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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '18522.q', 'ambient_counter': 17, 'ambient_order': 18522, 'ambient_tex': 'C_7^3:C_3^2:S_3', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 1, 'counter': 49, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '18522.q.3087.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '3087.a1', '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': 3087, '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': '6.2', 'subgroup_hash': 2, 'subgroup_order': 6, 'subgroup_tex': 'C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '18522.q', 'aut_centralizer_order': None, 'aut_label': '3087.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '3087.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['441.a1', '441.b1', '441.d1', '1029.a1'], 'contains': ['6174.b1', '9261.a1'], 'core': '18522.a1', 'coset_action_label': None, 'count': 3087, 'diagramx': [2510, -1, 5550, -1], 'generators': [3427, 10068], 'label': '18522.q.3087.a1', 'mobius_quo': None, 'mobius_sub': -3, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3087.a1', 'old_label': '3087.a1', 'projective_image': '18522.q', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3087.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [5]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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], [2, 1, 1], [3, 1, 2], [6, 1, 2]], 'center_label': '6.2', 'center_order': 6, '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': ['2.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 6]], 'label': '6.2', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6', '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': 6, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5]], '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': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2]], 'representations': {'PC': {'code': 21, 'gens': [1], 'pres': [2, -2, -3, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [73]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 56]}, 'Perm': {'d': 5, 'gens': [24, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6', 'transitive_degree': 6, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 6, 'aut_exponent': 252, 'aut_gen_orders': [3, 18], 'aut_gens': [[1, 2, 6, 18, 378, 2646], [9835, 15758, 6, 288, 378, 2646], [479, 17472, 17380, 18152, 6264, 6642]], 'aut_group': None, 'aut_hash': 4061174464532135472, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 444528, 'aut_permdeg': 63, 'aut_perms': [1561256972709163435746238807376563724991638141008314849961959810734853799986551633637189, 1318306919327821762059867839945419029851837343882720720683293788282423758295691715071918], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 63, 1, 1], [3, 294, 4, 1], [3, 343, 1, 2], [6, 3087, 1, 2], [7, 9, 8, 1], [7, 27, 8, 1], [7, 54, 1, 1], [14, 189, 8, 2], [21, 882, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_7^3.\\He_3.Q_8.C_6', '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': 252, 'autcentquo_group': None, 'autcentquo_hash': 4061174464532135472, 'autcentquo_nilpotent': False, 'autcentquo_order': 444528, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_7^3.\\He_3.Q_8.C_6', 'cc_stats': [[1, 1, 1], [2, 63, 1], [3, 294, 4], [3, 343, 2], [6, 3087, 2], [7, 9, 8], [7, 27, 8], [7, 54, 1], [14, 189, 16], [21, 882, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '18522.q', 'commutator_count': 1, 'commutator_label': '9261.e', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '7.1', '7.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 17, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 63, 1, 1], [3, 294, 1, 4], [3, 343, 2, 1], [6, 3087, 2, 1], [7, 9, 2, 4], [7, 27, 2, 4], [7, 54, 1, 1], [14, 189, 2, 8], [21, 882, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 7411824, 'exponent': 42, 'exponents_of_order': [3, 3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[9, 0, 16], [18, 0, 8], [27, 0, 16], [54, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '18522.q', 'hash': 4981235326282517820, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [6, 3, 21, 21, 7, 7], 'inner_gens': [[1, 14044, 4440, 7320, 8100, 10584], [707, 2, 462, 16806, 15444, 6048], [10273, 12200, 6, 1044, 9450, 2646], [18421, 11798, 2004, 18, 6804, 5292], [11179, 6482, 12102, 14760, 378, 2646], [10585, 17768, 6, 15894, 378, 2646]], 'inner_hash': 4981235326282517820, 'inner_nilpotent': False, 'inner_order': 18522, 'inner_split': True, 'inner_tex': 'C_7^3:C_3^2:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': 9, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 2], [2, 4], [3, 4], [9, 16], [18, 8], [27, 16], [54, 1]], 'label': '18522.q', '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': 'C7^3:C3^2:S3', '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, 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': 13, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 51, 'number_divisions': 29, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 54, 'number_subgroup_classes': 156, 'number_subgroups': 22276, 'old_label': None, 'order': 18522, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 63], [3, 1862], [6, 6174], [7, 342], [14, 3024], [21, 7056]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [3, 4], 'outer_gen_pows': [10068, 10068], 'outer_gens': [[17393, 8068, 7400, 2620, 15444, 12582], [6967, 10892, 11468, 7600, 18306, 15552]], 'outer_group': '24.3', 'outer_hash': 3, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 8, 'outer_perms': [430, 11657], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\SL(2,3)', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 4], [6, 2], [18, 8], [36, 4], [54, 9]], 'representations': {'PC': {'code': '2476221336175889615170135909922109921182192538864327163383202322012963472977470164388170911', 'gens': [1, 2, 3, 4, 6, 7], 'pres': [7, -2, -3, -3, 3, -7, 7, 7, 23982, 196617, 93242, 4860, 55582, 204963, 235294, 4889, 108, 317524, 138926, 648, 340205, 324336, 66169, 15902, 518622, 148189, 14433]}, 'Perm': {'d': 21, 'gens': [18428720747012313890, 51090942109530359706, 20388914196081981350]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_7^3:C_3^2:S_3', 'transitive_degree': 21, 'wreath_data': None, 'wreath_product': False}
