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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '972.461', 'ambient_counter': 461, 'ambient_order': 972, 'ambient_tex': 'C_3^3:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 27, 'characteristic': False, 'core_order': 9, 'counter': 120, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '972.461.36.g1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '36.g1.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': 36, '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': '27.2', 'subgroup_hash': 2, 'subgroup_order': 27, 'subgroup_tex': 'C_3\\times C_9', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '972.461', 'aut_centralizer_order': 81, 'aut_label': '36.g1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '12.5', 'aut_weyl_index': 486, 'centralizer': '36.g1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.c1.b1', '18.g1.b1', '18.m1.b1', '18.be1.b1'], 'contains': ['108.a1.a1', '108.k1.b1', '108.l1.a1'], 'core': '108.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [9173, -1, 9812, -1, 3205, -1, 4307, -1], 'generators': [6121, 8920], 'label': '972.461.36.g1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.c1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.d1.b1', 'old_label': '36.g1.b1', 'projective_image': '972.461', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36.g1.b1', 'subgroup_fusion': None, 'weyl_group': '12.5'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '27.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [3, 3, 2, 2, 6], 'aut_gens': [[1, 3], [1, 23], [1, 12], [2, 24], [2, 4], [11, 15]], 'aut_group': '108.28', 'aut_hash': 28, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 108, 'aut_permdeg': 11, 'aut_perms': [4556304, 11088384, 1, 169567, 26386807], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [9, 1, 18, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '108.28', 'autcent_hash': 28, 'autcent_nilpotent': False, 'autcent_order': 108, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2: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], [3, 1, 8], [9, 1, 18]], 'center_label': '27.2', 'center_order': 27, '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': ['3.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4], [9, 1, 6, 3]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 4, 'exponent': 9, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '9.2', 'hash': 2, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 27]], 'label': '27.2', 'linC_count': 216, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 9, 'linQ_dim': 8, 'linQ_dim_count': 9, 'linR_count': 54, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C9', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 27, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 8], [9, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 3, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 23], [1, 12], [2, 24], [2, 4], [11, 15]], 'outer_group': '108.28', 'outer_hash': 28, 'outer_nilpotent': False, 'outer_order': 108, 'outer_permdeg': 11, 'outer_perms': [4556304, 11088384, 1, 169567, 26386807], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 3, 'primary_abelian_invariants': [3, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 3]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -3, 3, -3, 22]}, 'GLFp': {'d': 2, 'p': 19, 'gens': [34311, 75456]}, 'Perm': {'d': 12, 'gens': [357120, 79833600, 80884]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 9], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_9', 'transitive_degree': 27, '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': 3, 'aut_exponent': 18, 'aut_gen_orders': [6, 6, 3, 3, 6, 3], 'aut_gens': [[99145, 7993, 75931, 6031, 8920], [31369, 42997, 41053, 79831, 61399], [99217, 44941, 40945, 10849, 61399], [99253, 7993, 75931, 6031, 8920], [99361, 7993, 75931, 5923, 8920], [101197, 7993, 75931, 43789, 61399], [99145, 7993, 75931, 61444, 8920]], 'aut_group': '5832.oe', 'aut_hash': 9197326188807043623, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5832, 'aut_permdeg': 30, 'aut_perms': [8472518236072252984838161186059, 240773310949267130022039380669406, 216123862350258811536897590891034, 1840993342204034413676313861420, 22058967574069636531792743760206, 6468822066140490723191084280], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 27, 1, 1], [2, 81, 1, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 2, 1], [3, 6, 3, 1], [3, 12, 3, 1], [3, 18, 1, 1], [3, 36, 1, 1], [6, 6, 1, 1], [6, 9, 2, 1], [6, 18, 3, 1], [6, 27, 2, 1], [6, 54, 1, 2], [6, 54, 2, 1], [6, 81, 2, 1], [9, 18, 2, 1], [9, 36, 2, 1], [18, 54, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3.S_3^3', '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': '5832.oe', 'autcentquo_hash': 9197326188807043623, 'autcentquo_nilpotent': False, 'autcentquo_order': 5832, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^3.S_3^3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 27, 1], [2, 81, 1], [3, 2, 2], [3, 3, 2], [3, 4, 1], [3, 6, 5], [3, 12, 3], [3, 18, 1], [3, 36, 1], [6, 6, 1], [6, 9, 2], [6, 18, 3], [6, 27, 2], [6, 54, 4], [6, 81, 2], [9, 18, 2], [9, 36, 2], [18, 54, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '972.461', 'commutator_count': 1, 'commutator_label': '243.51', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 461, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['162.19', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 27, 1, 1], [2, 81, 1, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 1, 3], [3, 6, 2, 1], [3, 12, 1, 3], [3, 18, 1, 1], [3, 36, 1, 1], [6, 6, 1, 1], [6, 9, 2, 1], [6, 18, 1, 3], [6, 27, 2, 1], [6, 54, 1, 2], [6, 54, 2, 1], [6, 81, 2, 1], [9, 18, 1, 2], [9, 36, 1, 2], [18, 54, 1, 2]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 54432, 'exponent': 18, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 3]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '108.39', 'hash': 461, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 3, 3, 18, 3], 'inner_gens': [[99145, 9829, 76039, 5959, 8920], [101197, 7993, 75931, 41035, 8920], [99361, 7993, 75931, 5923, 8920], [99217, 77875, 76039, 6031, 61399], [99145, 7993, 75931, 61444, 8920]], 'inner_hash': 461, 'inner_nilpotent': False, 'inner_order': 972, 'inner_split': True, 'inner_tex': 'C_3^3:S_3^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 10], [3, 8], [4, 4], [6, 10], [12, 3]], 'label': '972.461', 'linC_count': 12, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 12, 'linQ_dim': 8, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3:S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 39, 'number_divisions': 33, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 187, 'number_subgroup_classes': 277, 'number_subgroups': 3396, 'old_label': None, 'order': 972, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 111], [3, 134], [6, 510], [9, 108], [18, 108]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [5833], 'outer_gens': [[99181, 79819, 40837, 77851, 8920]], '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': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 4], [6, 10], [12, 5]], 'representations': {'PC': {'code': 12648026085625634511530461410974381378890807, 'gens': [1, 2, 3, 4, 7], 'pres': [7, 2, 3, 3, 2, 3, 3, 3, 57, 4664, 8571, 430, 605, 80, 10084, 4526, 1488, 137, 9077, 1791]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [5869, 61399, 40837, 5941, 5995, 42997, 99145]}, 'Perm': {'d': 12, 'gens': [1, 11654904, 52377384, 95091144, 3, 95080320, 134322504]}}, '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': 'C_3^3:S_3^2', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
