-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '486.173', 'ambient_counter': 173, 'ambient_order': 486, 'ambient_tex': '(C_3^2\\times C_9):C_6', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 3, 'counter': 48, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '486.173.27.a1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '27.a1.c1', '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': 27, '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': '18.3', 'subgroup_hash': 3, 'subgroup_order': 18, 'subgroup_tex': 'C_3\\times S_3', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '486.173', 'aut_centralizer_order': 54, 'aut_label': '27.a1', 'aut_quo_index': None, 'aut_stab_index': 81, 'aut_weyl_group': '12.4', 'aut_weyl_index': 4374, 'centralizer': '162.e1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['9.a1.a1'], 'contains': ['54.e1.c1', '81.a1.a1', '81.b1.c1'], 'core': '162.a1.c1', 'coset_action_label': None, 'count': 27, 'diagramx': [9818, -1, 983, -1, 9663, -1, 619, -1], 'generators': [3, 2, 330], 'label': '486.173.27.a1.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.a1.c1', 'old_label': '27.a1.c1', 'projective_image': '486.173', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '27.a1.c1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 6], [5, 12], [17, 6]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', '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': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 1, 2], [3, 2, 3], [6, 3, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '18.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 12], [13, 6]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 6], [2, 3]], 'label': '18.3', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*S3', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 9, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 9, 'number_subgroups': 14, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 8], [6, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5, 6]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 5451, 'gens': [1, 3], 'pres': [3, -2, -3, -3, 6, 110]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [11780110, 20974441]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'GU'}, {'d': 2, 'q': 4, 'gens': [20, 130, 194], 'family': 'COPlus'}, {'d': 2, 'q': 2, 'gens': [20, 130, 138], 'family': 'CU'}, {'d': 1, 'q': 9, 'gens': [93882, 62619, 1930], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 687, 1374]}, 'Perm': {'d': 6, 'gens': [450, 147, 243]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times S_3', 'transitive_degree': 6, 'wreath_data': ['C_3', 'C_2', '2T1'], 'wreath_product': True}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [6, 18, 6, 3, 3, 3, 3, 3, 3], 'aut_gens': [[1, 6, 18, 54], [157, 6, 36, 114], [407, 12, 198, 390], [179, 6, 198, 384], [325, 6, 18, 216], [337, 6, 18, 216], [169, 168, 18, 60], [373, 330, 180, 264], [199, 6, 18, 54], [325, 6, 18, 54]], 'aut_group': '52488.sz', 'aut_hash': 4618096886160246644, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 52488, 'aut_permdeg': 99, 'aut_perms': [462071569669439836928027087487463375609991955761999485028977461229628174985017596274689997897013199326382115580393736228028176585371540066464769893845468746, 360401535624651496010944093015309931812082251696535955021690221297319468521218552469366645315903549896856938483735619921799528372883331987132431661081563851, 56660342019073944935251026444835612156243404284781512296826091336599099925517781897807353654676097339436619408516449750495622666515410440285609910741287063, 72547673785568881145656506795891852198675792324550245540169867643963578725929334851966505429599555095499728913377582517877649283533492520665154306997300, 9129501061462203852198436618184965229119901839171931673721506892059927692074658006655186555505100424246823182843272064740852257376041555761133104762234078, 959841313701365015006554541257841187166793572829835572533002519532666646291069975800870342835134200815566438958569536874614478406797988220520078728164966, 826940140042338766061870874230585468771934931368011414100902552779125090533895454130179445883647326058733829371408893042936181271596005044799323712127140640, 826457132400424244336439916872301040274050262732135084814745393095859791718674516241377634843814695980649993582538825081832494643923342451188241053800093088, 1515439177771969507534301320794407341057410518019669447790604514637801173477595810753308207632682367666391154514808960296767728392509196874727840177026232], 'aut_phi_ratio': 324.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 1, 1], [3, 2, 3, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 9, 2, 1], [3, 18, 2, 1], [3, 18, 6, 1], [6, 81, 2, 1], [9, 6, 9, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^5.S_3^3', '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': 18, 'autcentquo_group': '52488.sz', 'autcentquo_hash': 4618096886160246644, 'autcentquo_nilpotent': False, 'autcentquo_order': 52488, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^5.S_3^3', 'cc_stats': [[1, 1, 1], [2, 81, 1], [3, 2, 4], [3, 6, 3], [3, 9, 2], [3, 18, 8], [6, 81, 2], [9, 6, 9]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '486.173', 'commutator_count': 1, 'commutator_label': '81.11', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 173, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 1, 4], [3, 6, 1, 3], [3, 9, 2, 1], [3, 18, 2, 4], [6, 81, 2, 1], [9, 6, 3, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1092, 'exponent': 18, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '54.13', 'hash': 173, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 3, 3, 9], 'inner_gens': [[1, 12, 360, 450], [13, 6, 18, 54], [199, 6, 18, 54], [145, 6, 18, 54]], 'inner_hash': 173, 'inner_nilpotent': False, 'inner_order': 486, 'inner_split': True, 'inner_tex': '(C_3^2\\times C_9):C_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 12], [6, 12]], 'label': '486.173', 'linC_count': 81, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 9, 'linQ_dim': 20, 'linQ_dim_count': 9, 'linR_count': 27, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C3^2*C9):C6', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 30, 'number_divisions': 18, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 59, 'number_subgroup_classes': 102, 'number_subgroups': 1196, 'old_label': None, 'order': 486, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 81], [3, 188], [6, 162], [9, 54]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 3, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 6, 342, 84], [1, 6, 180, 414], [5, 12, 180, 144], [1, 12, 18, 54], [5, 336, 180, 468]], 'outer_group': '108.28', 'outer_hash': 28, 'outer_nilpotent': False, 'outer_order': 108, 'outer_permdeg': 11, 'outer_perms': [4001760, 7267710, 1, 18367, 16116145], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [4, 4], [6, 3], [18, 3]], 'representations': {'PC': {'code': 4729215162704215933968643167, 'gens': [1, 3, 4, 5], 'pres': [6, -2, -3, -3, -3, 3, -3, 12, 218, 8643, 4113, 13504, 5950, 118, 11669]}, 'Perm': {'d': 30, 'gens': [77045088581362252582261067545, 10150520661976002260956949005200, 19309740689350546317541865888667, 28464720981880092964601879043604, 1840644697362472214723741462040, 37621082160661303505654350009320]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_3^2\\times C_9):C_6', 'transitive_degree': 81, 'wreath_data': None, 'wreath_product': False}
