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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '350.9', 'ambient_counter': 9, 'ambient_order': 350, 'ambient_tex': 'C_5:D_{35}', 'central': False, 'central_factor': False, 'centralizer_order': 175, 'characteristic': False, 'core_order': 35, 'counter': 15, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '350.9.10.a1.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '10.a1.f1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '10.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 10, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '35.1', 'subgroup_hash': 1, 'subgroup_order': 35, 'subgroup_tex': 'C_{35}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '350.9', 'aut_centralizer_order': 3500, 'aut_label': '10.a1', 'aut_quo_index': 1, 'aut_stab_index': 6, 'aut_weyl_group': '24.9', 'aut_weyl_index': 21000, 'centralizer': '2.a1.a1', 'complements': ['35.a1.a1', '35.a1.b1', '35.a1.d1', '35.a1.c1', '35.a1.e1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '5.a1.f1'], 'contains': ['50.a1.a1', '70.a1.f1'], 'core': '10.a1.f1', 'coset_action_label': None, 'count': 1, 'diagramx': [5080, 5274, 9802, 6778, 6463, 7697, 1122, 6613], 'generators': [142, 50], 'label': '350.9.10.a1.f1', 'mobius_quo': 1, 'mobius_sub': 5, 'normal_closure': '10.a1.f1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['50.a1.a1', '70.a1.f1'], 'normalizer': '1.a1.a1', 'old_label': '10.a1.f1', 'projective_image': '350.9', 'quotient_action_image': '2.1', 'quotient_action_kernel': '5.1', 'quotient_action_kernel_order': 5, 'quotient_fusion': None, 'short_label': '10.a1.f1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '35.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1], [6], [3]], 'aut_group': '24.9', 'aut_hash': 9, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 24, 'aut_permdeg': 9, 'aut_perms': [42000, 867], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [7, 1, 6, 1], [35, 1, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{12}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.9', 'autcent_hash': 9, 'autcent_nilpotent': True, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{12}', '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], [5, 1, 4], [7, 1, 6], [35, 1, 24]], 'center_label': '35.1', 'center_order': 35, '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': ['5.1', '7.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [7, 1, 6, 1], [35, 1, 24, 1]], 'element_repr_type': 'PC', 'elementary': 35, 'eulerian_function': 1, 'exponent': 35, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [5, 7], 'faithful_reps': [[1, 0, 24]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '35.1', 'hash': 1, 'hyperelementary': 35, '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': 24, 'irrQ_dim': 24, 'irrR_degree': 2, 'irrep_stats': [[1, 35]], 'label': '35.1', 'linC_count': 24, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 12, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C35', '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': 35, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 35, 'order_factorization_type': 11, 'order_stats': [[1, 1], [5, 4], [7, 6], [35, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[6], [3]], 'outer_group': '24.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [42000, 867], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{12}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [5, 7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [6, 1], [24, 1]], 'representations': {'PC': {'code': 203, 'gens': [1], 'pres': [2, -5, -7, 10]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [72212]}, 'Perm': {'d': 12, 'gens': [159667200, 4320]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [35], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{35}', 'transitive_degree': 35, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 840, 'aut_gen_orders': [12, 20, 20], 'aut_gens': [[1, 2, 10], [215, 78, 192], [101, 212, 202], [225, 4, 64]], 'aut_group': None, 'aut_hash': 482492227677916899, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 504000, 'aut_permdeg': 175, 'aut_perms': [700598358285431837819187525711887508225206924243846839019788847165818724844010066201421500225281173265599408303207219458473401941283254231341635917372059296321674158155814068822980811185685439296309197476001819074484195253063267273390278129472259181767940129048606150271793776916171457208198202436272489316472933122612, 1067269288582825612183900843601562226079915893798045368426369017721331583359599559832344572537314107887081782367968724410957781557482460850653671844765706605722106592827351523730476012768622296973866076741094467350944305313915321931247312761405864874665160206354623462272524683400189891784433328309272565561138938682819, 437353279488507747215123333881686307732066668680924712440513683594044793896058092921426535495762088238935683020210998173952389951147209888675899841878315078749849878020680986203560596483051299881567646316721035427931174010639597449001363957500109141615712916000113914819178538283132237411721887547450073606348609939811], 'aut_phi_ratio': 4200.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 175, 1, 1], [5, 2, 12, 1], [7, 2, 3, 1], [35, 2, 72, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^2:C_4.S_5\\times F_7', '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': 840, 'autcentquo_group': None, 'autcentquo_hash': 482492227677916899, 'autcentquo_nilpotent': False, 'autcentquo_order': 504000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^2:C_4.S_5\\times F_7', 'cc_stats': [[1, 1, 1], [2, 175, 1], [5, 2, 12], [7, 2, 3], [35, 2, 72]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '350.9', 'commutator_count': 1, 'commutator_label': '175.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '5.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 175, 1, 1], [5, 2, 2, 6], [7, 2, 3, 1], [35, 2, 12, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 56, 'exponent': 70, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '350.9', 'hash': 9, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [2, 5, 35], 'inner_gens': [[1, 8, 340], [5, 2, 10], [21, 2, 10]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 350, 'inner_split': True, 'inner_tex': 'C_5:D_{35}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 87]], 'label': '350.9', 'linC_count': 2880, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 15, 'linQ_dim': 14, 'linQ_dim_count': 15, 'linR_count': 2880, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5:D35', '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 89, 'number_divisions': 15, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 32, 'number_subgroups': 464, 'old_label': None, 'order': 350, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 175], [5, 24], [7, 6], [35, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [2, 4, 24, 12, 24], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 60], [1, 4, 220], [1, 214, 192], [1, 6, 330], [1, 214, 338]], 'outer_group': '1440.1683', 'outer_hash': 1683, 'outer_nilpotent': False, 'outer_order': 1440, 'outer_permdeg': 27, 'outer_perms': [1291780674555363076995479544, 6944509515767057892681024984, 6298483160206003154231734708, 6944509515767057892681024988, 3019285671845808623776733764], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_3\\times \\GL(2,5)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 6], [6, 1], [24, 6]], 'representations': {'PC': {'code': 144468044517443, 'gens': [1, 2, 3], 'pres': [4, -2, -5, -5, -7, 65, 4082, 70, 4803]}, 'Perm': {'d': 17, 'gens': [1313905058047, 23544365644800, 42019983573120, 973]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5:D_{35}', 'transitive_degree': 175, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 5], 'aut_gens': [[1, 2], [3, 6], [5, 2]], 'aut_group': '20.3', 'aut_hash': 3, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20, 'aut_permdeg': 5, 'aut_perms': [17, 89], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_5', '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': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [5, 2, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 10, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 1, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '10.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 8], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 2]], 'label': '10.1', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D5', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 8, 'old_label': None, 'order': 10, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 5], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [1], 'outer_gens': [[1, 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': 5, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 1]], 'representations': {'PC': {'code': 227, 'gens': [1, 2], 'pres': [2, -2, -5, 33]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [23707240, 26329372]}, 'Lie': [{'d': 2, 'q': 4, 'gens': [81, 94], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 129]}, 'Perm': {'d': 5, 'gens': [23, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
