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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '9216.kh', 'ambient_counter': 268, 'ambient_order': 9216, 'ambient_tex': 'C_2^6.D_6^2', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 2304, 'counter': 26, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '9216.kh.4.H', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.h1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2304.wg', 'subgroup_hash': 2376812632324799625, 'subgroup_order': 2304, 'subgroup_tex': 'S_3\\times C_2^2.\\GL(2,\\mathbb{Z}/4)', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '9216.kh', 'aut_centralizer_order': None, 'aut_label': '4.H', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 3, 'contained_in': None, 'contains': None, 'core': '4.H', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [87178291200, 87091200, 174436452933, 93525425340, 87185940244, 127370880, 120113280, 93405312000, 93405312774, 6310855505], 'label': '9216.kh.4.H', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.H', 'normal_contained_in': [], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '4.h1', 'projective_image': '4608.lt', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.H', 'subgroup_fusion': None, 'weyl_group': '4608.lt'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [12, 12, 12, 12, 6, 6, 4], 'aut_gens': [[93405331342, 127370880, 87265415194, 87178291200, 180623533182, 47537280, 93445607254, 4024352, 174356582400, 40279680], [93405317759, 127370880, 180623882880, 180583635994, 6234664553, 7257600, 93492412343, 83857952, 174356582400, 87091200], [93405317490, 40279680, 180670694400, 180583635994, 6230674284, 7257600, 93532692023, 120113549, 174356582400, 127370880], [93405317759, 40279680, 6354424474, 6227020800, 87261768318, 123379200, 93492418774, 47537549, 174356582400, 87091200], [93405331220, 40279680, 180711006874, 180583603200, 6310506702, 123379200, 93492412343, 47569952, 174356582400, 87091200], [174356601742, 87091200, 6267300480, 6227053594, 180707370203, 47537280, 174483968854, 79866272, 93405312000, 127370880], [93405317490, 127370880, 6314112000, 6227053594, 87178678764, 3991680, 93445600938, 47569952, 174356582400, 40279680], [174356601620, 127370880, 180623915674, 180583603200, 87185948469, 120113280, 174443682743, 43545869, 93405312000, 87091200]], 'aut_group': '2592.jt', 'aut_hash': 7469300832140606591, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 18432, 'aut_permdeg': 102, 'aut_perms': [863854108696358709931404187266613857192582969715322273235312630191278575048057047393651111863155427978752411151965303861646074638307461845484461107894560416184517, 438280911522829317954743952459818038541838331439786558771098729189540598695419864520766064694415088266271340364200873182258612726957436941405793698303822529002751, 697961055710142235025470721588992563653259118349532252866750961358874655539535773014775829591453437234440203974130147429555361160210789792770813445426465112061813, 706296139741222781165341343304932990441306370920110454020068886747407098722534764746159556283649780984020777982752930130282590133489602157265623102544763972663921, 645770976480566489963470444403693600278454577702427004995365959984017964440348260497437750619168040899588170501983001573273387537388588964787327214379725074406773, 595004749848789486179617850642804321975710448038564438361071899449787402011732793944086903698349764680254138082411004866720369968666061713693675755983759689535645, 549440142359742233021681889550291250939845505047832116117194726930016915127614670233421398506953751811459655309574508360072221614560322880728372872227181408851212], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 2, 2, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 4, 1, 1], [2, 6, 1, 2], [2, 6, 2, 2], [2, 9, 2, 1], [2, 12, 1, 2], [2, 18, 1, 1], [2, 18, 2, 1], [2, 36, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 4, 1, 1], [4, 12, 1, 2], [4, 24, 4, 2], [4, 36, 1, 1], [4, 72, 4, 2], [6, 2, 1, 1], [6, 4, 1, 1], [6, 4, 2, 1], [6, 6, 1, 2], [6, 8, 1, 2], [6, 8, 2, 1], [6, 12, 1, 1], [6, 12, 2, 1], [6, 16, 1, 1], [6, 16, 2, 3], [6, 24, 1, 1], [6, 24, 2, 2], [6, 32, 2, 2], [6, 48, 2, 2], [12, 8, 1, 1], [12, 16, 2, 1], [12, 24, 1, 1], [12, 32, 2, 1], [12, 48, 2, 1], [12, 48, 4, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2:(C_2^2\\times F_9)', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '2304.oh', 'autcentquo_hash': 2462485762897143514, 'autcentquo_nilpotent': False, 'autcentquo_order': 2304, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^4:D_6^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 3, 4], [2, 4, 1], [2, 6, 6], [2, 9, 2], [2, 12, 2], [2, 18, 3], [2, 36, 1], [3, 2, 1], [3, 8, 1], [3, 16, 1], [4, 4, 1], [4, 12, 2], [4, 24, 8], [4, 36, 1], [4, 72, 8], [6, 2, 1], [6, 4, 3], [6, 6, 2], [6, 8, 4], [6, 12, 3], [6, 16, 7], [6, 24, 5], [6, 32, 4], [6, 48, 4], [12, 8, 1], [12, 16, 2], [12, 24, 1], [12, 32, 2], [12, 48, 10]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '1152.156550', 'commutator_count': 1, 'commutator_label': '144.193', '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', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 579, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['384.5721', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 3, 1, 4], [2, 4, 1, 1], [2, 6, 1, 6], [2, 9, 1, 2], [2, 12, 1, 2], [2, 18, 1, 3], [2, 36, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 4, 1, 1], [4, 12, 1, 2], [4, 24, 2, 4], [4, 36, 1, 1], [4, 72, 2, 4], [6, 2, 1, 1], [6, 4, 1, 3], [6, 6, 1, 2], [6, 8, 1, 2], [6, 8, 2, 1], [6, 12, 1, 3], [6, 16, 1, 3], [6, 16, 2, 2], [6, 24, 1, 3], [6, 24, 2, 1], [6, 32, 1, 2], [6, 32, 2, 1], [6, 48, 1, 2], [6, 48, 2, 1], [12, 8, 1, 1], [12, 16, 2, 1], [12, 24, 1, 1], [12, 32, 2, 1], [12, 48, 2, 5]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 161280, 'exponent': 12, 'exponents_of_order': [8, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 1]], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '288.1028', 'hash': 2376812632324799625, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 2, 2, 2, 4, 3, 6, 6, 3, 2], 'inner_gens': [[93405331342, 127370880, 6314144794, 6227020800, 87218221182, 47537280, 93445600938, 4024352, 174356582400, 40279680], [93405331342, 127370880, 87265415194, 87178291200, 180583979262, 123379200, 93445607254, 79866272, 174356582400, 40279680], [174356601742, 127370880, 87265415194, 87178291200, 6227396862, 3991680, 174396877654, 47569952, 93405312000, 40279680], [174356601742, 127370880, 87265415194, 87178291200, 6266950782, 47537280, 174396877654, 4024352, 93405312000, 40279680], [174356601742, 87091200, 6354424474, 6227020800, 180623533182, 120113280, 174396877916, 83825549, 93405312000, 40279680], [93405331342, 40279680, 87305694874, 87178291200, 180591236862, 47537280, 93492418774, 123411872, 174356582400, 87091200], [93405317490, 127370880, 6314144794, 6227020800, 87218232684, 79833600, 93445607254, 123411872, 174356582400, 40279680], [93405331342, 40279680, 87305694874, 87178291200, 180667090523, 79833600, 93492418774, 4024352, 174356582400, 87091200], [93405331342, 127370880, 180670727194, 180583603200, 6266950782, 47537280, 93445607254, 4024352, 174356582400, 40279680], [93405331342, 127370880, 87265415194, 87178291200, 180623533182, 79833600, 93445607254, 123411872, 174356582400, 40279680]], 'inner_hash': 1579924085759091414, 'inner_nilpotent': False, 'inner_order': 1152, 'inner_split': True, 'inner_tex': 'D_6.\\GL(2,\\mathbb{Z}/4)', 'inner_used': [1, 2, 3, 4, 5, 6, 7], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 16], [2, 28], [3, 16], [4, 16], [6, 12], [8, 3], [12, 4], [24, 1]], 'label': '2304.wg', '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': 'S3*C2^2.GL(2,Z/4)', '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, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 57, 'number_characteristic_subgroups': 85, 'number_conjugacy_classes': 96, 'number_divisions': 75, 'number_normal_subgroups': 131, 'number_subgroup_autclasses': 1876, 'number_subgroup_classes': 3366, 'number_subgroups': 42208, 'old_label': None, 'order': 2304, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 191], [3, 26], [4, 832], [6, 646], [12, 608]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[93405331342, 87091200, 87305694874, 87178291200, 180623533182, 120113280, 93445607254, 83857952, 174356582400, 40279680], [174356601742, 127370880, 87265382400, 87178323994, 6266950782, 47537280, 174396877654, 4024352, 93405312000, 40279680], [93405331342, 127370880, 87265415194, 87178291200, 180623546701, 47537280, 93445600938, 3991949, 174356582400, 40279680], [93405317759, 87091200, 87305694874, 87178291200, 180623534718, 120113280, 93445600823, 83857952, 174356582400, 40279680]], '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': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [3, 8], [4, 14], [6, 12], [8, 5], [12, 6], [16, 1], [24, 1]], 'representations': {'PC': {'code': '5123629597591496725216672616139309254408258528952717534945839613233177495184556303393', 'gens': [1, 3, 4, 6, 7, 8], 'pres': [10, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 20, 34682, 38083, 113, 4004, 8645, 755, 585, 1296, 256, 169287, 53777, 21637, 1727, 237, 172808, 21638, 268, 153609, 19239]}, 'Perm': {'d': 15, 'gens': [93405331342, 127370880, 87265415194, 87178291200, 180623533182, 47537280, 93445607254, 4024352, 174356582400, 40279680]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_2^2.\\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 72, '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': '32.51', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 6, 4, 6, 4, 6, 6, 2], 'aut_gens': [[174479997767, 87178291200, 180710620679, 47537280, 180663800454, 40279680, 127370880, 174444070748, 174356582400, 6266937660, 174403756854, 7620480], [174440443887, 6227057733, 87258521924, 7257600, 87305335407, 127370880, 40279680, 174403774564, 174356582400, 180587232774, 174444073407, 7620480], [174363840734, 87178291200, 180671080748, 120113280, 180627512414, 40279680, 87091200, 174483609490, 174356582400, 6266973767, 174476369687, 123779013], [93525462153, 6227057733, 180663833924, 43545600, 180671094153, 127370880, 40279680, 93452492159, 93405312000, 87181920774, 93532320014, 83462400], [174436416774, 6227020800, 87258502145, 123379200, 87222235847, 127370880, 87091200, 174403779505, 174356582400, 180707382153, 174476333580, 362880], [93448894473, 180583640133, 87225489025, 120113280, 87265745340, 127370880, 40279680, 93485522999, 93405312000, 6350799047, 93532320014, 7657413], [174360610953, 87178291200, 180710630284, 79833600, 180630814473, 40279680, 127370880, 174444050999, 174356582400, 6266938334, 174436815767, 83499333], [174440443847, 180583640133, 6270957844, 120113280, 6274231367, 87091200, 127370880, 174476367068, 174356582400, 87185947887, 174436815093, 39916800], [93528691974, 87178291200, 6354043265, 47537280, 6307253447, 40279680, 127370880, 93492788785, 93405312000, 180623556873, 93452522553, 7620480]], 'aut_group': '10000.cc', 'aut_hash': 1661811110069203530, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 221184, 'aut_permdeg': 38, 'aut_perms': [11920328761505811191037570362282908297726359, 131500559726640109891518078537954974045899480, 30878447328711892373216177373263205530426748, 110814542996264502738751999943069909051770897, 350847747060892247057800558052114897750434708, 72405279747181096056860644573279511657142471, 478839317402233426261072348262438188345465416, 5696882116915019331809084662062325838310], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 4, 2, 1], [2, 4, 3, 1], [2, 6, 2, 1], [2, 6, 3, 2], [2, 9, 2, 1], [2, 12, 2, 2], [2, 12, 3, 3], [2, 18, 2, 1], [2, 18, 3, 1], [2, 24, 2, 1], [2, 24, 3, 1], [2, 36, 2, 1], [2, 36, 3, 2], [2, 72, 2, 1], [2, 72, 3, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 4, 3, 1], [4, 6, 2, 1], [4, 8, 3, 1], [4, 12, 3, 3], [4, 18, 2, 1], [4, 24, 2, 1], [4, 24, 3, 5], [4, 36, 3, 2], [4, 48, 3, 2], [4, 72, 2, 1], [4, 72, 3, 4], [4, 144, 3, 2], [6, 2, 1, 1], [6, 4, 3, 1], [6, 6, 1, 2], [6, 8, 1, 1], [6, 8, 2, 1], [6, 8, 3, 1], [6, 12, 2, 1], [6, 12, 3, 1], [6, 16, 1, 1], [6, 16, 3, 1], [6, 24, 2, 2], [6, 24, 3, 2], [6, 32, 2, 1], [6, 32, 3, 2], [6, 48, 2, 1], [6, 48, 3, 2], [6, 64, 2, 1], [6, 64, 3, 1], [6, 96, 2, 1], [6, 96, 3, 1], [12, 8, 3, 1], [12, 12, 2, 1], [12, 16, 3, 1], [12, 24, 3, 2], [12, 32, 3, 1], [12, 48, 2, 1], [12, 48, 3, 4], [12, 64, 3, 2], [12, 96, 3, 3], [12, 128, 3, 1], [12, 192, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^4:D_4:C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '6912.go', 'autcentquo_hash': 530348522389483268, 'autcentquo_nilpotent': False, 'autcentquo_order': 6912, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_6\\times S_4^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 3, 4], [2, 4, 5], [2, 6, 8], [2, 9, 2], [2, 12, 13], [2, 18, 5], [2, 24, 5], [2, 36, 8], [2, 72, 5], [3, 2, 1], [3, 8, 1], [3, 16, 1], [4, 4, 3], [4, 6, 2], [4, 8, 3], [4, 12, 9], [4, 18, 2], [4, 24, 17], [4, 36, 6], [4, 48, 6], [4, 72, 14], [4, 144, 6], [6, 2, 1], [6, 4, 3], [6, 6, 2], [6, 8, 6], [6, 12, 5], [6, 16, 4], [6, 24, 10], [6, 32, 8], [6, 48, 8], [6, 64, 5], [6, 96, 5], [12, 8, 3], [12, 12, 2], [12, 16, 3], [12, 24, 6], [12, 32, 3], [12, 48, 14], [12, 64, 6], [12, 96, 9], [12, 128, 3], [12, 192, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '4608.lt', 'commutator_count': 1, 'commutator_label': '288.1041', 'complements_known': False, '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', '3.1', '3.1'], 'composition_length': 12, 'conjugacy_classes_known': True, 'counter': 268, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['24.12', 1], ['6.1', 1], ['64.138', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 3, 1, 4], [2, 4, 1, 5], [2, 6, 1, 8], [2, 9, 1, 2], [2, 12, 1, 13], [2, 18, 1, 5], [2, 24, 1, 5], [2, 36, 1, 8], [2, 72, 1, 5], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 4, 1, 3], [4, 6, 1, 2], [4, 8, 1, 3], [4, 12, 1, 9], [4, 18, 1, 2], [4, 24, 1, 17], [4, 36, 1, 6], [4, 48, 1, 6], [4, 72, 1, 14], [4, 144, 1, 6], [6, 2, 1, 1], [6, 4, 1, 3], [6, 6, 1, 2], [6, 8, 1, 6], [6, 12, 1, 5], [6, 16, 1, 4], [6, 24, 1, 10], [6, 32, 1, 8], [6, 48, 1, 8], [6, 64, 1, 5], [6, 96, 1, 5], [12, 8, 1, 3], [12, 12, 1, 2], [12, 16, 1, 3], [12, 24, 1, 6], [12, 32, 1, 3], [12, 48, 1, 14], [12, 64, 1, 6], [12, 96, 1, 9], [12, 128, 1, 3], [12, 192, 1, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 12, 'exponents_of_order': [10, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 4]], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '1152.157863', 'hash': 8684473499629194244, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 2, 4, 3, 4, 2, 2, 12, 3, 2, 12, 2], 'inner_gens': [[174479997767, 180583603200, 6346781939, 3991680, 6314475654, 87091200, 40279680, 174400525148, 174356582400, 87185911740, 174483590454, 3628800], [93528727367, 87178291200, 6354038279, 47537280, 6307218054, 40279680, 127370880, 93492800348, 93405312000, 180623520060, 93452486454, 7620480], [93525426054, 6227020800, 180710620679, 7257600, 180627547847, 40279680, 87091200, 93492799004, 93405312000, 87178654140, 93525062414, 3628800], [174436452167, 87178291200, 180703363079, 47537280, 180710611974, 87091200, 40279680, 174400525148, 174356582400, 6234641340, 174444036534, 123742080], [93412605767, 6227020800, 180703364339, 43545600, 180663800454, 87091200, 127370880, 93449254748, 93405312000, 87261753660, 93452486454, 39916800], [174360610247, 87178291200, 180710620679, 79833600, 180630778374, 40279680, 127370880, 174444070748, 174356582400, 6266937660, 174436778934, 83462400], [174404155847, 87178291200, 180671066759, 123379200, 180663800454, 40279680, 127370880, 174483624668, 174356582400, 6227383740, 174403756854, 83462400], [174400164167, 180583603200, 6354043305, 83825280, 6346771974, 40279680, 87091200, 174444070748, 174356582400, 87178690953, 174400491600, 123742080], [174479997767, 180583603200, 6354038279, 47537280, 6307218054, 40279680, 127370880, 174444070748, 174356582400, 87218208060, 174403756854, 7620480], [93412605767, 180583603200, 87265754759, 120113280, 87222200454, 40279680, 87091200, 93532342705, 93405312000, 6266937660, 93525062454, 123742080], [174400164167, 180583603200, 6270938699, 7257600, 6307218054, 87091200, 127370880, 174476365724, 174356582400, 87185911740, 174403756854, 362880], [174400164167, 87178291200, 180630787079, 120113280, 180671058054, 127370880, 40279680, 174436813148, 174356582400, 6350762940, 174483590454, 7620480]], 'inner_hash': 7796095685647803946, 'inner_nilpotent': False, 'inner_order': 4608, 'inner_split': None, 'inner_tex': 'C_2^5.D_6^2', 'inner_used': [1, 2, 3, 5, 8, 11], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 32], [2, 56], [3, 32], [4, 40], [6, 40], [8, 14], [12, 20], [16, 2], [24, 4]], 'label': '9216.kh', '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': 'C2^6.D6^2', 'ngens': 12, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 96, 'number_characteristic_subgroups': 89, 'number_conjugacy_classes': 240, 'number_divisions': 240, 'number_normal_subgroups': 945, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 9216, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 1119], [3, 26], [4, 2976], [6, 1878], [12, 3216]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 3], 'outer_gen_pows': [87109068, 0, 93405312060, 174360574080, 0], 'outer_gens': [[93525462153, 6227057733, 180663833924, 43545600, 180671094153, 127370880, 40279680, 93452492159, 93405312000, 87181920774, 93532320014, 83462400], [174440443847, 87178328133, 180663809159, 7257600, 180710647367, 127370880, 40279680, 174403791068, 174356582400, 6230686473, 174444073367, 7620480], [174363840774, 180583603200, 6314484359, 120113280, 6270930054, 40279680, 87091200, 174483613849, 174356582400, 87218244873, 174476369727, 123779013], [174436416774, 6227020800, 87258502145, 123379200, 87222235847, 127370880, 87091200, 174403779505, 174356582400, 180707382153, 174476333580, 362880], [174479961660, 180583603200, 6354046570, 47537280, 6307254153, 40279680, 127370880, 174444050999, 174356582400, 87218244207, 174403792953, 7620480]], 'outer_group': '48.51', 'outer_hash': 51, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 24, 'outer_perms': [9201488537217066197096, 564180177037686258498690, 438114961659895570516472, 88286689152226335980259, 204246273305494220017063], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_6', 'pc_rank': None, 'perfect': False, 'permutation_degree': None, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 56], [3, 32], [4, 40], [6, 40], [8, 14], [12, 20], [16, 2], [24, 4]], 'representations': {'PC': {'code': '36321656101395025653574854187727635643665740686120276878459457957369872249061871714277849401522948280349662587012373866411785790394054863769073967279305344', 'gens': [1, 2, 4, 6, 8, 10, 11, 12], 'pres': [12, -2, -2, -3, -2, -3, -2, -2, -2, -2, 2, 2, 2, 14112, 166705, 61, 62498, 5210, 28227, 181167, 103851, 135, 1456, 98501, 298097, 22061, 22073, 209, 5082, 373267, 214303, 5059, 655, 283, 518421, 224673, 380170, 304150, 209122, 31726, 5350, 912395, 456215, 103715, 62255, 10439]}, 'Perm': {'d': 15, 'gens': [174479997767, 87178291200, 180710620679, 47537280, 180663800454, 40279680, 127370880, 174444070748, 174356582400, 6266937660, 174403756854, 7620480]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 16, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^6.D_6^2', 'transitive_degree': None, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
