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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '40320.a', 'ambient_counter': 1, 'ambient_order': 40320, 'ambient_tex': 'S_8', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 1, 'counter': 106, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '40320.a.1120.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '1120.d1.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': 1120, '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': '36.12', 'subgroup_hash': 12, 'subgroup_order': 36, 'subgroup_tex': 'C_6\\times S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '40320.a', 'aut_centralizer_order': None, 'aut_label': '1120.d1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '6720.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['112.b1.a1', '560.g1.a1'], 'contains': ['2240.a1.a1', '2240.b1.a1', '2240.d1.a1', '3360.e1.a1', '3360.g1.a1'], 'core': '40320.a1.a1', 'coset_action_label': None, 'count': 560, 'diagramx': [424, -1, 9083, -1, 9505, -1, 534, -1], 'generators': [13504, 33905, 24, 5], 'label': '40320.a.1120.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '560.g1.a1', 'old_label': '1120.d1.a1', 'projective_image': '40320.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1120.d1.a1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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gps_groups • Show schema
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{'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': 6, 'aut_gen_orders': [2, 2, 6], 'aut_gens': [[1, 6], [5, 6], [1, 30], [31, 6]], 'aut_group': '24.14', 'aut_hash': 14, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [7, 127, 856], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_6', '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': 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, 3, 2], [3, 1, 2], [3, 2, 3], [6, 1, 2], [6, 2, 3], [6, 3, 4]], 'center_label': '6.2', 'center_order': 6, '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', '2.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '36.12', 'hash': 12, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 30], [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, 12], [2, 6]], 'label': '36.12', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 7, 'linQ_dim': 4, 'linQ_dim_count': 7, 'linR_count': 7, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6*S3', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 18, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 22, 'number_subgroups': 36, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 8], [6, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[19, 6], [5, 6]], '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': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 2]], 'representations': {'PC': {'code': 403784533, 'gens': [1, 3], 'pres': [4, -2, -3, -2, -3, 8, 362, 34, 387]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16858573, 35931481]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 687, 1718]}, 'Perm': {'d': 8, 'gens': [720, 1, 30, 5760]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times S_3', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 840, 'aut_gen_orders': [8, 2], 'aut_gens': [[5913, 5040], [24151, 3752], [6201, 5040]], 'aut_group': '40320.a', 'aut_hash': 6644662513446815922, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 40320, 'aut_permdeg': 28, 'aut_perms': [277321134931820842725130040744, 186848318957116539924564770], 'aut_phi_ratio': 4.375, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 28, 1, 1], [2, 105, 1, 1], [2, 210, 1, 1], [2, 420, 1, 1], [3, 112, 1, 1], [3, 1120, 1, 1], [4, 420, 1, 1], [4, 1260, 1, 2], [4, 2520, 1, 1], [5, 1344, 1, 1], [6, 1120, 1, 2], [6, 1680, 1, 1], [6, 3360, 1, 2], [7, 5760, 1, 1], [8, 5040, 1, 1], [10, 4032, 1, 1], [12, 3360, 1, 1], [15, 2688, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_8', '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': 840, 'autcentquo_group': '40320.a', 'autcentquo_hash': 6644662513446815922, 'autcentquo_nilpotent': False, 'autcentquo_order': 40320, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_8', 'cc_stats': [[1, 1, 1], [2, 28, 1], [2, 105, 1], [2, 210, 1], [2, 420, 1], [3, 112, 1], [3, 1120, 1], [4, 420, 1], [4, 1260, 2], [4, 2520, 1], [5, 1344, 1], [6, 1120, 2], [6, 1680, 1], [6, 3360, 2], [7, 5760, 1], [8, 5040, 1], [10, 4032, 1], [12, 3360, 1], [15, 2688, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '40320.a', 'commutator_count': 1, 'commutator_label': '20160.a', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '20160.a'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 28, 1, 1], [2, 105, 1, 1], [2, 210, 1, 1], [2, 420, 1, 1], [3, 112, 1, 1], [3, 1120, 1, 1], [4, 420, 1, 1], [4, 1260, 1, 2], [4, 2520, 1, 1], [5, 1344, 1, 1], [6, 1120, 1, 2], [6, 1680, 1, 1], [6, 3360, 1, 2], [7, 5760, 1, 1], [8, 5040, 1, 1], [10, 4032, 1, 1], [12, 3360, 1, 1], [15, 2688, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 24666, 'exponent': 840, 'exponents_of_order': [7, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3, 5, 7], 'faithful_reps': [[7, 1, 2], [14, 1, 2], [20, 1, 2], [21, 1, 2], [28, 1, 2], [35, 1, 2], [42, 1, 1], [56, 1, 2], [64, 1, 2], [70, 1, 2], [90, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '40320.a', 'hash': 6644662513446815922, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 840, 'inner_gen_orders': [8, 2], 'inner_gens': [[5913, 720], [10233, 5040]], 'inner_hash': 6644662513446815922, 'inner_nilpotent': False, 'inner_order': 40320, 'inner_split': True, 'inner_tex': 'S_8', 'inner_used': [1, 2], 'irrC_degree': 7, 'irrQ_degree': 7, 'irrQ_dim': 7, 'irrR_degree': 7, 'irrep_stats': [[1, 2], [7, 2], [14, 2], [20, 2], [21, 2], [28, 2], [35, 2], [42, 1], [56, 2], [64, 2], [70, 2], [90, 1]], 'label': '40320.a', 'linC_count': 2, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 2, 'linQ_dim': 7, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'S8', 'ngens': 2, '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, 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, 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': 22, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 22, 'number_divisions': 22, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 296, 'number_subgroup_classes': 296, 'number_subgroups': 151221, 'old_label': None, 'order': 40320, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 763], [3, 1232], [4, 5460], [5, 1344], [6, 10640], [7, 5760], [8, 5040], [10, 4032], [12, 3360], [15, 2688]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': None, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [7, 2], [14, 2], [20, 2], [21, 2], [28, 2], [35, 2], [42, 1], [56, 2], [64, 2], [70, 2], [90, 1]], 'representations': {'Lie': [{'d': 6, 'q': 2, 'gens': [34630353537, 1344290976, 17197729794], 'family': 'SOPlus'}, {'d': 6, 'q': 2, 'family': 'GOPlus'}, {'d': 6, 'q': 2, 'family': 'PSOPlus'}, {'d': 6, 'q': 2, 'family': 'PGOPlus'}, {'d': 6, 'q': 2, 'family': 'CSOPlus'}, {'d': 6, 'q': 2, 'family': 'COPlus'}], 'Perm': {'d': 8, 'gens': [5913, 5040]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_8', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
