964886840, 1225611938867400, 1604085651118806]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times D_8:C_8', 'transitive_degree': 32, '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': 3, 'aut_exponent': 16, 'aut_gen_orders': [2, 4, 4, 4, 4, 8], 'aut_gens': [[1, 4, 16, 128], [3, 716, 600, 896], [769, 4, 856, 640], [673, 580, 16, 128], [769, 516, 848, 640], [1, 556, 28, 40], [933, 932, 16, 384]], 'aut_group': None, 'aut_hash': 79916486570037805, 'aut_nilpotency_class': 6, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 288, 'aut_perms': [257426493764658284129274345122854960070795564276271665892586829754102032618419818409178327374801993938388254719909728479962211842839740795683281460777906595277103782084614366963810574939290242209817915982296848017259410287552186989054837652092575623755721179140688224155137597478277559810586067913890256293242420730039220686556640523705968984335029462788633678301004334288299368141011162649717844835896015854202628321617459582144131614475541197349165169860143303282368397460239251120169168202674965159595005658427077612558184520727695003568983567679518118494411346891329245968540565526, 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209450987211000848941985001741724864703908088712520368109463789203984096664714719558618382655539294769846254866845235395964357384716195421655545395047936259396410916548938541378848951516829049322291550589984108104366923311786713613344312876531436038421789495768837259027617328543305471813376498393222362259339946896586022165667146638484963142959280295308411763546551802820684437385242846224330216429807098644828733500449753307299765724394542503090474547679769009934929945566928936217045575956202655399561760717007507439253129837261241561071466513781882315495214906762284942124014717499, 514220147590949785404094743327785600368175731743746176854388010427977872243621289978687690581603280895394914409021141872471473594538360953629598369534761461948403851832854813751575101798641135897967206660958898716712369139699996220741907775936643461002614255795938172038892529450167377083507487689494427806876745027765646902842594599093213116124728939690857180941087856650707442540705856114481485356142656577015368494265856886807375572713872302908773331713128338181264094031327141042617562041860967296930421882252421705774484226078278327557762780025009417702529347169965334391176958985, 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213043553545280607970306202270065685155411456843944840529076376752285371036315694876483655551070974488488122606568037191853173555957568011886051642287876236800968450177651341378417236430747504423139385208632724933662728765217008105091302850878235378785234574139585997696802646845711501136888135892676960244706643889631900728302425016923058375927700786385469541708095106948008120359789369147438722617680558165663283869536267988327607299369801183210454577111272990615229058985164917661508540216231931264222383885329028272353412118396612104631229979793039877350195062898529173933860356920], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 32, 1, 3], [4, 4, 1, 4], [4, 8, 1, 1], [4, 32, 1, 1], [4, 64, 1, 1], [4, 64, 4, 1], [8, 8, 1, 4], [8, 8, 4, 1], [8, 16, 2, 1], [8, 32, 2, 1], [16, 32, 4, 1], [16, 32, 8, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2\\times C_4:D_4).C_2^6.C_2', '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': 8, 'autcentquo_group': '2048.cpu', 'autcentquo_hash': 3144402051166287857, 'autcentquo_nilpotent': True, 'autcentquo_order': 2048, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4^2.C_2^5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 32, 3], [4, 4, 4], [4, 8, 1], [4, 32, 1], [4, 64, 5], [8, 8, 8], [8, 16, 2], [8, 32, 2], [16, 32, 12]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '512.1866', 'commutator_count': 1, 'commutator_label': '128.576', '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', '2.1', '2.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 2245, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 32, 1, 3], [4, 4, 1, 4], [4, 8, 1, 1], [4, 32, 1, 1], [4, 64, 1, 1], [4, 64, 2, 2], [8, 8, 1, 4], [8, 8, 4, 1], [8, 16, 2, 1], [8, 32, 2, 1], [16, 32, 2, 2], [16, 32, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 48, 'exponent': 16, 'exponents_of_order': [10], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[16, 1, 2]], 'familial': False, 'frattini_label': '256.2825', 'frattini_quotient': '4.2', 'hash': 2101610243620390082, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 16, 'inner_gen_orders': [4, 4, 8, 4], 'inner_gens': [[1, 556, 28, 40], [105, 4, 752, 384], [933, 932, 16, 384], [745, 772, 784, 128]], 'inner_hash': 8985888352205780980, 'inner_nilpotent': True, 'inner_order': 512, 'inner_split': False, 'inner_tex': 'C_2^3.C_2\\wr C_4', 'inner_used': [1, 2, 3], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 10], [4, 21], [8, 2], [16, 2]], 'label': '1024.dii', 'linC_count': 2, 'linC_degree': 16, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 2, 'linQ_dim': 16, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 16, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C4:D8.D8', 'ngens': 2, 'nilpotency_class': 7, 'nilpotent': True, 'normal_counts': [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': 25, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 43, 'number_divisions': 28, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 279, 'number_subgroup_classes': 313, 'number_subgroups': 2901, 'old_label': None, 'order': 1024, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 103], [4, 376], [8, 160], [16, 384]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 8, 64, 8], 'outer_gens': [[3, 716, 600, 896], [769, 4, 856, 640], [673, 580, 16, 128], [769, 516, 848, 640]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 32, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 9], [8, 8], [16, 3]], 'representations': {'PC': {'code': '33440014478390050552912017046848909401624210676892595164840054554654072041162920565732080483829', 'gens': [1, 3, 5, 8], 'pres': [10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 20, 16682, 14112, 82, 20803, 1404, 2814, 9424, 144, 23525, 18255, 1465, 175, 53766, 20176, 10106, 626, 3207, 35857, 7707, 1967, 237, 5768, 34578, 17308, 4368, 268]}, 'Perm': {'d': 32, 'gens': [195152660219665819875502796339963392, 118943267166205684098373210499907247]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4:D_8.D_8', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
