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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '18144.f', 'ambient_counter': 6, 'ambient_order': 18144, 'ambient_tex': 'C_3^2:D_6\\times \\GL(3,2)', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 54, 'counter': 14, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '18144.f.14.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '14.b1', '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': 14, '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': '1296.3005', 'subgroup_hash': 3005, 'subgroup_order': 1296, 'subgroup_tex': 'C_2\\times \\He_3:S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '18144.f', 'aut_centralizer_order': None, 'aut_label': '14.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '3024.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['7.a1'], 'contains': ['28.b1', '28.e1', '42.e1', '42.f1', '42.g1', '56.c1'], 'core': '336.a1', 'coset_action_label': None, 'count': 14, 'diagramx': [5927, -1, 5246, -1], 'generators': [108549439111854, 31734726, 14641230, 87676284270, 1, 4843440, 186838367095, 21103405120104], 'label': '18144.f.14.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1', 'old_label': '14.b1', 'projective_image': '3024.bc', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '14.b1', 'subgroup_fusion': None, 'weyl_group': '432.746'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [3, 4, 6, 8, 2, 4, 2, 3, 3, 6, 2, 12], 'aut_gens': [[1, 2, 6, 36, 216], [3, 2, 6, 900, 216], [1, 506, 6, 1048, 216], [1247, 868, 114, 468, 1080], [381, 1010, 654, 542, 1080], [1, 144, 6, 110, 1080], [1, 940, 6, 472, 216], [19, 1010, 762, 180, 1080], [1, 72, 6, 614, 216], [507, 866, 6, 900, 216], [85, 74, 30, 792, 972], [1, 4, 6, 180, 216], [17, 508, 6, 256, 972]], 'aut_group': '20736.mt', 'aut_hash': 370528852606498682, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20736, 'aut_permdeg': 108, 'aut_perms': [112804936918561285719297247094534528322315124392721023174041220870904152310611119425168459748563227849033762655057414721229270355280761947529931902162661424200239089289439848, 2416944541952711094164634971231335884587997117908480819935750411569058724569054952675334478261251421533326095037600882674390288027285578671273749353642910670143384186214400, 827364305542744681114128625379183683871305259234664947194122097457217965174381783980901911750779889133577383208371083955305667279696023084942958620287004491832539573330880969, 835758124137117059620933776221797971665682396811053799068282432128703550448558151614853766010273934539787992556538144261752303261857472678150796226156142899621141356834571923, 48673296412153189378816931208406917351640588095664634187975135775852999519428902703077148096973597624833032185379370078579874823953718101207751076246936854004291964154424, 1146291953051031554453188821371105809130781414468975483837532768036685935310676940736058874865674490402177509166348126404381358423870683124619804718325007591945965625742086, 475341300312437450186834934271163104979058807028957886270106618969286113094534344680605691987368364158603389792784180474768764101021433342687688711524310673665504519810318073, 21638353419978861787144912606679926168865191548672797613269924436796957582865077934805345624361115793663409051238826417439858028418271965315817293406923475647686082727686, 250578243865010161050709791112706820798648756750846001283539697445254557778879678187732063287230011198871051534553466730458476182392930938997743404882485243706806706029888310, 817393912306197048326639320781472067343669246369368573190245965833092934031643098814527335939688553664718864641643789826854805532570022292955997045265257384607047918296494098, 2303397784269919144982698285600972657323859275618417399097954238934189676319572659852296454322973863657933720466654188022657035299660808916162051644089209225948526712999686, 356743368151784017634454593674616115214009890145609511908826221135633231994448975429547109257780051853346148764371342049704956915114911673453191169349021758631708502042701110], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 54, 2, 1], [3, 1, 2, 1], [3, 6, 4, 1], [3, 8, 1, 1], [3, 8, 2, 1], [3, 24, 8, 1], [4, 54, 2, 1], [6, 1, 2, 1], [6, 3, 2, 2], [6, 6, 4, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 18, 4, 2], [6, 24, 8, 1], [6, 54, 4, 1], [12, 54, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2:(D_6\\times \\GL(2,3))', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '10368.qn', 'autcentquo_hash': 3718250448986069408, 'autcentquo_nilpotent': False, 'autcentquo_order': 10368, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [2, 54, 2], [3, 1, 2], [3, 6, 4], [3, 8, 3], [3, 24, 8], [4, 54, 2], [6, 1, 2], [6, 3, 4], [6, 6, 4], [6, 8, 3], [6, 18, 8], [6, 24, 8], [6, 54, 4], [12, 54, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '216.165', 'commutator_count': 2, 'commutator_label': '324.130', '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': 3005, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['648.575', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 54, 1, 2], [3, 1, 2, 1], [3, 6, 1, 4], [3, 8, 1, 1], [3, 8, 2, 1], [3, 24, 1, 8], [4, 54, 1, 2], [6, 1, 2, 1], [6, 3, 2, 2], [6, 6, 1, 4], [6, 8, 1, 1], [6, 8, 2, 1], [6, 18, 1, 8], [6, 24, 1, 8], [6, 54, 2, 2], [12, 54, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 62690355, 'exponent': 12, 'exponents_of_order': [4, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[9, 0, 4]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '432.762', 'hash': 3005, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3, 3, 6, 2], 'inner_gens': [[1, 4, 30, 828, 216], [5, 2, 6, 468, 216], [13, 2, 6, 684, 972], [721, 866, 654, 36, 216], [1, 2, 762, 36, 216]], 'inner_hash': 165, 'inner_nilpotent': False, 'inner_order': 216, 'inner_split': True, 'inner_tex': 'C_3^2:S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 9, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 4], [2, 26], [3, 12], [6, 12], [9, 8]], 'label': '1296.3005', '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': True, 'name': 'C2*He3:S4', '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': 22, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 62, 'number_divisions': 52, 'number_normal_subgroups': 77, 'number_subgroup_autclasses': 162, 'number_subgroup_classes': 623, 'number_subgroups': 6770, 'old_label': None, 'order': 1296, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 115], [3, 242], [4, 108], [6, 614], [12, 216]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [3, 2, 2, 2, 4, 4], 'outer_gen_pows': [1, 0, 0, 0, 0, 0], 'outer_gens': [[1, 4, 30, 1262, 216], [19, 4, 6, 36, 1080], [19, 2, 30, 684, 216], [19, 2, 6, 36, 216], [19, 1010, 6, 614, 216], [1, 580, 6, 902, 216]], 'outer_group': '96.189', 'outer_hash': 189, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 10, 'outer_perms': [127728, 929784, 1, 889561, 1816087, 2434344], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times \\GL(2,3)', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 26], [3, 4], [6, 12], [12, 2], [18, 4]], 'representations': {'PC': {'code': 6148329798653481561174274377287738013122540921806640213, 'gens': [1, 2, 3, 5, 7], 'pres': [8, 2, 3, 2, 3, 2, 3, 2, 3, 65, 722, 66, 771, 33124, 9372, 4580, 2668, 116, 6917, 22477, 9094, 1542, 166]}, 'Perm': {'d': 15, 'gens': [12948732736, 127, 304, 101084981760, 19327034160, 201344698800, 126, 7]}}, 'schur_multiplier': [6, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times \\He_3:S_4', 'transitive_degree': 54, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 168, 'aut_gen_orders': [21, 6, 12, 24], 'aut_gens': [[4808208, 8029471, 400154583628800, 12459600, 12526254, 732298125081600, 12561486], [4808208, 24527785, 45943438464000, 31664574, 12526254, 1509230577254400, 8253054], [4808208, 14737375, 2217822434150400, 188286, 12526254, 86399913600000, 121728], [4808208, 31785366, 400154583628800, 27672865, 12606864, 87657292800, 4762830], [14641230, 4984944, 1804678285132800, 188287, 24316086, 45774829900800, 121728]], 'aut_group': None, 'aut_hash': 2612532850181749679, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 290304, 'aut_permdeg': 39, 'aut_perms': [6186802126571664466628393932000264478708973240, 1781851109660560709434177068959217943591260430, 13722025152332875074719089139022491204176612008, 1042118284422156949093499798920579161548226569], 'aut_phi_ratio': 56.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [2, 21, 1, 2], [2, 189, 2, 1], [3, 1, 2, 1], [3, 6, 4, 1], [3, 56, 1, 1], [3, 56, 2, 1], [3, 336, 4, 1], [4, 42, 1, 2], [4, 378, 2, 1], [6, 1, 2, 1], [6, 6, 4, 1], [6, 9, 4, 1], [6, 21, 2, 2], [6, 56, 1, 1], [6, 56, 2, 1], [6, 126, 4, 2], [6, 189, 4, 1], [6, 336, 4, 1], [6, 504, 2, 1], [6, 504, 4, 1], [7, 24, 2, 1], [12, 42, 2, 2], [12, 252, 4, 2], [12, 378, 4, 1], [14, 24, 2, 1], [14, 216, 4, 1], [21, 24, 4, 1], [21, 144, 8, 1], [42, 24, 4, 1], [42, 144, 8, 1], [42, 216, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times \\SO(3,7)\\times \\AGL(2,3)', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 168, 'autcentquo_group': None, 'autcentquo_hash': 5821230525008183488, 'autcentquo_nilpotent': False, 'autcentquo_order': 145152, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SO(3,7)\\times \\AGL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [2, 21, 2], [2, 189, 2], [3, 1, 2], [3, 6, 4], [3, 56, 3], [3, 336, 4], [4, 42, 2], [4, 378, 2], [6, 1, 2], [6, 6, 4], [6, 9, 4], [6, 21, 4], [6, 56, 3], [6, 126, 8], [6, 189, 4], [6, 336, 4], [6, 504, 6], [7, 24, 2], [12, 42, 4], [12, 252, 8], [12, 378, 4], [14, 24, 2], [14, 216, 4], [21, 24, 4], [21, 144, 8], [42, 24, 4], [42, 144, 8], [42, 216, 8]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '3024.bc', 'commutator_count': 1, 'commutator_label': None, 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '168.42'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['168.42', 1], ['2.1', 1], ['54.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 21, 1, 2], [2, 189, 1, 2], [3, 1, 2, 1], [3, 6, 1, 4], [3, 56, 1, 1], [3, 56, 2, 1], [3, 336, 1, 4], [4, 42, 1, 2], [4, 378, 1, 2], [6, 1, 2, 1], [6, 6, 1, 4], [6, 9, 2, 2], [6, 21, 2, 2], [6, 56, 1, 1], [6, 56, 2, 1], [6, 126, 1, 8], [6, 189, 2, 2], [6, 336, 1, 4], [6, 504, 1, 2], [6, 504, 2, 2], [7, 24, 2, 1], [12, 42, 2, 2], [12, 252, 1, 8], [12, 378, 2, 2], [14, 24, 2, 1], [14, 216, 2, 2], [21, 24, 4, 1], [21, 144, 2, 4], [42, 24, 4, 1], [42, 144, 2, 4], [42, 216, 4, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 7579656, 'exponent': 84, 'exponents_of_order': [5, 4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[9, 0, 8], [18, 0, 4], [21, 0, 4], [24, 0, 4]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '6048.bk', 'hash': 8317723421914599461, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [1, 2, 3, 2, 3, 2, 3], 'inner_gens': [[4808208, 8029471, 400154583628800, 12459600, 12526254, 732298125081600, 12561486], [4808208, 8029471, 400154583628800, 12459600, 31734726, 732298125081600, 24477144], [4808208, 8029471, 400154583628800, 12459600, 12526254, 68173902720000, 12561486], [4808208, 8029471, 400154583628800, 12459600, 31734726, 732298125081600, 24477144], [4808208, 14737375, 400154583628800, 188286, 12526254, 732298125081600, 19058064], [4808208, 8029471, 1486925867750400, 12459600, 12526254, 732298125081600, 12561486], [4808208, 14782687, 400154583628800, 223584, 8207664, 732298125081600, 12561486]], 'inner_hash': 3930612955062600336, 'inner_nilpotent': False, 'inner_order': 3024, 'inner_split': False, 'inner_tex': 'C_3:S_3\\times \\GL(3,2)', 'inner_used': [2, 3, 5, 6, 7], 'irrC_degree': 9, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 18, 'irrep_stats': [[1, 4], [2, 8], [3, 16], [6, 20], [7, 4], [8, 4], [9, 16], [12, 8], [14, 8], [16, 8], [18, 8], [21, 8], [24, 8]], 'label': '18144.f', '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': False, 'name': 'C3^2:D6*GL(3,2)', 'ngens': 7, '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, 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': 40, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 120, 'number_divisions': 80, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': 449, 'number_subgroup_classes': 1902, 'number_subgroups': 117217, 'old_label': None, 'order': 18144, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 439], [3, 1538], [4, 840], [6, 6446], [7, 48], [12, 3696], [14, 912], [21, 1248], [42, 2976]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2], 'outer_gen_pows': [0, 0, 1090688080204800, 8029471, 8029471], 'outer_gens': [[4808208, 8029470, 400154583628800, 12459601, 12526254, 732298125081600, 12561486], [14641230, 18869809, 400154583628800, 12459600, 14605920, 732298125081600, 12561486], [4808208, 8029471, 2199688870579200, 12459600, 4843440, 1090600422912000, 12561486], [4808208, 8029471, 400154583628800, 12459600, 12606864, 732298125081600, 14605920], [4808208, 8029471, 400154583628800, 12459600, 14605920, 732298125081600, 27743094]], 'outer_group': '96.226', 'outer_hash': 226, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [7, 120, 856, 16680, 5160], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^2\\times S_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [6, 12], [7, 4], [8, 4], [12, 16], [14, 8], [16, 8], [36, 8], [42, 4], [48, 4]], 'representations': {'Perm': {'d': 18, 'gens': [4808208, 8029471, 400154583628800, 12459600, 12526254, 732298125081600, 12561486]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:D_6\\times \\GL(3,2)', 'transitive_degree': 126, 'wreath_data': None, 'wreath_product': False}
