Formats: - HTML - YAML - JSON - 2026-07-23T10:19:50.381277
  • gps_subgroup_search • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '41472.jo', 'ambient_counter': 249, 'ambient_order': 41472, 'ambient_tex': 'C_3^4:(Q_8^2.C_2^3)', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 324, 'counter': 268, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '41472.jo.32.cu1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '32.cu1', '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': 32, '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': '1296.3501', 'subgroup_hash': 3501, 'subgroup_order': 1296, 'subgroup_tex': 'C_3^2\\wr C_2.Q_8', 'supersolvable': False, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '41472.jo', 'aut_centralizer_order': None, 'aut_label': '32.cu1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '20736.a1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['16.w1', '16.cz1', '16.da1', '16.de1', '16.dg1', '16.dk1', '16.dl1'], 'contains': ['64.bn1', '64.bp1', '64.bq1', '288.dt1', '288.dx1'], 'core': '128.c1', 'coset_action_label': None, 'count': 16, 'diagramx': None, 'generators': [19870708156126921, 3475999542034374, 39049845055664575, 205261218739998720, 685808788671024600, 64163386385286, 83622221265558960, 5505528523587984], 'label': '41472.jo.32.cu1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.c1', 'old_label': '32.cu1', 'projective_image': '41472.jo', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.cu1', 'subgroup_fusion': None, 'weyl_group': None}
  • gps_groups • Show schema
    {'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': 24, 'aut_gen_orders': [4, 8, 8, 4, 12], 'aut_gens': [[1, 4, 12, 144, 432], [1015, 52, 852, 576, 1008], [1189, 56, 616, 144, 1152], [225, 48, 448, 1008, 1152], [147, 100, 500, 720, 576], [441, 104, 884, 144, 432]], 'aut_group': None, 'aut_hash': 2723700116397070647, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20736, 'aut_permdeg': 648, 'aut_perms': 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'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 81, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [3, 16, 4, 1], [4, 18, 1, 1], [4, 162, 1, 1], [4, 162, 4, 1], [6, 36, 2, 1], [6, 72, 1, 1], [12, 36, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_4^2.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': None, 'autcentquo_hash': 2723700116397070647, 'autcentquo_nilpotent': False, 'autcentquo_order': 20736, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.D_4^2.C_2^2', 'cc_stats': [[1, 1, 1], [2, 9, 2], [2, 81, 1], [3, 4, 2], [3, 8, 1], [3, 16, 4], [4, 18, 1], [4, 162, 5], [6, 36, 2], [6, 72, 1], [12, 36, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1296.3501', 'commutator_count': 1, 'commutator_label': '162.52', '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': 3501, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 81, 1, 1], [3, 4, 1, 2], [3, 8, 1, 1], [3, 16, 1, 4], [4, 18, 1, 1], [4, 162, 1, 1], [4, 162, 2, 2], [6, 36, 1, 2], [6, 72, 1, 1], [12, 36, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 12, 'exponents_of_order': [4, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1296.3501', 'hash': 3501, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 3, 12, 3, 3], 'inner_gens': [[1, 52, 136, 432, 288], [97, 4, 12, 144, 432], [33, 4, 12, 720, 576], [1009, 4, 1164, 144, 432], [577, 4, 300, 144, 432]], 'inner_hash': 3501, 'inner_nilpotent': False, 'inner_order': 1296, 'inner_split': False, 'inner_tex': 'C_3^2\\wr C_2.Q_8', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 2], [4, 8], [8, 2], [16, 4]], 'label': '1296.3501', 'linC_count': 16, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 8, 'linQ_dim': 12, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^2wrC2.Q8', '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': 13, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 24, 'number_divisions': 20, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 88, 'number_subgroup_classes': 174, 'number_subgroups': 4900, 'old_label': None, 'order': 1296, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 99], [3, 80], [4, 828], [6, 144], [12, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [108, 2, 38, 72], 'outer_gens': [[111, 104, 12, 1152, 288], [3, 96, 112, 1008, 1152], [37, 104, 20, 1152, 288], [1, 8, 132, 864, 144]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 4], [8, 4], [16, 4]], 'representations': {'PC': {'code': 937960966385290460747021815643911329369909296707477052010942208874791, 'gens': [1, 3, 4, 7, 8], 'pres': [8, 2, 2, 3, 2, 2, 3, 3, 3, 16, 1250, 106, 4355, 971, 91, 5124, 2412, 116, 4805, 2317, 24198, 8078, 3390, 710, 18439, 27663, 3103, 2343]}, 'Perm': {'d': 15, 'gens': [7355945653, 13986206640, 20797746503, 20797746480, 106978152960, 188857635120, 56, 333]}}, 'schur_multiplier': [3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2\\wr C_2.Q_8', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 48, 'aut_gen_orders': [12, 8, 8, 4, 12, 4], 'aut_gens': [[149539467511629505, 277798901855431729, 525790413758095927, 377749867068727255], [22140996856117441, 831175841327853288, 530540011646271511, 1079901839518213945], [1102060395491570664, 346316118822537487, 906095913876022369, 2175187699857111871], [175826909012121481, 1323312292687131649, 528322127221592089, 2087601560693334864], [1173965143894953126, 1984830112143065790, 1124141907608095927, 1361455562400445854], [1851661813608176448, 2110779328357929750, 1124141907608095927, 1078205680121534911], [1129781028479494345, 467495351572279105, 701363353812441487, 776481454060342951]], 'aut_group': None, 'aut_hash': 8218745305919483472, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 331776, 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'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 18, 1, 2], [2, 72, 4, 1], [2, 81, 1, 2], [2, 144, 2, 1], [3, 16, 1, 1], [3, 64, 1, 1], [4, 36, 1, 2], [4, 72, 2, 1], [4, 324, 1, 4], [4, 648, 2, 3], [4, 648, 4, 1], [4, 1296, 2, 1], [6, 16, 1, 1], [6, 64, 1, 1], [6, 144, 1, 2], [6, 576, 2, 2], [6, 576, 4, 1], [8, 324, 2, 2], [8, 432, 2, 2], [8, 648, 2, 1], [8, 1296, 4, 1], [8, 2592, 4, 1], [12, 288, 1, 2], [12, 576, 2, 1], [24, 864, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_4^2.C_2^6.C_2^2', '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': 48, 'autcentquo_group': '41472.m', 'autcentquo_hash': 7917341340725978817, 'autcentquo_nilpotent': False, 'autcentquo_order': 41472, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)^2.D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 18, 2], [2, 72, 4], [2, 81, 2], [2, 144, 2], [3, 16, 1], [3, 64, 1], [4, 36, 2], [4, 72, 2], [4, 324, 4], [4, 648, 10], [4, 1296, 2], [6, 16, 1], [6, 64, 1], [6, 144, 2], [6, 576, 8], [8, 324, 4], [8, 432, 4], [8, 648, 2], [8, 1296, 4], [8, 2592, 4], [12, 288, 2], [12, 576, 2], [24, 864, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '20736.w', 'commutator_count': 1, 'commutator_label': '2592.lu', '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', '3.1', '3.1', '3.1'], 'composition_length': 13, 'conjugacy_classes_known': True, 'counter': 249, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['20736.w', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 18, 1, 2], [2, 72, 1, 4], [2, 81, 1, 2], [2, 144, 1, 2], [3, 16, 1, 1], [3, 64, 1, 1], [4, 36, 1, 2], [4, 72, 1, 2], [4, 324, 1, 4], [4, 648, 1, 10], [4, 1296, 1, 2], [6, 16, 1, 1], [6, 64, 1, 1], [6, 144, 1, 2], [6, 576, 1, 8], [8, 324, 2, 2], [8, 432, 2, 2], [8, 648, 1, 2], [8, 1296, 1, 4], [8, 2592, 1, 4], [12, 288, 1, 2], [12, 576, 1, 2], [24, 864, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2742740582400, 'exponent': 24, 'exponents_of_order': [9, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 1, 4], [32, 0, 2], [32, 1, 1], [64, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '41472.jo', 'hash': 2915094534016945548, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [8, 8, 2, 4], 'inner_gens': [[149539467511629505, 776706548802148441, 1124141907608095927, 1406944610127003055], [174928605379744009, 277798901855431729, 1616167960412761129, 794121313147355071], [93657516308959831, 1528141399327728415, 525790413758095927, 1320486296129542975], [1873460898826737847, 996653958206556655, 88435964694331465, 377749867068727255]], 'inner_hash': 4402493192939309901, 'inner_nilpotent': False, 'inner_order': 20736, 'inner_split': True, 'inner_tex': 'C_3^4:(Q_8^2:C_2^2)', 'inner_used': [1, 2, 4], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 12], [4, 20], [8, 2], [16, 8], [32, 6], [64, 8]], 'label': '41472.jo', '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': 'C3^4:(Q8^2.C2^3)', 'ngens': 4, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 41, 'number_characteristic_subgroups': 28, 'number_conjugacy_classes': 72, 'number_divisions': 66, 'number_normal_subgroups': 116, 'number_subgroup_autclasses': 1579, 'number_subgroup_classes': 3627, 'number_subgroups': 494594, 'old_label': None, 'order': 41472, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 775], [3, 80], [4, 10584], [6, 4976], [8, 19872], [12, 1728], [24, 3456]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1790666911979260710, 279369462202027225, 1908728163059937247, 749094526370501311], [152782500385794295, 277809544226880055, 525790413758095927, 377748913270641984], [149539467511629505, 277798901855431728, 525790413758095927, 377749867068727255], [1790666911979260711, 279369462202027224, 1908728163059937247, 749094526370501310]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [7, 288, 127, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 12], [8, 6], [16, 8], [32, 2], [64, 10]], 'representations': {'PC': {'code': '1394811045689100718296644388088594488764098502067124516755135476791515791672947635023607900897316783578704710448263574766946819803006465596559953146405128625266018670914328494487719553247318591941966949536710969841885915455517526674673263030001593106702169576249856514661102779041607838670906556916267179657183454394553856679963', 'gens': [1, 2, 4, 7, 10, 11, 13], 'pres': [13, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 180128, 1002717, 66, 738974, 234964, 295779, 788544, 432981, 146, 1800244, 754537, 88170, 186, 1599941, 1109490, 370063, 1381750, 1692619, 633392, 379333, 36822, 55581, 266, 2412807, 540820, 269601, 387758, 213883, 4648, 306, 4492808, 44949, 134818, 142319, 37500, 1945, 599049, 299542, 1248035, 599088, 305821, 6314, 40647, 1660, 1098250, 1867031, 256292, 128177, 25230, 51555, 426, 239627, 2995224, 20005, 10034, 47487, 45004, 1730572, 1254681, 854502, 427283, 362400, 146093, 38622, 42691]}, 'Perm': {'d': 20, 'gens': [149539467511629505, 277798901855431729, 525790413758095927, 377749867068727255]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:(Q_8^2.C_2^3)', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}