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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1377.8', 'ambient_counter': 8, 'ambient_order': 1377, 'ambient_tex': '\\He_3.C_{51}', 'central': True, 'central_factor': False, 'centralizer_order': 1377, 'characteristic': True, 'core_order': 17, 'counter': 19, 'cyclic': True, 'direct': True, 'hall': 17, 'label': '1377.8.81.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '81.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '81.8', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 8, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 81, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3.\\He_3', 'simple': True, 'solvable': True, 'special_labels': ['C4'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '17.1', 'subgroup_hash': 1, 'subgroup_order': 17, 'subgroup_tex': 'C_{17}', 'supersolvable': True, 'sylow': 17}
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gps_subgroup_data • Show schema
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{'ambient': '1377.8', 'aut_centralizer_order': 324, 'aut_label': '81.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '16.1', 'aut_weyl_index': 324, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['27.a1.a1', '27.b1.a1', '27.c1.a1'], 'contains': ['1377.a1.a1'], 'core': '81.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [899, 648, 841, 1970, 3424, 4772, 2841, 4644], 'generators': [81], 'label': '1377.8.81.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '81.a1.a1', 'normal_contained_in': ['27.a1.a1'], 'normal_contains': ['1377.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '81.a1.a1', 'projective_image': '81.8', 'quotient_action_image': '1.1', 'quotient_action_kernel': '81.8', 'quotient_action_kernel_order': 81, 'quotient_fusion': None, 'short_label': '81.a1.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': '17.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 16, 'aut_gen_orders': [16], 'aut_gens': [[1], [3]], 'aut_group': '16.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 16, 'aut_perms': [1401602636313], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [17, 1, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{16}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 16, 'autcent_group': '16.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{16}', '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], [17, 1, 16]], 'center_label': '17.1', 'center_order': 17, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['17.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [17, 1, 16, 1]], 'element_repr_type': 'PC', 'elementary': 17, 'eulerian_function': 1, 'exponent': 17, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [17], 'faithful_reps': [[1, 0, 16]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '17.1', 'hash': 1, 'hyperelementary': 17, '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': 16, 'irrQ_dim': 16, 'irrR_degree': 2, 'irrep_stats': [[1, 17]], 'label': '17.1', 'linC_count': 16, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 8, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C17', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 17, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 17, 'order_factorization_type': 1, 'order_stats': [[1, 1], [17, 16]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [16], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '16.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 16, 'outer_perms': [1401602636313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{16}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 17, 'pgroup': 17, 'primary_abelian_invariants': [17], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [16, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -17]}, 'Lie': [{'d': 1, 'q': 17, 'gens': [22324392524313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 17, 'gens': [4931]}, 'Perm': {'d': 17, 'gens': [334764638208000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [17], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{17}', 'transitive_degree': 17, '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': '153.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 48, 'aut_gen_orders': [12, 24, 6, 48], 'aut_gens': [[1, 3, 9], [920, 3, 1260], [925, 462, 906], [925, 924, 912], [4, 924, 1182]], 'aut_group': None, 'aut_hash': 5951716982031371295, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5184, 'aut_permdeg': 52, 'aut_perms': [8007210222266211679209132446203272258731176947109265471494268768462, 29997071044975757183909537945684668371954199820665577176669066959473, 31611751660579409526699058817990553287211166272951080991959805444891, 23510956637062307684792900864499506744562082520357010649072077616078], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 9, 2, 1], [9, 3, 6, 1], [9, 9, 4, 1], [17, 1, 16, 1], [51, 1, 32, 1], [51, 3, 32, 1], [51, 9, 32, 1], [153, 3, 96, 1], [153, 9, 64, 1]], 'aut_supersolvable': True, 'aut_tex': '\\He_3.C_{48}.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 48, 'autcent_group': '144.30', 'autcent_hash': 30, 'autcent_nilpotent': True, 'autcent_order': 144, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_3\\times C_{48}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 3, 2], [3, 9, 2], [9, 3, 6], [9, 9, 4], [17, 1, 16], [51, 1, 32], [51, 3, 32], [51, 9, 32], [153, 3, 96], [153, 9, 64]], 'center_label': '51.1', 'center_order': 51, 'central_product': True, 'central_quotient': '27.3', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1', '17.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['81.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 9, 2, 1], [9, 3, 6, 1], [9, 9, 2, 2], [17, 1, 16, 1], [51, 1, 32, 1], [51, 3, 32, 1], [51, 9, 32, 1], [153, 3, 96, 1], [153, 9, 32, 2]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 216, 'exponent': 153, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3, 17], 'faithful_reps': [[3, 0, 96]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '153.2', 'hash': 8, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 3], 'inner_gens': [[1, 462, 474], [919, 3, 9], [922, 3, 9]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 27, 'inner_split': True, 'inner_tex': '\\He_3', 'inner_used': [1, 2, 3], 'irrC_degree': 3, 'irrQ_degree': 288, 'irrQ_dim': 288, 'irrR_degree': 6, 'irrep_stats': [[1, 153], [3, 136]], 'label': '1377.8', 'linC_count': 96, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 34, 'linQ_degree_count': 1, 'linQ_dim': 34, 'linQ_dim_count': 1, 'linR_count': 48, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'He3.C51', 'ngens': 5, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 12, 'number_conjugacy_classes': 289, 'number_divisions': 14, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 28, 'number_subgroups': 64, 'old_label': None, 'order': 1377, 'order_factorization_type': 31, 'order_stats': [[1, 1], [3, 26], [9, 54], [17, 16], [51, 416], [153, 864]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 48, 'outer_gen_orders': [2, 2, 48], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 6, 315], [2, 6, 9], [1, 6, 180]], 'outer_group': '192.935', 'outer_hash': 935, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 23, 'outer_perms': [2439325392333218664, 1124000727777607680000, 19984155209275203], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{48}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 44, 'pgroup': 0, 'primary_abelian_invariants': [3, 3, 17], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 1], [16, 1], [18, 1], [32, 4], [96, 1], [288, 1]], 'representations': {'PC': {'code': 15476630733353753055519, 'gens': [1, 2, 3], 'pres': [5, -3, -3, 3, -3, -17, 4621, 7112, 57, 78]}, 'Perm': {'d': 44, 'gens': [1113382054045330298473822076827514801318422118400000000, 491952856373189362873831983977585598509705584656384000, 590482044366309459597965173064663040000, 120832157984842263734676011776550579041278516609024000, 334764638208000]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 51], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\He_3.C_{51}', 'transitive_degree': 459, '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': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [6, 6, 3, 3], 'aut_gens': [[1, 3, 9], [1, 6, 18], [32, 6, 9], [1, 57, 69], [55, 3, 9]], 'aut_group': '324.117', 'aut_hash': 117, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 324, 'aut_permdeg': 12, 'aut_perms': [3669867, 43607760, 15966720, 332207400], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 9, 2, 1], [9, 3, 6, 1], [9, 9, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3:D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '9.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 9, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 3, 2], [3, 9, 2], [9, 3, 6], [9, 9, 4]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '27.3', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 9, 2, 1], [9, 3, 6, 1], [9, 9, 2, 2]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 12, 'exponent': 9, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [[3, 0, 6]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '9.2', 'hash': 8, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 3], 'inner_gens': [[1, 30, 66], [55, 3, 9], [34, 3, 9]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 27, 'inner_split': True, 'inner_tex': '\\He_3', 'inner_used': [1, 2, 3], 'irrC_degree': 3, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 6, 'irrep_stats': [[1, 9], [3, 8]], 'label': '81.8', 'linC_count': 6, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 1, 'linQ_dim': 18, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3.He3', 'ngens': 2, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 17, 'number_divisions': 7, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 14, 'number_subgroups': 32, 'old_label': None, 'order': 81, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 26], [9, 54]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 3, 72], [1, 6, 18]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 27, 'pgroup': 3, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 1], [18, 1]], 'representations': {'PC': {'code': 36993564, 'gens': [1, 2, 3], 'pres': [4, 3, 3, 3, 3, 241, 794, 46]}, 'GLZN': {'d': 2, 'p': 54, 'gens': [158437, 157789, 6309571, 2991853]}, 'GLZq': {'d': 2, 'q': 27, 'gens': [19765, 199339, 196849, 19927]}, 'Perm': {'d': 27, 'gens': [7550102686338507196661760000, 3323121639686209319422786084, 711549292653507, 806634631153204606767248884]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3.\\He_3', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
