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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '15552.dp', 'ambient_counter': 94, 'ambient_order': 15552, 'ambient_tex': 'C_6^4.D_6', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 432, 'counter': 135, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '15552.dp.12.ck1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.ck1', 'outer_equivalence': True, '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': 12, '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': '1296.782', 'subgroup_hash': 782, 'subgroup_order': 1296, 'subgroup_tex': 'C_6^2:C_{36}', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '15552.dp', 'aut_centralizer_order': None, 'aut_label': '12.ck1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2592.B', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.f1', '6.o1'], 'contains': ['24.bh1', '36.g1', '36.hv1', '48.bk1'], 'core': '36.g1', 'coset_action_label': None, 'count': 2, 'diagramx': [1255, -1, 1475, -1], 'generators': [108, 1296, 7776, 9268, 144, 6048, 5184, 216], 'label': '15552.dp.12.ck1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.f1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1', 'old_label': '12.ck1', 'projective_image': '7776.dy', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.ck1', 'subgroup_fusion': None, 'weyl_group': '1296.3016'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '36.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 12, 6, 6], 'aut_gens': [[1, 36, 216], [67, 720, 1260], [131, 792, 468], [883, 684, 1044], [571, 828, 612]], 'aut_group': None, 'aut_hash': 1651852858775664145, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7776, 'aut_permdeg': 90, 'aut_perms': [955870206128153971032954413052878451770674660808598982722608379443890112298145817866621620941649674429039400540669144508060458330487206741, 91149202061334668788412289613422677821758485530326754023079214243446426765892673239258860730318517410041184499578657587672604600838088302, 979976613115722955996039483091762287256664508433237249211663497441621770575679120850667855217058656770938812706937004520060787848984747353, 602274049376973829757724145598651821056651706194259696881625752394290232613974525522895832455099038807087421991347943504769673017487856406], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [4, 9, 2, 1], [4, 27, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2], [6, 6, 1, 3], [6, 6, 2, 3], [6, 6, 3, 2], [6, 6, 6, 2], [9, 12, 6, 1], [9, 24, 6, 1], [12, 9, 4, 1], [12, 27, 4, 1], [18, 12, 6, 1], [18, 24, 6, 1], [36, 36, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times (A_4\\times \\He_3).C_6.C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '1296.2931', 'autcentquo_hash': 2931, 'autcentquo_nilpotent': False, 'autcentquo_order': 1296, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2:S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 1, 2], [3, 2, 3], [3, 6, 3], [4, 9, 2], [4, 27, 2], [6, 1, 2], [6, 2, 3], [6, 3, 4], [6, 6, 27], [9, 12, 6], [9, 24, 6], [12, 9, 4], [12, 27, 4], [18, 12, 6], [18, 24, 6], [36, 36, 12]], 'center_label': '6.2', 'center_order': 6, 'central_product': False, 'central_quotient': '216.99', 'commutator_count': 1, 'commutator_label': '36.14', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 782, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [4, 9, 2, 1], [4, 27, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2], [6, 6, 1, 9], [6, 6, 2, 9], [9, 12, 6, 1], [9, 24, 6, 1], [12, 9, 4, 1], [12, 27, 4, 1], [18, 12, 6, 1], [18, 24, 6, 1], [36, 36, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 36, 'exponents_of_order': [4, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 6]], 'familial': False, 'frattini_label': '18.5', 'frattini_quotient': '72.44', 'hash': 782, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [6, 6, 6], 'inner_gens': [[1, 828, 612], [721, 36, 216], [1117, 36, 216]], 'inner_hash': 99, 'inner_nilpotent': False, 'inner_order': 216, 'inner_split': True, 'inner_tex': 'C_6^2:C_6', 'inner_used': [1, 2, 3], 'irrC_degree': 6, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 36], [2, 18], [3, 12], [6, 30]], 'label': '1296.782', 'linC_count': 6, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 17, 'linQ_dim': 12, 'linQ_dim_count': 3, 'linR_count': 18, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6^2:C36', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 33, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 96, 'number_divisions': 41, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 162, 'number_subgroup_classes': 206, 'number_subgroups': 1127, 'old_label': None, 'order': 1296, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 7], [3, 26], [4, 72], [6, 182], [9, 216], [12, 144], [18, 216], [36, 432]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[5, 36, 1116], [667, 36, 360]], 'outer_group': '36.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 36, 'outer_permdeg': 8, 'outer_perms': [750, 10081], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times S_3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [4, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 5], [3, 2], [4, 3], [6, 15], [12, 14]], 'representations': {'PC': {'code': 1995897511311534382533137579130860773877957277036605888309, 'gens': [1, 5, 7], 'pres': [8, 2, 2, 3, 3, 2, 3, 2, 3, 16, 41, 90, 33124, 15852, 6860, 116, 6917, 34278, 11102, 12622, 166, 64519, 18447, 8087]}, 'Perm': {'d': 26, 'gens': [104636688894360458035200, 16829874681133931649037623, 44460928512000, 98356, 482630400, 1037836800, 1891795534781531258880000, 33612173726610141831168000]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [36], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2:C_{36}', 'transitive_degree': 36, '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': '24.15', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [12, 6, 6, 18, 2, 18], 'aut_gens': [[1, 2, 12, 432, 2592], [6337, 2954, 2940, 14256, 8640], [3745, 11234, 14124, 2160, 12960], [14569, 866, 9564, 5616, 14256], [15529, 9650, 1812, 4752, 2160], [11329, 5402, 204, 5616, 14256], [8041, 3818, 12900, 4752, 2160]], 'aut_group': None, 'aut_hash': 4504041920383935304, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 31104, 'aut_permdeg': 128, 'aut_perms': [110893626837188355482891836783057133078766191823963720636982128101953321753655959822326990827218700143264179526129819529498628497684154683970666680630259560524262414452904899267861329725104176507670791640385676956805, 174270734476654264890971783882038927390937389017701427415790163557696652360603983544270145427873271534695058501427803686530529837278094796316278120365760015897317559542357212072128949436812337837178906404579986671346, 38538294357761333637547472318369254507919889978474078259870737649367239191539750343719538295618334639541158103538772724553516810211650470224035019870629419190576656946902205529432448316213611668368233671207071390006, 249492816706460950289842070108406455163975019208147389268807986880850020902619787140593168426946123654057461161825875807874332835962938936125604287073405909253918176779221170968416033811288748723118100332169909775412, 101136715229109793573007899301773428528451871725229604241870739711119797423611644025977164282418102015094110952669291497148967880208326464484108674000363200703187984523635776275053687638235448229380918257708969561776, 190611978506884363896981877081866185078732761322049873062036538261227397858321879651065213331634266974068224304474839201875398977170839418469664979047950267622444707315059483649643708662047950334076601032847378348604], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 6, 1, 1], [2, 18, 1, 1], [2, 54, 1, 1], [2, 54, 2, 1], [2, 108, 1, 2], [3, 2, 1, 2], [3, 3, 1, 2], [3, 4, 1, 1], [3, 6, 1, 5], [3, 12, 1, 3], [4, 18, 1, 1], [4, 54, 1, 1], [4, 54, 2, 1], [4, 108, 1, 4], [6, 2, 1, 2], [6, 3, 1, 6], [6, 4, 1, 3], [6, 4, 2, 1], [6, 6, 1, 27], [6, 6, 2, 6], [6, 12, 1, 41], [6, 12, 2, 18], [6, 36, 1, 1], [6, 54, 1, 4], [6, 54, 2, 2], [6, 108, 1, 7], [6, 108, 2, 6], [6, 216, 1, 3], [9, 72, 1, 3], [9, 144, 1, 3], [12, 36, 1, 1], [12, 54, 1, 4], [12, 54, 2, 2], [12, 108, 1, 11], [12, 108, 2, 6], [12, 216, 1, 9], [18, 72, 1, 3], [18, 144, 1, 6], [18, 144, 2, 3], [18, 432, 1, 3], [36, 432, 1, 3]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_6^2.C_3^3.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': '3888.fl', 'autcentquo_hash': 3044566718417071494, 'autcentquo_nilpotent': False, 'autcentquo_order': 3888, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^3.(C_6\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 3, 2], [2, 6, 1], [2, 18, 1], [2, 54, 3], [2, 108, 2], [3, 2, 2], [3, 3, 2], [3, 4, 1], [3, 6, 5], [3, 12, 3], [4, 18, 1], [4, 54, 3], [4, 108, 4], [6, 2, 2], [6, 3, 6], [6, 4, 5], [6, 6, 39], [6, 12, 77], [6, 36, 1], [6, 54, 8], [6, 108, 19], [6, 216, 3], [9, 72, 3], [9, 144, 3], [12, 36, 1], [12, 54, 8], [12, 108, 23], [12, 216, 9], [18, 72, 3], [18, 144, 12], [18, 432, 3], [36, 432, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '7776.dy', 'commutator_count': 1, 'commutator_label': '648.319', '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', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 94, '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, 1], [2, 3, 1, 2], [2, 6, 1, 1], [2, 18, 1, 1], [2, 54, 1, 3], [2, 108, 1, 2], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 1, 1], [3, 6, 2, 2], [3, 12, 1, 1], [3, 12, 2, 1], [4, 18, 1, 1], [4, 54, 1, 3], [4, 108, 1, 4], [6, 2, 1, 2], [6, 3, 2, 3], [6, 4, 1, 5], [6, 6, 1, 7], [6, 6, 2, 16], [6, 12, 1, 11], [6, 12, 2, 33], [6, 36, 1, 1], [6, 54, 2, 4], [6, 108, 1, 3], [6, 108, 2, 8], [6, 216, 1, 1], [6, 216, 2, 1], [9, 72, 1, 1], [9, 72, 2, 1], [9, 144, 1, 1], [9, 144, 2, 1], [12, 36, 1, 1], [12, 54, 2, 4], [12, 108, 1, 3], [12, 108, 2, 10], [12, 216, 1, 3], [12, 216, 2, 3], [18, 72, 1, 1], [18, 72, 2, 1], [18, 144, 1, 2], [18, 144, 2, 5], [18, 432, 1, 1], [18, 432, 2, 1], [36, 432, 1, 1], [36, 432, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': None, 'exponent': 36, 'exponents_of_order': [6, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 16], [12, 1, 2]], 'familial': False, 'frattini_label': '18.5', 'frattini_quotient': '864.4690', 'hash': 154246664385077618, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [12, 6, 18, 6, 6], 'inner_gens': [[1, 14834, 5676, 14256, 8640], [3745, 2, 10308, 2160, 12960], [14641, 8282, 12, 2592, 4752], [4321, 866, 13404, 432, 2592], [12097, 5186, 444, 432, 2592]], 'inner_hash': 1798508750155658558, 'inner_nilpotent': False, 'inner_order': 7776, 'inner_split': True, 'inner_tex': 'C_6^3.S_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 24], [2, 42], [3, 24], [4, 15], [6, 70], [12, 86]], 'label': '15552.dp', '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': None, 'name': 'C6^4.D6', 'ngens': 11, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 215, 'number_characteristic_subgroups': 118, 'number_conjugacy_classes': 261, 'number_divisions': 165, 'number_normal_subgroups': 126, 'number_subgroup_autclasses': 4900, 'number_subgroup_classes': 5948, 'number_subgroups': 123570, 'old_label': None, 'order': 15552, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 411], [3, 80], [4, 612], [6, 4368], [9, 648], [12, 4896], [18, 3240], [36, 1296]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[217, 2, 12, 432, 2592], [5185, 5186, 12, 2160, 12960]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [3, 8], [4, 19], [6, 30], [8, 6], [12, 42], [24, 34]], 'representations': {'PC': {'code': '1026443151632060123927052002535544065244452809555291157660508939457902085553996039225111557807986862058839821704878857460329710861608677926835344895952995998490464982369590193069349173', 'gens': [1, 2, 4, 8, 10], 'pres': [11, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3, 163196, 326349, 56, 9638, 249747, 226790, 3325, 124, 244204, 388755, 2336, 158, 500549, 106144, 5571, 258, 22182, 1254535, 95058, 19048, 17475, 23822, 260, 1026440, 85555, 42809, 28564, 21447, 950409, 712820, 43602, 9953, 30754, 328, 209098, 627285, 69739, 8766, 28379]}, 'Perm': {'d': 26, 'gens': [32988353322738031068141488, 16211572098992913983171889, 47231787976459017058648937]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 64, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^4.D_6', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
