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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '17496.no', 'ambient_counter': 353, 'ambient_order': 17496, 'ambient_tex': 'C_3^4.S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 36, 'characteristic': False, 'core_order': 1, 'counter': 1899, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '17496.no.972.fp1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '972.fp1', 'outer_equivalence': True, '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': 972, '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': '17496.no', 'aut_centralizer_order': None, 'aut_label': '972.fp1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '486.z1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['324.hj1', '324.hv1', '324.hx1', '324.if1', '486.i1', '486.dy1', '486.el1'], 'contains': ['1944.cu1', '2916.bp1', '2916.bv1'], 'core': '17496.a1', 'coset_action_label': None, 'count': 81, 'diagramx': [7863, -1, 7583, -1], 'generators': [18, 2, 2088], 'label': '17496.no.972.fp1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.l1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '81.c1', 'old_label': '972.fp1', 'projective_image': '17496.no', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '972.fp1', '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': '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [6, 9, 18, 18], 'aut_gens': [[1, 6, 36, 108, 324, 5832], [265, 3342, 3924, 216, 11520, 11664], [9265, 9366, 4032, 108, 6300, 5832], [12481, 12594, 4068, 216, 12060, 5832], [1513, 3390, 144, 108, 8208, 11664]], 'aut_group': None, 'aut_hash': 913560679085993305, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 52488, 'aut_permdeg': 63, 'aut_perms': [1649144805927183336094559723979304776325487989649703244687804644699236013138874409663771, 1800917396716742162596897419820289524414873988694086636519858343235084267956795728642207, 163033986164406336507459141811317480637308742861361004741250997197316889129623720972243, 1806743368637040215442984994364262466619290814317228243600913170154179827037419963424950], 'aut_phi_ratio': 9.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 27, 1, 2], [2, 81, 1, 3], [2, 243, 1, 1], [3, 2, 1, 3], [3, 4, 1, 3], [3, 6, 1, 2], [3, 8, 1, 1], [3, 9, 1, 2], [3, 12, 1, 4], [3, 18, 1, 3], [3, 24, 1, 2], [3, 36, 1, 1], [3, 54, 1, 3], [3, 108, 1, 3], [6, 6, 1, 2], [6, 12, 1, 1], [6, 18, 1, 2], [6, 27, 1, 2], [6, 36, 1, 2], [6, 54, 1, 5], [6, 81, 1, 6], [6, 108, 1, 2], [6, 162, 1, 18], [6, 243, 1, 6], [6, 324, 1, 6], [6, 486, 1, 6], [9, 36, 3, 3], [9, 54, 1, 3], [9, 72, 3, 3], [9, 108, 1, 3], [18, 108, 3, 3], [18, 162, 1, 6], [18, 324, 1, 3], [18, 486, 1, 3]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3.C_3^4.C_2^2\\times S_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': None, 'autcentquo_hash': 913560679085993305, 'autcentquo_nilpotent': False, 'autcentquo_order': 52488, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^3.C_3^4.C_2^2\\times S_3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 27, 2], [2, 81, 3], [2, 243, 1], [3, 2, 3], [3, 4, 3], [3, 6, 2], [3, 8, 1], [3, 9, 2], [3, 12, 4], [3, 18, 3], [3, 24, 2], [3, 36, 1], [3, 54, 3], [3, 108, 3], [6, 6, 2], [6, 12, 1], [6, 18, 2], [6, 27, 2], [6, 36, 2], [6, 54, 5], [6, 81, 6], [6, 108, 2], [6, 162, 18], [6, 243, 6], [6, 324, 6], [6, 486, 6], [9, 36, 9], [9, 54, 3], [9, 72, 9], [9, 108, 3], [18, 108, 9], [18, 162, 6], [18, 324, 3], [18, 486, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '17496.no', 'commutator_count': 1, 'commutator_label': '729.139', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 353, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2916.ef', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 27, 1, 2], [2, 81, 1, 3], [2, 243, 1, 1], [3, 2, 1, 3], [3, 4, 1, 3], [3, 6, 2, 1], [3, 8, 1, 1], [3, 9, 2, 1], [3, 12, 2, 2], [3, 18, 1, 1], [3, 18, 2, 1], [3, 24, 2, 1], [3, 36, 1, 1], [3, 54, 1, 1], [3, 54, 2, 1], [3, 108, 1, 1], [3, 108, 2, 1], [6, 6, 1, 2], [6, 12, 1, 1], [6, 18, 2, 1], [6, 27, 2, 1], [6, 36, 2, 1], [6, 54, 1, 5], [6, 81, 2, 3], [6, 108, 1, 2], [6, 162, 1, 6], [6, 162, 2, 6], [6, 243, 2, 3], [6, 324, 1, 2], [6, 324, 2, 2], [6, 486, 1, 2], [6, 486, 2, 2], [9, 36, 1, 3], [9, 36, 2, 3], [9, 54, 1, 1], [9, 54, 2, 1], [9, 72, 1, 3], [9, 72, 2, 3], [9, 108, 1, 1], [9, 108, 2, 1], [18, 108, 1, 3], [18, 108, 2, 3], [18, 162, 1, 2], [18, 162, 2, 2], [18, 324, 1, 1], [18, 324, 2, 1], [18, 486, 1, 1], [18, 486, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 22643712, 'exponent': 18, 'exponents_of_order': [7, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 0, 6], [24, 1, 3]], 'familial': False, 'frattini_label': '27.5', 'frattini_quotient': '648.731', 'hash': 2844203660225196988, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 6, 3, 3, 18, 3], 'inner_gens': [[1, 1542, 36, 216, 7668, 11664], [2641, 6, 2124, 108, 9468, 11664], [1, 4182, 36, 108, 432, 5832], [217, 6, 36, 108, 324, 5832], [16309, 14406, 252, 108, 324, 11664], [11665, 11670, 36, 108, 11988, 5832]], 'inner_hash': 2844203660225196988, 'inner_nilpotent': False, 'inner_order': 17496, 'inner_split': True, 'inner_tex': 'C_3^4.S_3^3', 'inner_used': [1, 2, 5], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 24], [2, 36], [4, 18], [6, 12], [8, 3], [12, 24], [18, 8], [24, 9], [36, 4]], 'label': '17496.no', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^4.S3^3', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 120, 'number_characteristic_subgroups': 91, 'number_conjugacy_classes': 138, 'number_divisions': 96, 'number_normal_subgroups': 91, 'number_subgroup_autclasses': 2227, 'number_subgroup_classes': 2563, 'number_subgroups': 125996, 'old_label': None, 'order': 17496, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 543], [3, 728], [6, 10392], [9, 1458], [18, 4374]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [0], 'outer_gens': [[49, 6, 3924, 108, 4248, 5832]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 18], [6, 4], [8, 7], [12, 12], [16, 1], [18, 8], [24, 11], [36, 4], [48, 3]], 'representations': {'PC': {'code': '219817183634451366252557497331690105329890745176766431869120788570247337636156590999661074758318162235715200708524462226740599', 'gens': [1, 3, 5, 6, 7, 10], 'pres': [10, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 20, 46262, 62202, 82, 166563, 43453, 17724, 8284, 12965, 536766, 86956, 110486, 13686, 886, 206, 267847, 185777, 19227, 31957, 1967, 317, 349928, 58348, 1166409, 194429, 3669]}, 'Perm': {'d': 21, 'gens': [5262884789913774722, 7798238974511072905, 2703685318984178784]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4.S_3^3', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
