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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '2160.dm', 'ambient_counter': 91, 'ambient_order': 2160, 'ambient_tex': 'S_3^3:C_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 30, 'characteristic': False, 'core_order': 30, 'counter': 54, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2160.dm.12.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.f1.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': 12, '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': '180.28', 'subgroup_hash': 28, 'subgroup_order': 180, 'subgroup_tex': 'C_5\\times S_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2160.dm', 'aut_centralizer_order': 6, 'aut_label': '12.f1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '144.143', 'aut_weyl_index': 24, 'centralizer': '72.f1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.f1.a1', '6.d1.a1'], 'contains': ['24.c1.a1', '24.e1.a1', '24.l1.a1', '36.b1.a1', '36.d1.a1', '60.f1.a1'], 'core': '72.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [5956, -1, 1566, -1, 6134, -1, 2413, -1], 'generators': [54, 25866086400, 30, 328472, 288152], 'label': '2160.dm.12.f1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '12.f1.a1', 'projective_image': '432.741', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.f1.a1', 'subgroup_fusion': None, 'weyl_group': '36.10'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '20.5', '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, 4, 2, 2, 3, 3], 'aut_gens': [[1, 2, 12], [6, 121, 80], [9, 10, 12], [1, 2, 156], [9, 130, 132], [1, 2, 48], [9, 62, 12], [5, 62, 12]], 'aut_group': '288.877', 'aut_hash': 877, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 10, 'aut_perms': [379560, 167160, 9, 887880, 16, 1612800, 1574040], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 2, 1], [2, 9, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [5, 1, 4, 1], [6, 6, 2, 1], [10, 3, 8, 1], [10, 9, 4, 1], [15, 2, 8, 1], [15, 4, 4, 1], [30, 6, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times \\SOPlus(4,2)', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '72.40', 'autcentquo_hash': 40, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)', 'cc_stats': [[1, 1, 1], [2, 3, 2], [2, 9, 1], [3, 2, 2], [3, 4, 1], [5, 1, 4], [6, 6, 2], [10, 3, 8], [10, 9, 4], [15, 2, 8], [15, 4, 4], [30, 6, 8]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '36.10', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 28, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 2], [2, 9, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [5, 1, 4, 1], [6, 6, 1, 2], [10, 3, 4, 2], [10, 9, 4, 1], [15, 2, 4, 2], [15, 4, 4, 1], [30, 6, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 30, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '180.28', 'hash': 28, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 3], 'inner_gens': [[1, 10, 12], [5, 2, 132], [1, 62, 12]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'S_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 20], [2, 20], [4, 5]], 'label': '180.28', 'linC_count': 100, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 20, 'linQ_dim': 8, 'linQ_dim_count': 20, 'linR_count': 56, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5*S3^2', 'ngens': 5, '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': 12, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 45, 'number_divisions': 18, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 44, 'number_subgroups': 120, 'old_label': None, 'order': 180, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 15], [3, 8], [5, 4], [6, 12], [10, 60], [15, 32], [30, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 121, 116], [1, 10, 96]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5], [8, 4], [16, 1]], 'representations': {'PC': {'code': 142380383727052823, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -3, -3, -5, 101, 26, 122, 1328, 78]}, 'GLZN': {'d': 2, 'p': 22, 'gens': [53245, 133343, 10891, 10933, 19886]}, 'Perm': {'d': 11, 'gens': [1, 362880, 5904, 3991680, 3]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times S_3^2', 'transitive_degree': 30, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [4, 12, 6, 24, 4], 'aut_gens': [[288206, 48, 288183, 160649, 13455751025, 10110, 5094, 288152], [160679, 30, 338600, 328479, 13455918873, 298281, 116639, 160625], [288158, 48, 328502, 45359, 7743904505, 5790, 5046, 288152], [328479, 48, 288183, 44638, 25866130984, 5790, 5046, 328473], [155590, 48, 298263, 55, 13455590401, 338582, 45311, 155584], [160679, 48, 338582, 328497, 13455918873, 298263, 116639, 160625]], 'aut_group': None, 'aut_hash': 7882798221068001515, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3456, 'aut_permdeg': 19, 'aut_perms': [97623060562171011, 83673582473787954, 89387170365081612, 27838675343844895, 97518440353051011], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 2, 1], [2, 9, 1, 1], [2, 18, 2, 1], [2, 27, 1, 1], [3, 2, 1, 1], [3, 4, 2, 1], [3, 8, 2, 1], [4, 18, 1, 1], [4, 54, 1, 1], [5, 1, 4, 1], [6, 12, 2, 3], [6, 18, 1, 1], [6, 24, 2, 1], [6, 36, 2, 1], [10, 3, 4, 1], [10, 6, 8, 1], [10, 9, 4, 1], [10, 18, 8, 1], [10, 27, 4, 1], [12, 36, 1, 1], [15, 2, 4, 1], [15, 4, 8, 1], [15, 8, 8, 1], [20, 18, 4, 1], [20, 54, 4, 1], [30, 12, 8, 3], [30, 18, 4, 1], [30, 24, 8, 1], [30, 36, 8, 1], [60, 36, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_6\\times S_3\\wr C_2).C_2^3', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '864.4662', 'autcentquo_hash': 4662, 'autcentquo_nilpotent': False, 'autcentquo_order': 864, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9:D_6', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 6, 2], [2, 9, 1], [2, 18, 2], [2, 27, 1], [3, 2, 1], [3, 4, 2], [3, 8, 2], [4, 18, 1], [4, 54, 1], [5, 1, 4], [6, 12, 6], [6, 18, 1], [6, 24, 2], [6, 36, 2], [10, 3, 4], [10, 6, 8], [10, 9, 4], [10, 18, 8], [10, 27, 4], [12, 36, 1], [15, 2, 4], [15, 4, 8], [15, 8, 8], [20, 18, 4], [20, 54, 4], [30, 12, 24], [30, 18, 4], [30, 24, 8], [30, 36, 8], [60, 36, 4]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '432.741', 'commutator_count': 1, 'commutator_label': '54.13', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 91, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['6.1', 1], ['72.40', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 2], [2, 9, 1, 1], [2, 18, 1, 2], [2, 27, 1, 1], [3, 2, 1, 1], [3, 4, 1, 2], [3, 8, 1, 2], [4, 18, 1, 1], [4, 54, 1, 1], [5, 1, 4, 1], [6, 12, 1, 6], [6, 18, 1, 1], [6, 24, 1, 2], [6, 36, 1, 2], [10, 3, 4, 1], [10, 6, 4, 2], [10, 9, 4, 1], [10, 18, 4, 2], [10, 27, 4, 1], [12, 36, 1, 1], [15, 2, 4, 1], [15, 4, 4, 2], [15, 8, 4, 2], [20, 18, 4, 1], [20, 54, 4, 1], [30, 12, 4, 6], [30, 18, 4, 1], [30, 24, 4, 2], [30, 36, 4, 2], [60, 36, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 833280, 'exponent': 60, 'exponents_of_order': [4, 3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 0, 16]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '2160.dm', 'hash': 5240928682237879649, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3, 3, 2, 2, 3, 2, 2], 'inner_gens': [[288206, 30, 328520, 116591, 13455706985, 10128, 5094, 288152], [288158, 48, 288183, 160679, 13455751025, 10110, 5046, 288152], [25, 48, 288183, 45311, 13455635705, 10110, 5064, 1], [726, 30, 10128, 160649, 13455751025, 288201, 7, 720], [774, 48, 10110, 160649, 13455751025, 288183, 55, 720], [288176, 48, 288183, 111550, 13455701944, 10110, 10824, 288152], [288206, 30, 288201, 155590, 13455745984, 5808, 5094, 288152], [288206, 48, 328502, 116609, 13455706985, 10110, 5094, 288152]], 'inner_hash': 741, 'inner_nilpotent': False, 'inner_order': 432, 'inner_split': False, 'inner_tex': 'S_3^3:C_2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 8, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 16, 'irrep_stats': [[1, 40], [2, 30], [4, 45], [8, 20]], 'label': '2160.dm', 'linC_count': 768, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 256, 'linQ_dim': 10, 'linQ_dim_count': 256, 'linR_count': 576, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'S3^3:C10', '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, 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': 36, 'number_characteristic_subgroups': 36, 'number_conjugacy_classes': 135, 'number_divisions': 54, 'number_normal_subgroups': 56, 'number_subgroup_autclasses': 236, 'number_subgroup_classes': 384, 'number_subgroups': 3440, 'old_label': None, 'order': 2160, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 87], [3, 26], [4, 72], [5, 4], [6, 210], [10, 348], [12, 36], [15, 104], [20, 288], [30, 840], [60, 144]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [288872, 0], 'outer_gens': [[160679, 48, 293943, 288176, 25866374552, 334262, 44638, 160625], [288206, 48, 328502, 160649, 7744019825, 5790, 10854, 288152]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 17], [8, 10], [16, 9], [32, 4]], 'representations': {'PC': {'code': '50237945519281595157768678547456823850464971197908070564677', 'gens': [1, 2, 5, 6, 8], 'pres': [8, 2, 2, 2, 5, 3, 2, 3, 3, 353, 41, 66, 10412, 820, 30725, 15373, 7221, 141, 31366, 15694, 6742, 92167, 815]}, 'Perm': {'d': 14, 'gens': [288206, 48, 288183, 160649, 13455751025, 10110, 5094, 288152]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3^3:C_{10}', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
