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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1536.10766179', 'ambient_counter': 10766179, 'ambient_order': 1536, 'ambient_tex': '(C_2\\times C_{16}).D_{24}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 128, 'counter': 35, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1536.10766179.6.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '6.g1.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': 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': '256.148', 'subgroup_hash': 148, 'subgroup_order': 256, 'subgroup_tex': 'C_8.(C_4\\times C_8)', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1536.10766179', 'aut_centralizer_order': None, 'aut_label': '6.g1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '96.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.g1.a1', '3.a1.a1'], 'contains': ['12.c1.a1', '12.j1.a1', '12.k1.a1'], 'core': '12.c1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [1, 102], 'label': '1536.10766179.6.g1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.g1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '6.g1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.g1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.3', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 16, 'aut_gen_orders': [4, 16, 8, 4, 16, 8, 4], 'aut_gens': [[1, 8, 128], [67, 88, 128], [29, 76, 192], [209, 140, 192], [167, 140, 192], [239, 204, 192], [183, 200, 128], [129, 60, 192]], 'aut_group': None, 'aut_hash': 3087661677154175411, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 128, 'aut_perms': [270867454311828888360787483790186613214151603189774425557378306766010409904294887818845705432141422556315345496383805302343340635455946578390504812955143887208449727443410445869524338459276490635329365794222750807853, 80549687084073649338622180449913440163345580410517727766659672086647189746011340915625206871704915099626473202912123709764835158130355516728195628560332319547166864677754911844566355524294961145743722892927887684148, 64981142853024070616929669017057174490465075974338214411906471707045849819360906758105446512851508114547532392910592418133735298123548649278948354886412947989378625654994312163037143152095588143216534183456949629288, 309841283787189790375714319029135703243422750174856103584436306703886860130106661498558977676294563192586472645604907323894253953246342381026681775295989973869882012928353891315057278649582252281685218181241807487797, 336550008930554385110015950050845871507555912809350692930353570356899455620834896919666173591061210716981504046469626848257530368443808677507414415764433421465581139801324107931556039816251615110464465123354264666995, 320422483010094301301896531163880541177603754961348636409690155638159450710435920427667373396429652584737493307817998903321317110795252392972957680679554924064822616952120028456798194142119134544903378517207387345369, 232388320819902963801378677263145286446096063170800393460024738022726221417642921878036560884074831412291852036219117332908565241465490525492215486956765882469992765035164117957158478328490503105051824958527909176051], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 1, 2], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 8, 1], [8, 1, 8, 1], [8, 2, 2, 4], [8, 2, 4, 1], [16, 2, 16, 2], [16, 8, 16, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2^4\\times C_8).C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.2167', 'autcent_hash': 2167, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_4^2:C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': '32.43', 'autcentquo_hash': 43, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_8:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [4, 1, 4], [4, 2, 10], [8, 1, 8], [8, 2, 12], [16, 2, 32], [16, 8, 16]], 'center_label': '16.5', 'center_order': 16, 'central_product': False, 'central_quotient': '16.7', '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': 148, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 4], [8, 1, 4, 2], [8, 2, 2, 6], [16, 2, 8, 4], [16, 8, 4, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 16, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '64.83', 'frattini_quotient': '4.2', 'hash': 148, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 8, 1], 'inner_gens': [[1, 184, 128], [209, 8, 128], [1, 8, 128]], 'inner_hash': 7, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': False, 'inner_tex': 'D_8', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 56]], 'label': '256.148', 'linC_count': 512, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 24, 'linQ_dim': 18, 'linQ_dim_count': 24, 'linR_count': 160, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C8.(C4*C8)', 'ngens': 2, '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': 19, 'number_characteristic_subgroups': 37, 'number_conjugacy_classes': 88, 'number_divisions': 30, 'number_normal_subgroups': 63, 'number_subgroup_autclasses': 70, 'number_subgroup_classes': 105, 'number_subgroups': 163, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 7], [4, 24], [8, 32], [16, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4, 4], 'outer_gen_pows': [0, 32, 8, 32, 0, 208], 'outer_gens': [[71, 136, 128], [5, 44, 192], [75, 72, 128], [5, 72, 128], [71, 168, 128], [125, 76, 192]], 'outer_group': '256.27523', 'outer_hash': 27523, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 18, 'outer_perms': [386329, 2533146832360344, 1156002822403062, 23449, 400335207535129, 2263778327054255], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3.C_2^5', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 36, 'pgroup': 2, 'primary_abelian_invariants': [4, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 10], [8, 2], [16, 4]], 'representations': {'PC': {'code': 1282631685137669895794745937471, 'gens': [1, 4, 8], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 16, 41, 402, 5891, 91, 4484, 116, 4613, 141]}, 'GLZN': {'d': 2, 'p': 51, 'gens': [3316300, 1724476, 2122432, 132667, 3876, 132664, 120360, 5040776]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [450837, 67566, 63555, 115203, 294921, 557073, 573953, 828161]}, 'Perm': {'d': 36, 'gens': [11021296973019583170724015005769168545250, 22761963387812078347259137570089503217133, 33466822406589806331480597874291250744880, 33762055205689987965519450049263163128263, 44102598855054034474753197282886280951543, 54740819685100883750008755674768618318040, 65085700351349289489925834444376352590760, 75723136061592889557031439900193318075600]}}, 'schur_multiplier': [2, 2], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4, 8], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_8.(C_4\\times C_8)', 'transitive_degree': 64, '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': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 24, 'aut_gen_orders': [2, 4, 4, 4, 8, 24, 12, 12, 12, 8, 12], 'aut_gens': [[1, 2, 16, 32], [1349, 222, 16, 800], [1153, 1154, 16, 176], [1293, 1158, 784, 1512], [385, 10, 16, 1184], [21, 210, 784, 1384], [525, 986, 784, 1000], [325, 222, 784, 40], [473, 202, 784, 232], [1425, 982, 784, 440], [453, 794, 16, 160], [533, 774, 784, 616]], 'aut_group': None, 'aut_hash': 4072702346274684291, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 49152, 'aut_permdeg': 112, 'aut_perms': [35282538380668457301359147336491636369785974166548549798424761844917286540419138113332919007914104268181985028104350281485158699939239751520704259601288147932032306682044699451845942, 43267328183471402879669543964938307722394554471474508397424456141379046310911327839247952077859994651284121439603190261610029950674147334012419350336639660966662854395482185493996032, 193262064503053604209938104751680994178872531873872002269065876152670659949620053183118745740386248888910168128591662428602617153374268063674797365913960311745436431469066138890044192, 57780148058590697417647683966791543202983046044366703358780266556189910521090923409331477462173144116087187151327915205107935953482458644042298669237115367729262802429308759041478471, 90151907709674081243140283427424023999041784269269884797892632005748449884503976222345567195673448994677216863801136906111200663517778909651545896330983083164442106180753553743729206, 10279425075909427758490662897773079216430865299624288571205320173238152403109164195384082662713206787977554771673492641640998161035466135660944916454831645116202371043128018655816250, 98197774431372822593967602479081939913815940801203646312108792373727114541407959958943202827362889281334364865518661141244842657381515849827687758816763850253119435603026162682070329, 163287936603159520195870028562931130467594953288809968355729178527883730593534564818934054701703479119666036219324600980371613413765901267199939990521035800548274283194527760790994753, 141489165295385462381561768835314044042838283311702732827769843179949587886183251204465978007773768375695895567317842341562876190659790962914991459083772928848087662638390492654790343, 91263582129803683881244258785240941264934260910484676884198183491306325987592002603501486984861131755395302149514919709135665355981276986362856109422307117669960922105882934113746796, 192404336908860342454818380582855278834486058527560483436860195818835474639454101846729160701715116323808205064174567417639972897734756348360571994953427127143768780441749978050121192], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 1, 2], [2, 48, 1, 1], [3, 2, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 4, 2], [4, 48, 1, 1], [4, 48, 2, 1], [6, 2, 1, 1], [6, 2, 2, 3], [8, 1, 4, 2], [8, 2, 2, 6], [8, 2, 32, 1], [8, 48, 2, 2], [12, 2, 2, 4], [12, 2, 8, 2], [16, 2, 16, 4], [16, 48, 4, 2], [24, 2, 4, 8], [24, 2, 64, 1], [48, 2, 32, 4]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^2.C_2^6.C_2^6)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.56082', 'autcent_hash': 56082, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '192.1514', 'autcentquo_hash': 1514, 'autcentquo_nilpotent': False, 'autcentquo_order': 192, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{12}:C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 48, 1], [3, 2, 1], [4, 1, 4], [4, 2, 10], [4, 48, 3], [6, 2, 7], [8, 1, 8], [8, 2, 44], [8, 48, 4], [12, 2, 24], [16, 2, 64], [16, 48, 8], [24, 2, 96], [48, 2, 128]], 'center_label': '16.5', 'center_order': 16, 'central_product': False, 'central_quotient': '96.110', 'commutator_count': 1, 'commutator_label': '48.23', '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', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 10766179, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 48, 1, 1], [3, 2, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 4], [4, 48, 1, 3], [6, 2, 1, 3], [6, 2, 2, 2], [8, 1, 4, 2], [8, 2, 2, 6], [8, 2, 4, 8], [8, 48, 2, 2], [12, 2, 2, 4], [12, 2, 4, 4], [16, 2, 8, 8], [16, 48, 4, 2], [24, 2, 4, 8], [24, 2, 8, 8], [48, 2, 16, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 21504, 'exponent': 48, 'exponents_of_order': [9, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '64.83', 'frattini_quotient': '24.14', 'hash': 10766179, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 8, 1, 6], 'inner_gens': [[1, 974, 16, 560], [581, 2, 16, 32], [1, 2, 16, 32], [1041, 2, 16, 32]], 'inner_hash': 110, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': None, 'inner_tex': 'C_2\\times D_{24}', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 376]], 'label': '1536.10766179', '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': '(C2*C16).D24', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 55, 'number_characteristic_subgroups': 131, 'number_conjugacy_classes': 408, 'number_divisions': 84, 'number_normal_subgroups': 161, 'number_subgroup_autclasses': 440, 'number_subgroup_classes': 538, 'number_subgroups': 2234, 'old_label': None, 'order': 1536, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 55], [3, 2], [4, 168], [6, 14], [8, 288], [12, 48], [16, 512], [24, 192], [48, 256]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4, 4, 4, 4, 4, 4, 4, 4], 'outer_gen_pows': [1032, 1032, 8, 0, 0, 0, 1032, 1024], 'outer_gens': [[733, 2, 784, 824], [325, 1178, 16, 48], [269, 594, 784, 1512], [1409, 1154, 784, 184], [1417, 990, 784, 552], [1101, 590, 16, 928], [325, 1178, 16, 48], [325, 222, 784, 40]], 'outer_group': '512.10493095', 'outer_hash': 6090485185678436248, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 128, 'outer_perms': [357114946107248561278594867683203306520271286292039216698874178701871875594060160289185600765978610756166869562409450030790238141527820686446988888850315624593268216174538487492158572800999221449817088287718647015429, 295473191041033428406551301984146808921856181134162854266782610872803054865840898929125189035070710963276375349715482095372141461465142732562014118755327796586984837103996913907707591488725429956926879115419160423548, 313072471094956548401903512105218307000738680490889713416191158527462236635342090481265249926693980767674201141541230974350621115625736889312252050729320595690700949178650119641886447124582685333301662581190331726681, 12374317715081418769248747198911430897698327798077091476973165058638310622538148649401991595244334183688904368682032332936519225354686610928458328118465962439616149414352645918233263616601358140172959857307841085066, 216365287001684296270780666349196760226937788095835568347680669252423143751138352009984609521751328917257293467311429336499095315274541010758856316606353544826905274510800808104909465298809097808725330944111443985090, 195190738665245318053860298337477452635002249957703692815540853617134619850630078338314901553888908644490123360982215470876581604892988430531636519503243683787827495511201355962219931894288290186726823262505610130461, 295473191041033428406551301984146808921856181134162854266782610872803054865840898929125189035070710963276375349715482095372141461465142732562014118755327796586984837103996913907707591488725429956926879115419160423548, 18436326022554677910218958386590044956931792050429418541441052793120667251422645412925300397860894682851949342342305593952618042389846403609066648825253724994305542219758039373584191671263048512913955385594786538752], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3.C_2^6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 39, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 20], [8, 20], [16, 16], [32, 8]], 'representations': {'PC': {'code': '7515572109682409418542953934247070566514542435236567020258434049', 'gens': [1, 2, 5, 6], 'pres': [10, -2, -2, -2, -2, -2, 2, -2, -2, -2, -3, 9600, 19481, 51, 11882, 82, 31043, 33605, 175, 76166, 206, 51207, 237, 115208, 268, 102409]}, 'Perm': {'d': 39, 'gens': [568553198245185592668926237044341106571864425, 1142508515627228489966037607894349568681640080, 1683677055123621834534924619081494913750829064, 2220956930425879810524842533037778542703993600, 2758123892843970031203069840360136999430171280, 2234720683518649569567091420208305637996102400, 3291946184487484550005012448449310226027530880, 3814981435663182197846561489600189604083996160, 1211327281083389705000925733947659476577627520, 3]}}, 'schur_multiplier': [2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times C_{16}).D_{24}', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
