-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1008.644', 'ambient_counter': 644, 'ambient_order': 1008, 'ambient_tex': 'C_{28}.C_6^2', 'central': False, 'central_factor': True, 'centralizer_order': 6, 'characteristic': False, 'core_order': 336, 'counter': 6, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '1008.644.3.b1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '3.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '3.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 3, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '336.128', 'subgroup_hash': 128, 'subgroup_order': 336, 'subgroup_tex': 'D_{28}:C_6', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1008.644', 'aut_centralizer_order': 2, 'aut_label': '3.b1', 'aut_quo_index': 1, 'aut_stab_index': 3, 'aut_weyl_group': '2016.dc', 'aut_weyl_index': 6, 'centralizer': '168.a1', 'complements': ['336.b1', '336.a1'], 'conjugacy_class_count': 3, 'contained_in': ['1.a1'], 'contains': ['6.b1', '6.d1', '6.f1', '9.a1', '21.b1'], 'core': '3.b1', 'coset_action_label': None, 'count': 3, 'diagramx': [8697, 4059, 8198, 3950], 'generators': [3, 144, 252, 518, 18, 504], 'label': '1008.644.3.b1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '3.b1', 'normal_contained_in': ['1.a1'], 'normal_contains': ['6.b1', '6.d1', '6.f1', '9.a1'], 'normalizer': '1.a1', 'old_label': '3.b1', 'projective_image': '504.178', 'quotient_action_image': '3.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '3.b1', 'subgroup_fusion': None, 'weyl_group': '168.47'}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 6], 'aut_gens': [[1, 2, 12], [169, 266, 60], [169, 266, 301], [84, 231, 301]], 'aut_group': '2016.dc', 'aut_hash': 2868159204135431305, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2016, 'aut_permdeg': 20, 'aut_perms': [62470578644208383, 910177249980063148, 472837259703152009], 'aut_phi_ratio': 21.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 14, 3, 1], [3, 7, 1, 2], [4, 2, 3, 1], [4, 7, 2, 1], [6, 7, 1, 2], [6, 14, 3, 2], [7, 6, 1, 1], [12, 7, 2, 2], [12, 14, 3, 2], [14, 6, 1, 1], [28, 12, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_4\\times F_7', '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': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 14, 3], [3, 7, 2], [4, 2, 3], [4, 7, 2], [6, 7, 2], [6, 14, 6], [7, 6, 1], [12, 7, 4], [12, 14, 6], [14, 6, 1], [28, 12, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '168.47', 'commutator_count': 1, 'commutator_label': '14.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 128, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 14, 1, 3], [3, 7, 2, 1], [4, 2, 1, 3], [4, 7, 2, 1], [6, 7, 2, 1], [6, 14, 2, 3], [7, 6, 1, 1], [12, 7, 4, 1], [12, 14, 2, 3], [14, 6, 1, 1], [28, 12, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5824, 'exponent': 84, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[12, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '168.47', 'hash': 128, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 6, 14], 'inner_gens': [[1, 2, 180], [1, 2, 36], [169, 314, 12]], 'inner_hash': 47, 'inner_nilpotent': False, 'inner_order': 168, 'inner_split': True, 'inner_tex': 'C_2^2\\times F_7', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 24], [2, 6], [6, 4], [12, 1]], 'label': '336.128', 'linC_count': 24, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 4, 'linQ_dim': 10, 'linQ_dim_count': 4, 'linR_count': 12, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D28:C6', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 35, 'number_divisions': 23, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 80, 'number_subgroups': 332, 'old_label': None, 'order': 336, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 43], [3, 14], [4, 20], [6, 98], [7, 6], [12, 112], [14, 6], [28, 36]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [84, 0], 'outer_gens': [[85, 86, 12], [84, 87, 13]], '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': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 1], [6, 4], [8, 1], [12, 1]], 'representations': {'PC': {'code': 907919732237965353966637920137533853, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -3, -2, -2, -7, 1008, 31, 4323, 441, 663, 69, 1090, 1636, 88, 2603, 881]}, 'GLZN': {'d': 2, 'p': 35, 'gens': [661025, 43051, 42891, 343022, 343188, 1243404]}, 'Perm': {'d': 15, 'gens': [6314117455, 13438, 15531, 13971242880, 22828, 105986724480]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{28}:C_6', 'transitive_degree': 56, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '72.50', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [6, 12, 3, 6, 6], 'aut_gens': [[1, 6, 36], [325, 404, 396], [1, 10, 54], [757, 766, 594], [127, 492, 486], [347, 420, 486]], 'aut_group': None, 'aut_hash': 8196202756283457289, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12096, 'aut_permdeg': 41, 'aut_perms': [7360894791054666626849181633504003231803674584657, 10375966136066697977128820087995610907831846122, 25627526109299723579056488451781431601266087901671, 7813696000801634132771409195993060748856822004082, 9191790073483816646293206734104312802481734604588], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 14, 3, 1], [3, 1, 2, 1], [3, 7, 3, 2], [4, 2, 3, 1], [4, 7, 2, 1], [6, 1, 2, 1], [6, 7, 3, 2], [6, 14, 6, 1], [6, 14, 9, 2], [7, 6, 1, 1], [12, 2, 6, 1], [12, 7, 4, 1], [12, 7, 6, 2], [12, 14, 9, 2], [14, 6, 1, 1], [21, 6, 2, 1], [28, 12, 3, 1], [42, 6, 2, 1], [84, 12, 6, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_{21}\\times A_4).C_6.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '48.51', 'autcent_hash': 51, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times D_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 14, 3], [3, 1, 2], [3, 7, 6], [4, 2, 3], [4, 7, 2], [6, 1, 2], [6, 7, 6], [6, 14, 24], [7, 6, 1], [12, 2, 6], [12, 7, 16], [12, 14, 18], [14, 6, 1], [21, 6, 2], [28, 12, 3], [42, 6, 2], [84, 12, 6]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '168.47', 'commutator_count': 1, 'commutator_label': '14.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 644, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['336.128', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 14, 1, 3], [3, 1, 2, 1], [3, 7, 2, 3], [4, 2, 1, 3], [4, 7, 2, 1], [6, 1, 2, 1], [6, 7, 2, 3], [6, 14, 2, 12], [7, 6, 1, 1], [12, 2, 2, 3], [12, 7, 4, 4], [12, 14, 2, 9], [14, 6, 1, 1], [21, 6, 2, 1], [28, 12, 1, 3], [42, 6, 2, 1], [84, 12, 2, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 23296, 'exponent': 84, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[12, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '504.178', 'hash': 644, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 6, 14], 'inner_gens': [[1, 6, 972], [1, 6, 396], [73, 654, 36]], 'inner_hash': 47, 'inner_nilpotent': False, 'inner_order': 168, 'inner_split': True, 'inner_tex': 'C_2^2\\times F_7', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 48, 'irrR_degree': 24, 'irrep_stats': [[1, 72], [2, 18], [6, 12], [12, 3]], 'label': '1008.644', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C28.C6^2', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 26, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 105, 'number_divisions': 55, 'number_normal_subgroups': 114, 'number_subgroup_autclasses': 80, 'number_subgroup_classes': 240, 'number_subgroups': 1032, 'old_label': None, 'order': 1008, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 43], [3, 44], [4, 20], [6, 380], [7, 6], [12, 376], [14, 6], [21, 12], [28, 36], [42, 12], [84, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6, 6], 'outer_gen_pows': [0, 18, 0], 'outer_gens': [[5, 6, 36], [505, 10, 54], [775, 276, 54]], 'outer_group': '72.46', 'outer_hash': 46, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 8, 'outer_perms': [720, 5761, 28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 32], [4, 1], [6, 4], [8, 4], [12, 5], [24, 1]], 'representations': {'PC': {'code': 41838527015480364060633338272207867187836649945018045, 'gens': [1, 3, 5], 'pres': [7, -2, -3, -2, -3, -2, -2, -7, 14, 58, 1200, 34024, 2328, 970, 102, 39317, 5563, 2294, 124, 42342, 4724, 1203]}, 'GLZN': {'d': 2, 'p': 35, 'gens': [661025, 43051, 686016, 42891, 343022, 343188, 1243404]}, 'Perm': {'d': 18, 'gens': [5337514828936, 404339141606400, 777033521510400, 403525, 725760, 1156002822374400, 1103]}}, 'schur_multiplier': [2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{28}.C_6^2', 'transitive_degree': 168, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], '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': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
