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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1992.1', 'ambient_counter': 1, 'ambient_order': 1992, 'ambient_tex': 'C_3:C_{664}', 'central': True, 'central_factor': False, 'centralizer_order': 1992, 'characteristic': True, 'core_order': 2, 'counter': 15, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1992.1.996.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '996.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '996.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 996, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3:C_{332}', 'simple': True, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1992.1', 'aut_centralizer_order': 1968, 'aut_label': '996.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 1968, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['12.a1.a1', '332.a1.a1', '498.a1.a1'], 'contains': ['1992.a1.a1'], 'core': '996.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3540, 3473, 3827, 4340, 5456, 6622, 5852, 6622], 'generators': [4], 'label': '1992.1.996.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '996.a1.a1', 'normal_contained_in': ['12.a1.a1', '332.a1.a1', '498.a1.a1'], 'normal_contains': ['1992.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '996.a1.a1', 'projective_image': '996.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '996.1', 'quotient_action_kernel_order': 996, 'quotient_fusion': None, 'short_label': '996.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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '2.1', 'center_order': 2, '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': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '664.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 246, 'aut_gen_orders': [2, 2, 2, 246], 'aut_gens': [[1, 8], [5, 8], [1, 1336], [3, 8], [1329, 272]], 'aut_group': '1968.201', 'aut_hash': 201, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1968, 'aut_permdeg': 94, 'aut_perms': [17463842137501446057462699372629563242369338615115663374879947935960534361760847735593172757845929888446792335360163699207, 463843429323168143444473124721017739350290866090283727900220721246891910625812247698882326306640402778395864344673599883824, 405872089845726216141590962202596025515894785993282712092791080183025260760886418311636032066184622660246238346386282077056, 43698935833826685623084654948245412804257858858198847096651888008912976309887189746848107630967380484007803826781368245039670533710511251873373808], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 1, 2, 1], [6, 2, 1, 1], [8, 3, 4, 1], [12, 2, 2, 1], [83, 1, 82, 1], [166, 1, 82, 1], [249, 2, 82, 1], [332, 1, 164, 1], [498, 2, 82, 1], [664, 3, 328, 1], [996, 2, 164, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{246}:C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 82, 'autcent_group': '328.15', 'autcent_hash': 15, 'autcent_nilpotent': True, 'autcent_order': 328, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{82}', '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, 1, 1], [3, 2, 1], [4, 1, 2], [6, 2, 1], [8, 3, 4], [12, 2, 2], [83, 1, 82], [166, 1, 82], [249, 2, 82], [332, 1, 164], [498, 2, 82], [664, 3, 328], [996, 2, 164]], 'center_label': '332.2', 'center_order': 332, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '83.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['24.1', 1], ['83.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 1, 2, 1], [6, 2, 1, 1], [8, 3, 4, 1], [12, 2, 2, 1], [83, 1, 82, 1], [166, 1, 82, 1], [249, 2, 82, 1], [332, 1, 164, 1], [498, 2, 82, 1], [664, 3, 328, 1], [996, 2, 164, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1008, 'exponent': 1992, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 41], 'factors_of_order': [2, 3, 83], 'faithful_reps': [[2, 0, 164]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '498.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 1336], [665, 8]], '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': 328, 'irrQ_dim': 328, 'irrR_degree': None, 'irrep_stats': [[1, 664], [2, 332]], 'label': '1992.1', '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': True, 'metacyclic': True, 'monomial': True, 'name': 'C3:C664', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 996, 'number_divisions': 14, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 20, 'old_label': None, 'order': 1992, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [8, 12], [12, 4], [83, 82], [166, 82], [249, 164], [332, 164], [498, 164], [664, 984], [996, 328]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 82, 'outer_gen_orders': [2, 2, 82], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[5, 8], [3, 8], [1, 1688]], 'outer_group': '328.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 328, 'outer_permdeg': 47, 'outer_perms': [5502622159812088949850305428800254892961651752960000000000, 2658271574788448768043625811014615890319638528000000000, 1437642729082795807916967057028356205707264000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{82}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 94, 'pgroup': 0, 'primary_abelian_invariants': [8, 83], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 2], [82, 2], [164, 3], [328, 2]], 'representations': {'PC': {'code': 66257961073843625295400195, 'gens': [1, 4], 'pres': [5, -2, -2, -2, -3, -83, 10, 26, 26723, 78]}, 'GLFp': {'d': 2, 'p': 997, 'gens': [686781692594, 339922252082, 1335980]}, 'Perm': {'d': 94, 'gens': [4083265, 1169347623700585276696643729757774793654840054462242304485405341093556139412655451199158077142491080220522169349448965967628442336528920376985600, 8443680, 12501360, 3]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [664], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:C_{664}', 'transitive_degree': 1992, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '332.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 246, 'aut_gen_orders': [2, 2, 246], 'aut_gens': [[1, 4], [3, 4], [1, 668], [665, 628]], 'aut_group': '984.44', 'aut_hash': 44, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 984, 'aut_permdeg': 88, 'aut_perms': [164429581433574880927898820235310765346321305960647363096836930467507732753356245910922983792641443329875530947555411995361742405499024, 7921657362770265548304, 157584549832128763326364963693679534313203948602157701395625541723353754556711896311830442593645271395584402337550486294886547653937193], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 3, 2, 1], [6, 2, 1, 1], [83, 1, 82, 1], [166, 1, 82, 1], [249, 2, 82, 1], [332, 3, 164, 1], [498, 2, 82, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6\\times C_{82}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 82, 'autcent_group': '164.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 164, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{82}', '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, 1, 1], [3, 2, 1], [4, 3, 2], [6, 2, 1], [83, 1, 82], [166, 1, 82], [249, 2, 82], [332, 3, 164], [498, 2, 82]], 'center_label': '166.2', 'center_order': 166, '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', '2.1', '3.1', '83.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.1', 1], ['83.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 3, 2, 1], [6, 2, 1, 1], [83, 1, 82, 1], [166, 1, 82, 1], [249, 2, 82, 1], [332, 3, 164, 1], [498, 2, 82, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 504, 'exponent': 996, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 41], 'factors_of_order': [2, 3, 83], 'faithful_reps': [[2, 0, 82]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '498.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 668], [333, 4]], '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': 164, 'irrQ_dim': 328, 'irrR_degree': 4, 'irrep_stats': [[1, 332], [2, 166]], 'label': '996.1', 'linC_count': 82, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 84, 'linQ_degree_count': 2, 'linQ_dim': 86, 'linQ_dim_count': 4, 'linR_count': 123, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3:C332', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 498, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 16, 'old_label': None, 'order': 996, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 6], [6, 2], [83, 82], [166, 82], [249, 164], [332, 492], [498, 164]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 82, 'outer_gen_orders': [2, 82], 'outer_gen_pows': [0, 0], 'outer_gens': [[665, 992], [667, 260]], 'outer_group': '164.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 164, 'outer_permdeg': 45, 'outer_perms': [1405006117752879898543142606244511569936384000000000, 2658304211399778683952999635465399734955409408000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{82}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 90, 'pgroup': 0, 'primary_abelian_invariants': [4, 83], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [82, 2], [164, 3]], 'representations': {'PC': {'code': 3169035348579396595, 'gens': [1, 3], 'pres': [4, -2, -2, -3, -83, 8, 8018, 46]}, 'GLFp': {'d': 2, 'p': 167, 'gens': [633821644, 65319392]}, 'Perm': {'d': 90, 'gens': [129, 16695570072624210767034167688394623360733515163575864136345910335924039962404869510225723072235842668787507993136908442336528920420934400, 16, 840]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [332], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:C_{332}', 'transitive_degree': 996, 'wreath_data': None, 'wreath_product': False}
