-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1944.2345', 'ambient_counter': 2345, 'ambient_order': 1944, 'ambient_tex': 'C_3^2.S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 162, 'counter': 28, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1944.2345.6.j1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.j1.a1', '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': 6, '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': '324.114', 'subgroup_hash': 114, 'subgroup_order': 324, 'subgroup_tex': 'C_3^2:D_{18}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1944.2345', 'aut_centralizer_order': 3, 'aut_label': '6.j1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '648.604', 'aut_weyl_index': 9, 'centralizer': '648.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1', '3.b1.a1'], 'contains': ['12.b1.a1', '12.t1.a1', '12.w1.a1', '18.f1.a1', '18.n1.a1', '18.p1.a1'], 'core': '12.b1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [8647, -1, 8152, -1, 8127, -1, 7697, -1], 'generators': [1, 216, 720, 18, 648, 4], 'label': '1944.2345.6.j1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '6.j1.a1', 'projective_image': '1944.2345', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.j1.a1', 'subgroup_fusion': None, 'weyl_group': '216.101'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 18, 'aut_gen_orders': [6, 18, 6], 'aut_gens': [[1, 6, 36], [1, 6, 180], [5, 282, 36], [29, 6, 36]], 'aut_group': '648.604', 'aut_hash': 604, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 648, 'aut_permdeg': 14, 'aut_perms': [84391463845, 47634279732, 145878624], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 1, 2, 1], [3, 2, 1, 2], [3, 2, 2, 2], [3, 4, 1, 1], [3, 4, 2, 1], [6, 3, 2, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 9, 2, 1], [6, 18, 1, 1], [6, 18, 2, 1], [6, 27, 2, 1], [9, 2, 3, 1], [9, 2, 6, 1], [9, 4, 3, 1], [9, 4, 6, 1], [18, 6, 3, 1], [18, 6, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{18}:C_6\\times S_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': 18, 'autcentquo_group': '324.118', 'autcentquo_hash': 118, 'autcentquo_nilpotent': False, 'autcentquo_order': 324, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2.S_3^2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 1], [3, 1, 2], [3, 2, 6], [3, 4, 3], [6, 3, 2], [6, 6, 3], [6, 9, 2], [6, 18, 3], [6, 27, 2], [9, 2, 9], [9, 4, 9], [18, 6, 9]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '108.16', 'commutator_count': 1, 'commutator_label': '27.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 114, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.1', 1], ['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 1, 2, 1], [3, 2, 1, 2], [3, 2, 2, 2], [3, 4, 1, 1], [3, 4, 2, 1], [6, 3, 2, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 9, 2, 1], [6, 18, 1, 1], [6, 18, 2, 1], [6, 27, 2, 1], [9, 2, 3, 1], [9, 2, 6, 1], [9, 4, 3, 1], [9, 4, 6, 1], [18, 6, 3, 1], [18, 6, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 18, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 6]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '108.38', 'hash': 114, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 6, 9], 'inner_gens': [[1, 30, 36], [13, 6, 288], [1, 78, 36]], 'inner_hash': 16, 'inner_nilpotent': False, 'inner_order': 108, 'inner_split': True, 'inner_tex': 'S_3\\times D_9', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 8, 'irrep_stats': [[1, 12], [2, 30], [4, 12]], 'label': '324.114', 'linC_count': 102, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 20, 'linQ_dim': 10, 'linQ_dim_count': 20, 'linR_count': 84, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:D18', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 24, 'number_characteristic_subgroups': 26, 'number_conjugacy_classes': 54, 'number_divisions': 24, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 86, 'number_subgroup_classes': 86, 'number_subgroups': 424, 'old_label': None, 'order': 324, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 39], [3, 26], [6, 150], [9, 54], [18, 54]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[5, 30, 180]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 5], [6, 2], [8, 1], [12, 3], [24, 1]], 'representations': {'PC': {'code': 78838386770656433812823, 'gens': [1, 3, 5], 'pres': [6, -2, -3, -2, -3, -3, -3, 12, 542, 50, 579, 1456, 118, 1313]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [10015, 40831, 99325, 5941, 96397, 5995]}, 'Perm': {'d': 15, 'gens': [88216128000, 13015, 193999276800, 57658, 280259481600, 104227]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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:D_{18}', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [6, 6, 6, 18], 'aut_gens': [[1, 2, 12, 72, 648], [9, 1514, 1236, 360, 1296], [9, 1762, 1308, 1800, 1080], [5, 898, 420, 792, 648], [9, 1090, 1500, 288, 1728]], 'aut_group': None, 'aut_hash': 140594708777273934, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5832, 'aut_permdeg': 39, 'aut_perms': [12337762777656033828740674630239610862293674747, 12101632050148495162438944825984352753321240249, 5778710701141924853235509017158303977245356189, 12457774075883486116020226075037706395122152468], 'aut_phi_ratio': 9.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 3], [2, 81, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 1], [3, 12, 1, 1], [3, 18, 1, 1], [3, 36, 1, 1], [6, 6, 1, 1], [6, 18, 1, 3], [6, 36, 1, 1], [6, 54, 1, 6], [6, 108, 1, 2], [6, 162, 1, 2], [9, 6, 3, 1], [9, 12, 3, 1], [9, 36, 1, 1], [9, 72, 1, 1], [18, 18, 3, 2], [18, 36, 3, 1], [18, 54, 3, 1], [18, 108, 1, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_3\\times C_9):C_3^2:C_2^2\\times S_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': None, 'autcentquo_hash': 140594708777273934, 'autcentquo_nilpotent': False, 'autcentquo_order': 5832, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': '(C_3\\times C_9):C_3^2:C_2^2\\times S_3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 3], [2, 81, 2], [3, 2, 2], [3, 4, 1], [3, 6, 1], [3, 12, 1], [3, 18, 1], [3, 36, 1], [6, 6, 1], [6, 18, 3], [6, 36, 1], [6, 54, 6], [6, 108, 2], [6, 162, 2], [9, 6, 3], [9, 12, 3], [9, 36, 1], [9, 72, 1], [18, 18, 6], [18, 36, 3], [18, 54, 3], [18, 108, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '1944.2345', 'commutator_count': 1, 'commutator_label': '243.52', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2345, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['324.40', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 3], [2, 81, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 1], [3, 12, 1, 1], [3, 18, 1, 1], [3, 36, 1, 1], [6, 6, 1, 1], [6, 18, 1, 3], [6, 36, 1, 1], [6, 54, 1, 6], [6, 108, 1, 2], [6, 162, 1, 2], [9, 6, 3, 1], [9, 12, 3, 1], [9, 36, 1, 1], [9, 72, 1, 1], [18, 18, 3, 2], [18, 36, 3, 1], [18, 54, 3, 1], [18, 108, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 290304, 'exponent': 18, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 6]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '216.162', 'hash': 2345, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 6, 6, 9, 3], 'inner_gens': [[1, 10, 12, 72, 648], [5, 2, 60, 72, 1728], [1, 26, 12, 1224, 1512], [1, 2, 1452, 72, 648], [1, 1514, 1740, 72, 648]], 'inner_hash': 2345, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': True, 'inner_tex': 'C_3^2.S_3^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 12, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 12], [4, 6], [6, 16], [8, 1], [12, 8]], 'label': '1944.2345', 'linC_count': 48, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 16, 'linQ_dim': 20, 'linQ_dim_count': 16, 'linR_count': 48, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^2.S3^3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 39, 'number_characteristic_subgroups': 44, 'number_conjugacy_classes': 51, 'number_divisions': 39, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 418, 'number_subgroup_classes': 418, 'number_subgroups': 9008, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 255], [3, 80], [6, 960], [9, 162], [18, 486]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [36], 'outer_gens': [[1, 2, 12, 144, 1296]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 6], [6, 4], [8, 1], [12, 2], [18, 4], [36, 2]], 'representations': {'PC': {'code': 76924399814180294785330994043659433761933084437097007, 'gens': [1, 2, 4, 6, 8], 'pres': [8, -2, -2, -3, -2, -3, -3, -3, 3, 161, 41, 194, 971, 91, 972, 4925, 1909, 189, 2046, 55311, 8095, 2343]}, 'Perm': {'d': 30, 'gens': [994521850752453975085160365944, 76647781351621846167866971944, 1, 1307617669352396791932576775440, 3, 10422670299976225656879282363120, 19592464762042860437011485951840, 28761311992073966670442687507440]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2.S_3^3', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
