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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1404.117', 'ambient_counter': 117, 'ambient_order': 1404, 'ambient_tex': '\\He_3:D_{26}', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 9, 'counter': 57, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1404.117.39.a1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '39.a1.b1', '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': 39, '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': '1404.117', 'aut_centralizer_order': 36, 'aut_label': '39.a1', 'aut_quo_index': None, 'aut_stab_index': 156, 'aut_weyl_group': '12.4', 'aut_weyl_index': 5616, 'centralizer': '234.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1.b1', '13.a1.a1'], 'contains': ['78.a1.b1', '78.b1.b1', '78.c1.b1', '117.a1.a1', '117.b1.b1'], 'core': '156.a1.b1', 'coset_action_label': None, 'count': 39, 'diagramx': [3739, -1, 1751, -1, 2609, -1, 1289, -1], 'generators': [1, 6, 12, 4], 'label': '1404.117.39.a1.b1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '39.a1.b1', 'old_label': '39.a1.b1', 'projective_image': '468.43', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '39.a1.b1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 312, 'aut_gen_orders': [2, 12, 12, 52, 3, 3], 'aut_gens': [[1, 2, 12, 36], [1, 2, 24, 1384], [1, 494, 12, 1216], [1, 478, 12, 596], [1, 1142, 12, 512], [5, 2, 12, 60], [957, 14, 12, 48]], 'aut_group': '67392.c', 'aut_hash': 1849751742543539073, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 67392, 'aut_permdeg': 22, 'aut_perms': [873411360538625953817, 873020686947836176685, 668854062152839551711, 616842658632896360522, 649841719940822079630, 868674683204935303926], 'aut_phi_ratio': 156.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 13, 1, 1], [2, 117, 1, 1], [3, 1, 2, 1], [3, 6, 4, 1], [6, 9, 2, 1], [6, 13, 2, 1], [6, 78, 4, 1], [6, 117, 2, 1], [13, 2, 6, 1], [26, 18, 6, 1], [39, 2, 12, 1], [39, 12, 24, 1], [78, 18, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_{13}\\times C_3^2:\\GL(2,3)', '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': 312, 'autcentquo_group': '67392.c', 'autcentquo_hash': 1849751742543539073, 'autcentquo_nilpotent': False, 'autcentquo_order': 67392, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_{13}\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 13, 1], [2, 117, 1], [3, 1, 2], [3, 6, 4], [6, 9, 2], [6, 13, 2], [6, 78, 4], [6, 117, 2], [13, 2, 6], [26, 18, 6], [39, 2, 12], [39, 12, 24], [78, 18, 12]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '468.43', 'commutator_count': 1, 'commutator_label': '351.10', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '13.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 117, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['26.1', 1], ['54.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 13, 1, 1], [2, 117, 1, 1], [3, 1, 2, 1], [3, 6, 1, 4], [6, 9, 2, 1], [6, 13, 2, 1], [6, 78, 1, 4], [6, 117, 2, 1], [13, 2, 6, 1], [26, 18, 6, 1], [39, 2, 12, 1], [39, 12, 6, 4], [78, 18, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 15876, 'exponent': 78, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 3, 13], 'faithful_reps': [[6, 0, 24]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '468.43', 'hash': 117, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 78, 'inner_gen_orders': [2, 6, 1, 39], 'inner_gens': [[1, 10, 12, 516], [5, 2, 12, 924], [1, 2, 12, 36], [961, 554, 12, 36]], 'inner_hash': 43, 'inner_nilpotent': False, 'inner_order': 468, 'inner_split': False, 'inner_tex': 'C_{39}:D_6', 'inner_used': [1, 2, 4], 'irrC_degree': 6, 'irrQ_degree': 72, 'irrQ_dim': 72, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 20], [3, 8], [4, 24], [6, 24]], 'label': '1404.117', 'linC_count': 96, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 8, 'linQ_dim': 18, 'linQ_dim_count': 8, 'linR_count': 48, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'He3:D26', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 15, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 80, 'number_divisions': 24, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 50, 'number_subgroup_classes': 110, 'number_subgroups': 1594, 'old_label': None, 'order': 1404, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 139], [3, 26], [6, 590], [13, 12], [26, 108], [39, 312], [78, 216]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [6, 6, 6, 6, 3, 3], 'outer_gen_pows': [1, 6, 6, 7, 7, 6], 'outer_gens': [[1, 486, 12, 352], [1, 486, 24, 884], [1, 2, 12, 684], [1, 478, 12, 1352], [1, 946, 12, 840], [1, 2, 12, 360]], 'outer_group': '144.188', 'outer_hash': 188, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 9, 'outer_perms': [41070, 768, 31, 131809, 121728, 48], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [6, 4], [12, 2], [24, 4], [72, 2]], 'representations': {'PC': {'code': 1313437689348573504318799098148431719, 'gens': [1, 2, 4, 5], 'pres': [6, -2, -2, -3, -3, 3, -13, 121, 31, 146, 15484, 13870, 376, 118, 23339]}, 'Perm': {'d': 22, 'gens': [40284847, 4988539717185024000, 7441025543875123200, 56201167527208704000, 559571293, 109872559308563251200]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\He_3:D_{26}', 'transitive_degree': 117, 'wreath_data': None, 'wreath_product': False}
