-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '7776.dz', 'ambient_counter': 104, 'ambient_order': 7776, 'ambient_tex': 'C_6^3.S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 1296, 'characteristic': False, 'core_order': 9, 'counter': 972, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '7776.dz.432.bi1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '432.bi1', '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': 432, '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': '18.5', 'subgroup_hash': 5, 'subgroup_order': 18, 'subgroup_tex': 'C_3\\times C_6', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '7776.dz', 'aut_centralizer_order': None, 'aut_label': '432.bi1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '6.c1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['144.r1', '144.y1', '144.z1', '216.d1', '216.l1', '216.dx1', '216.fn1', '216.fz1'], 'contains': ['864.c1', '1296.k1', '1296.l1', '1296.m1'], 'core': '864.c1', 'coset_action_label': None, 'count': 3, 'diagramx': [3047, -1, 2536, -1], 'generators': [4536, 3026, 72], 'label': '7776.dz.432.bi1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '216.d1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1', 'old_label': '432.bi1', 'projective_image': '7776.dz', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '432.bi1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 3], [1, 17], [7, 3]], 'aut_group': '48.29', 'aut_hash': 29, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 8, 'aut_perms': [475, 23888], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 8, 1], [6, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', '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, 1], [3, 1, 8], [6, 1, 8]], 'center_label': '18.5', 'center_order': 18, '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', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [6, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '18.5', 'hash': 5, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 18]], 'label': '18.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 18, 'linQ_dim': 4, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C6', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 18, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 12, 'number_subgroups': 12, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 8], [6, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 17], [7, 3]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [475, 23888], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8]], 'representations': {'PC': {'code': 277, 'gens': [1, 2], 'pres': [3, -3, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931151, 16858245]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 1374]}, 'Perm': {'d': 8, 'gens': [5040, 240, 4]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_6', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [6, 18, 6, 12, 6], 'aut_gens': [[1, 6, 216, 1296], [4117, 2694, 1080, 1944], [4549, 5190, 4104, 5400], [973, 258, 216, 1296], [4705, 2478, 4968, 2160], [2773, 942, 1080, 6480]], 'aut_group': None, 'aut_hash': 1593371249671273754, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 15552, 'aut_permdeg': 126, 'aut_perms': [20110231380398025099525143368845347033043799537561335016237454155018069176568972526397576605198588067042393787656064233833557200342795514069159240368252663877911459094575599928917421681183223927284171366240943078, 10982157647754771903434670130085417923569644984844403473535829513347344778694148902129324711483833341003049177740381945776321239356954698910684617413258972848356458830448744428484459367811385049323190278378089151, 12146565549451129877076435825192174719357536747476264918322977419626810837279993904512672652086335853989718695534883527474851027986031344978117582327536082262155175291052549876695684617094828892414877836460374, 4384491630282153938114428462973644646102918949508666603420875788507709457616791397533210477470173125060215690777740818739853178409052260374311680081132605969739866570134067418336699449764723149292483100547539739, 23483719915270388620376546580665675584725346307366377163148455035993166180200803320153122172602649199702853286164034515752721094643398428973171871455261371699628630986869758098017010432944915363172759133188270564], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 54, 2, 1], [3, 2, 1, 2], [3, 3, 1, 2], [3, 4, 1, 1], [3, 6, 1, 5], [3, 12, 1, 3], [4, 9, 2, 1], [4, 27, 2, 1], [4, 54, 2, 3], [6, 2, 1, 2], [6, 3, 1, 6], [6, 4, 1, 1], [6, 6, 1, 23], [6, 12, 1, 33], [6, 54, 2, 2], [6, 108, 2, 3], [9, 72, 1, 3], [9, 144, 1, 3], [12, 18, 2, 1], [12, 27, 2, 4], [12, 54, 2, 9], [12, 108, 2, 9], [18, 72, 1, 3], [18, 144, 1, 3], [36, 216, 2, 3]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.C_3^3.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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, 3, 2], [2, 54, 2], [3, 2, 2], [3, 3, 2], [3, 4, 1], [3, 6, 5], [3, 12, 3], [4, 9, 2], [4, 27, 2], [4, 54, 6], [6, 2, 2], [6, 3, 6], [6, 4, 1], [6, 6, 23], [6, 12, 33], [6, 54, 4], [6, 108, 6], [9, 72, 3], [9, 144, 3], [12, 18, 2], [12, 27, 8], [12, 54, 18], [12, 108, 18], [18, 72, 3], [18, 144, 3], [36, 216, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '3888.fl', 'commutator_count': 1, 'commutator_label': '324.59', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 104, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 54, 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, 9, 2, 1], [4, 27, 2, 1], [4, 54, 1, 2], [4, 54, 2, 2], [6, 2, 1, 2], [6, 3, 2, 3], [6, 4, 1, 1], [6, 6, 1, 7], [6, 6, 2, 8], [6, 12, 1, 7], [6, 12, 2, 13], [6, 54, 2, 2], [6, 108, 1, 2], [6, 108, 2, 2], [9, 72, 1, 1], [9, 72, 2, 1], [9, 144, 1, 1], [9, 144, 2, 1], [12, 18, 2, 1], [12, 27, 4, 2], [12, 54, 2, 3], [12, 54, 4, 3], [12, 108, 1, 2], [12, 108, 2, 4], [12, 108, 4, 2], [18, 72, 1, 1], [18, 72, 2, 1], [18, 144, 1, 1], [18, 144, 2, 1], [36, 216, 2, 1], [36, 216, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 432, 'exponent': 36, 'exponents_of_order': [5, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 8], [12, 1, 1]], 'familial': False, 'frattini_label': '18.5', 'frattini_quotient': '432.745', 'hash': 7970830318610051341, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 18, 6, 6], 'inner_gens': [[1, 1758, 4320, 6264], [7753, 6, 4968, 3672], [4969, 4326, 216, 1296], [4105, 6702, 216, 1296]], 'inner_hash': 3044566718417071494, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': True, 'inner_tex': 'C_3^3.(C_6\\times S_4)', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 24], [2, 24], [3, 24], [4, 6], [6, 52], [12, 38]], 'label': '7776.dz', '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': 'C6^3.S3^2', '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, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 131, 'number_characteristic_subgroups': 60, 'number_conjugacy_classes': 168, 'number_divisions': 95, 'number_normal_subgroups': 68, 'number_subgroup_autclasses': 1274, 'number_subgroup_classes': 1418, 'number_subgroups': 20386, 'old_label': None, 'order': 7776, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 115], [3, 80], [4, 396], [6, 1424], [9, 648], [12, 3168], [18, 648], [36, 1296]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 6052], 'outer_gens': [[3025, 114, 1080, 6480], [109, 4062, 4320, 6264]], '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': 4, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 12], [3, 4], [4, 11], [6, 18], [8, 3], [12, 26], [24, 17]], 'representations': {'PC': {'code': '15993306671191420583852791292786957343751959800310183049503351831495444905594451833270075055613158911916517648320613938485', 'gens': [1, 3, 7, 9], 'pres': [10, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3, 20, 52742, 41142, 82, 11043, 18973, 113, 222004, 15014, 194, 8645, 302406, 57986, 27756, 12016, 206, 69127, 11547, 563768, 55108, 17858, 20298, 268, 345609, 100829, 28839, 12649]}, 'Perm': {'d': 26, 'gens': [1323155781313721134985640, 16186841650081106550576009]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^3.S_3^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
