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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '2916.ev', 'ambient_counter': 126, 'ambient_order': 2916, 'ambient_tex': 'C_9^2.S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 1, 'counter': 186, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2916.ev.162.l1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '162.l1.c1', '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': 162, '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': '18.3', 'subgroup_hash': 3, 'subgroup_order': 18, 'subgroup_tex': 'C_3\\times S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2916.ev', 'aut_centralizer_order': 27, 'aut_label': '162.l1', 'aut_quo_index': None, 'aut_stab_index': 81, 'aut_weyl_group': '12.4', 'aut_weyl_index': 2187, 'centralizer': '324.l1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['54.m1.c1', '54.n1.a1', '81.f1.c1'], 'contains': ['324.k1.c1', '486.f1.a1', '486.h1.b1'], 'core': '2916.a1.a1', 'coset_action_label': None, 'count': 27, 'diagramx': [2077, -1, 6391, -1, 8185, -1, 6681, -1], 'generators': [145, 108, 2172], 'label': '2916.ev.162.l1.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '6.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.h1.c1', 'old_label': '162.l1.c1', 'projective_image': '2916.ev', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '162.l1.c1', '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 6], [5, 12], [17, 6]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', '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': 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, 3, 1], [3, 1, 2], [3, 2, 3], [6, 3, 2]], 'center_label': '3.1', 'center_order': 3, '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', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '18.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 12], [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, 6], [2, 3]], 'label': '18.3', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*S3', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 9, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 9, 'number_subgroups': 14, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 8], [6, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5, 6]], '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': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 5451, 'gens': [1, 3], 'pres': [3, -2, -3, -3, 6, 110]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [11780110, 20974441]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'GU'}, {'d': 2, 'q': 4, 'gens': [20, 130, 194], 'family': 'COPlus'}, {'d': 2, 'q': 2, 'gens': [20, 130, 138], 'family': 'CU'}, {'d': 1, 'q': 9, 'gens': [93882, 62619, 1930], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 687, 1374]}, 'Perm': {'d': 6, 'gens': [450, 147, 243]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times S_3', 'transitive_degree': 6, 'wreath_data': ['C_3', 'C_2', '2T1'], 'wreath_product': True}
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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': [18, 3, 6], 'aut_gens': [[1, 2, 12, 36, 324], [1205, 262, 1968, 792, 1728], [1757, 62, 228, 1656, 1404], [1293, 142, 2172, 2880, 324]], 'aut_group': None, 'aut_hash': 1415438368598865940, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 26244, 'aut_permdeg': 108, 'aut_perms': [137342979149916079220739251304326734595566297902824269596750947599258432044861198433607852100130479792789929965013570084968071817663114232356611852110827958019922668001275672, 90043010124294963280031501447884637625465217917251357549485027598641062368039961974290857582813204615391443317684991825549087790917653951863754659563650676592586831409784795, 118470645692172679890084881117654532674096509691635847388690911682872660785867472588124845130274900176076522015603566046522995169054689456359026303595857799870019905786542126], 'aut_phi_ratio': 27.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [2, 81, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 6, 3, 1], [3, 54, 1, 1], [3, 108, 1, 1], [6, 54, 1, 1], [6, 54, 3, 1], [6, 162, 1, 1], [6, 486, 1, 1], [9, 2, 3, 1], [9, 6, 3, 3], [9, 12, 1, 1], [9, 12, 3, 1], [9, 12, 9, 1], [9, 108, 1, 3], [18, 54, 3, 3], [18, 162, 3, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_3\\times C_9).C_3^5.C_2^2', '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': None, 'autcentquo_hash': 1415438368598865940, 'autcentquo_nilpotent': False, 'autcentquo_order': 26244, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': '\\He_3:C_3.C_3^4.C_2^2', 'cc_stats': [[1, 1, 1], [2, 27, 1], [2, 81, 1], [2, 243, 1], [3, 2, 1], [3, 6, 4], [3, 54, 1], [3, 108, 1], [6, 54, 4], [6, 162, 1], [6, 486, 1], [9, 2, 3], [9, 6, 9], [9, 12, 13], [9, 108, 3], [18, 54, 9], [18, 162, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '2916.ev', 'commutator_count': 1, 'commutator_label': '729.397', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 126, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [2, 81, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 4], [3, 54, 1, 1], [3, 108, 1, 1], [6, 54, 1, 4], [6, 162, 1, 1], [6, 486, 1, 1], [9, 2, 3, 1], [9, 6, 3, 3], [9, 12, 1, 1], [9, 12, 3, 4], [9, 108, 1, 1], [9, 108, 2, 1], [18, 54, 3, 3], [18, 162, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 18, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 1, 18], [12, 1, 9]], 'familial': False, 'frattini_label': '27.2', 'frattini_quotient': '108.40', 'hash': 6714582029697048367, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [18, 6, 3, 9, 9], 'inner_gens': [[1, 286, 240, 36, 2808], [845, 2, 2076, 396, 2808], [133, 1214, 12, 36, 324], [1, 2882, 12, 36, 324], [757, 758, 12, 36, 324]], 'inner_hash': 6714582029697048367, 'inner_nilpotent': False, 'inner_order': 2916, 'inner_split': True, 'inner_tex': 'C_9^2.S_3^2', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 6, 'irrep_stats': [[1, 4], [2, 6], [4, 5], [6, 30], [12, 12]], 'label': '2916.ev', 'linC_count': 18, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 6, 'linQ_dim': 18, 'linQ_dim_count': 6, 'linR_count': 18, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C9^2.S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 27, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 57, 'number_divisions': 32, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 224, 'number_subgroup_classes': 262, 'number_subgroups': 9245, 'old_label': None, 'order': 2916, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 351], [3, 188], [6, 864], [9, 540], [18, 972]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 3, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [1465, 0], 'outer_gens': [[1633, 10, 2184, 36, 2808], [217, 2, 984, 144, 1296]], 'outer_group': '9.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 9, 'outer_permdeg': 6, 'outer_perms': [3, 144], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 3], [6, 6], [8, 1], [18, 8], [36, 4]], 'representations': {'PC': {'code': '14018937435406837908756861921671689613873177463828156009359945471775462207', 'gens': [1, 2, 4, 5, 7], 'pres': [8, -2, -2, -3, -3, -3, -3, 3, -3, 288, 4577, 41, 18338, 7683, 33227, 979, 7932, 3620, 156, 28525, 12981, 157254, 78638, 34798, 222, 124423, 62223]}, 'Perm': {'d': 27, 'gens': [6694988457505281749541106563, 6307183476091692578629100285]}}, '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_9^2.S_3^2', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
