-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '960.4348', 'ambient_counter': 4348, 'ambient_order': 960, 'ambient_tex': 'C_6.(D_4\\times C_{20})', 'central': False, 'central_factor': False, 'centralizer_order': 480, 'characteristic': True, 'core_order': 60, 'counter': 88, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '960.4348.16.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '16.f1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '16.13', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 13, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4:C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '60.13', 'subgroup_hash': 13, 'subgroup_order': 60, 'subgroup_tex': 'C_2\\times C_{30}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '960.4348', 'aut_centralizer_order': 1536, 'aut_label': '16.f1', 'aut_quo_index': 6, 'aut_stab_index': 1, 'aut_weyl_group': '8.2', 'aut_weyl_index': 1536, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '8.t1.a1', '8.u1.a1', '8.v1.a1'], 'contains': ['32.c1.a1', '32.f1.a1', '32.g1.a1', '48.f1.a1', '80.f1.a1'], 'core': '16.f1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7921, 5624, 1313, 3702, 3920, 6308, 1406, 4463], 'generators': [2, 640, 192, 480], 'label': '960.4348.16.f1.a1', 'mobius_quo': 2, 'mobius_sub': 0, 'normal_closure': '16.f1.a1', 'normal_contained_in': ['8.a1.a1'], 'normal_contains': ['32.f1.a1', '32.c1.a1', '32.g1.a1', '48.f1.a1', '80.f1.a1'], 'normalizer': '1.a1.a1', 'old_label': '16.f1.a1', 'projective_image': '48.39', 'quotient_action_image': '2.1', 'quotient_action_kernel': '8.3', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '16.f1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '60.13', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 12], 'aut_gens': [[1, 2], [30, 33], [1, 22], [30, 15]], 'aut_group': '48.35', 'aut_hash': 35, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 9, 'aut_perms': [1, 24, 46107], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [5, 1, 4, 1], [6, 1, 6, 1], [10, 1, 12, 1], [15, 1, 8, 1], [30, 1, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '48.35', 'autcent_hash': 35, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times D_6', '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, 3], [3, 1, 2], [5, 1, 4], [6, 1, 6], [10, 1, 12], [15, 1, 8], [30, 1, 24]], 'center_label': '60.13', 'center_order': 60, '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', '2.1', '3.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [5, 1, 4, 1], [6, 1, 2, 3], [10, 1, 4, 3], [15, 1, 8, 1], [30, 1, 8, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 24, 'exponent': 30, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '60.13', 'hash': 13, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 60]], 'label': '60.13', 'linC_count': 576, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 6, 'linQ_dim': 6, 'linQ_dim_count': 6, 'linR_count': 24, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C30', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 60, 'number_divisions': 16, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 20, 'number_subgroups': 20, 'old_label': None, 'order': 60, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 2], [5, 4], [6, 6], [10, 12], [15, 8], [30, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[30, 33], [1, 22], [30, 15]], 'outer_group': '48.35', 'outer_hash': 35, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 9, 'outer_perms': [1, 24, 46107], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 4], [8, 4]], 'representations': {'PC': {'code': 8521223, 'gens': [1, 2], 'pres': [4, -2, -2, -3, -5, 21, 46]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [24992906222183252, 58415888370979138]}, 'GLFp': {'d': 2, 'p': 31, 'gens': [178762, 893760]}, 'Perm': {'d': 12, 'gens': [39916800, 362880, 10080, 96]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 30], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{30}', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '80.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[1, 4, 16], [3, 4, 656], [9, 4, 656], [481, 4, 656], [1, 6, 656], [1, 12, 656], [1, 484, 656], [1, 4, 658], [1, 4, 664], [1, 4, 176], [1, 4, 272], [1, 4, 656], [1, 644, 16]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 12288, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 48.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [4, 2, 4, 1], [4, 4, 2, 2], [4, 6, 4, 2], [4, 12, 2, 2], [5, 1, 4, 1], [6, 2, 1, 7], [10, 1, 4, 7], [12, 4, 4, 3], [15, 2, 4, 1], [20, 2, 16, 1], [20, 4, 8, 2], [20, 6, 16, 2], [20, 12, 8, 2], [30, 2, 4, 7], [60, 4, 16, 3]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 2, 1], [4, 2, 4], [4, 4, 4], [4, 6, 8], [4, 12, 4], [5, 1, 4], [6, 2, 7], [10, 1, 28], [12, 4, 12], [15, 2, 4], [20, 2, 16], [20, 4, 16], [20, 6, 32], [20, 12, 16], [30, 2, 28], [60, 4, 48]], 'center_label': '40.14', 'center_order': 40, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '12.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4348, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['192.213', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [4, 2, 2, 2], [4, 4, 1, 2], [4, 4, 2, 1], [4, 6, 2, 4], [4, 12, 1, 2], [4, 12, 2, 1], [5, 1, 4, 1], [6, 2, 1, 7], [10, 1, 4, 7], [12, 4, 2, 4], [12, 4, 4, 1], [15, 2, 4, 1], [20, 2, 8, 2], [20, 4, 4, 2], [20, 4, 8, 1], [20, 6, 8, 4], [20, 12, 4, 2], [20, 12, 8, 1], [30, 2, 4, 7], [60, 4, 8, 4], [60, 4, 16, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 20832, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '120.45', 'hash': 4348, 'hyperelementary': 2, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [2, 2, 6], 'inner_gens': [[1, 4, 504], [1, 4, 176], [489, 804, 16]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 24, 'inner_split': None, 'inner_tex': 'C_2\\times D_6', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 140], [4, 20]], 'label': '960.4348', 'linC_count': 63488, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 1152, 'linQ_dim': 14, 'linQ_dim_count': 1680, 'linR_count': 832, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6.(D4*C20)', '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, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 52, 'number_characteristic_subgroups': 102, 'number_conjugacy_classes': 240, 'number_divisions': 66, 'number_normal_subgroups': 138, 'number_subgroup_autclasses': 248, 'number_subgroup_classes': 308, 'number_subgroups': 704, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 120], [5, 4], [6, 14], [10, 28], [12, 48], [15, 8], [20, 480], [30, 56], [60, 192]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '512.10494201', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 512, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2^7\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 20], [8, 13], [16, 12], [32, 1]], 'representations': {'PC': {'code': 13014056774510034983993612219648463667207, 'gens': [1, 3, 5], 'pres': [8, -2, -2, -2, -2, 2, -2, -3, -5, 16, 66, 20164, 1780, 116, 4245, 141, 9878, 222]}, 'GLZN': {'d': 2, 'p': 88, 'gens': [28163237, 30666285, 684443, 6133257, 685345, 49875609, 44295745, 2575427]}, 'Perm': {'d': 24, 'gens': [25903243617736530085200, 27027371704018176691200, 25903107681056773534080, 33, 40279680, 392327162440012800, 53952034933325168640000, 5760]}}, 'schur_multiplier': [2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6.(D_4\\times C_{20})', 'transitive_degree': 960, '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': 12, 'aut_gen_orders': [2, 2, 3, 2, 2], 'aut_gens': [[30, 377, 501], [120, 105, 65], [120, 377, 129], [65, 90, 120], [120, 253, 129], [120, 377, 501]], 'aut_group': '48.48', 'aut_hash': 48, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 6, 'aut_perms': [415, 450, 233, 403, 444], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [4, 1, 2, 1], [4, 2, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_4', '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': 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, 1, 1], [2, 2, 3], [4, 1, 2], [4, 2, 3]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [4, 1, 2, 1], [4, 2, 1, 3]], 'element_repr_type': 'GLFp', 'elementary': 2, 'eulerian_function': 28, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 13, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[30, 253, 129], [120, 377, 501], [120, 377, 501]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.13', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4:C2', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 9, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 20, 'number_subgroups': 23, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [377, 126], 'outer_gens': [[65, 377, 501], [65, 90, 30]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 1]], 'representations': {'PC': {'code': 8716, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 81, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [25793872, 22766861, 817574]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [30, 377, 501]}, 'Perm': {'d': 8, 'gens': [35278, 23616, 16582, 5167]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4:C_2', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
