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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1218.11', 'ambient_counter': 11, 'ambient_order': 1218, 'ambient_tex': 'C_3\\times D_{203}', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 87, 'counter': 5, 'cyclic': False, 'direct': None, 'hall': 174, 'label': '1218.11.7.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '7.a1.a1', 'outer_equivalence': False, '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': 7, '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': '174.2', 'subgroup_hash': 2, 'subgroup_order': 174, 'subgroup_tex': 'C_3\\times D_{29}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1218.11', 'aut_centralizer_order': 6, 'aut_label': '7.a1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '1624.43', 'aut_weyl_index': 42, 'centralizer': '406.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['14.a1.a1', '21.a1.a1', '203.a1.a1'], 'core': '14.a1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [6507, -1, 7526, -1, 8540, -1, 5790, -1], 'generators': [1, 812, 42], 'label': '1218.11.7.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '7.a1.a1', 'projective_image': '406.5', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '7.a1.a1', 'subgroup_fusion': None, 'weyl_group': '58.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, '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': 812, 'aut_gen_orders': [28, 58], 'aut_gens': [[1, 2], [1, 62], [169, 118]], 'aut_group': '1624.43', 'aut_hash': 43, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1624, 'aut_permdeg': 31, 'aut_perms': [28427838184161711685914586317360, 532454996034526474225826194971121], 'aut_phi_ratio': 29.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 29, 1, 1], [3, 1, 2, 1], [6, 29, 2, 1], [29, 2, 14, 1], [87, 2, 28, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_{29}', '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': 812, 'autcentquo_group': '812.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 812, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{29}', 'cc_stats': [[1, 1, 1], [2, 29, 1], [3, 1, 2], [6, 29, 2], [29, 2, 14], [87, 2, 28]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '58.1', 'commutator_count': 1, 'commutator_label': '29.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '29.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['58.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 29, 1, 1], [3, 1, 2, 1], [6, 29, 2, 1], [29, 2, 14, 1], [87, 2, 28, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 174, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 7, 29], 'factors_of_order': [2, 3, 29], 'faithful_reps': [[2, 0, 28]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '174.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 58, 'inner_gen_orders': [2, 29], 'inner_gens': [[1, 56], [121, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 58, 'inner_split': False, 'inner_tex': 'D_{29}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 56, 'irrQ_dim': 56, 'irrR_degree': 4, 'irrep_stats': [[1, 6], [2, 42]], 'label': '174.2', 'linC_count': 28, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 30, 'linQ_degree_count': 2, 'linQ_dim': 30, 'linQ_dim_count': 2, 'linR_count': 42, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*D29', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 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': 48, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 64, 'old_label': None, 'order': 174, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 29], [3, 2], [6, 58], [29, 28], [87, 56]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 14, 'outer_gen_orders': [2, 14], 'outer_gen_pows': [1, 0], 'outer_gens': [[1, 92], [1, 40]], 'outer_group': '28.4', 'outer_hash': 4, 'outer_nilpotent': True, 'outer_order': 28, 'outer_permdeg': 11, 'outer_perms': [40320, 3629673], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{14}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 32, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [28, 1], [56, 1]], 'representations': {'PC': {'code': 575331670199, 'gens': [1, 2], 'pres': [3, -2, -3, -29, 337, 22, 1514]}, 'GLFp': {'d': 2, 'p': 349, 'gens': [4675940647, 6163739621, 122150]}, 'Perm': {'d': 32, 'gens': [265558152069838740888854948258664, 3, 8753954958317600299503272776517304]}}, '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 D_{29}', 'transitive_degree': 87, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 2436, 'aut_gen_orders': [28, 42, 28, 42], 'aut_gens': [[1, 2], [181, 436], [1165, 218], [1009, 632], [1033, 416]], 'aut_group': None, 'aut_hash': 2327187107679239996, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 68208, 'aut_permdeg': 205, 'aut_perms': [251347488100031144434656232754628922852408324955740080002270504850610617189139300261552552020974272030169633395979550236512937297052984373357092113934567950890247202183490733280447292187425405887167310617066869235419726152679110662439014269082560743444345707807937671841927563207767167562667053074716097850860101410348892255792660641675372827418413813311909813078335800279080392366296214, 11488523642223069662107181171699473212618884160463165573099519756497217086406800240711594819936973607208744829286282786327265155510248919525737815929927080837769335205177996686793907354572867876845221263202024030812806662753514841204599179404433935596140993363675467156321756151826542395974842507248759002933628156489923700219541462877685346935137476985200163469310670976692289652432800, 39794676521659556728109185382219835763568140490379830983422244235633430914862820655367756838845211397415970175492669796194796691426053258986850920084532559593689163257382333721038187308711517355797356164153636150653372723090099453288517819424078798668465404235653959012423780223437131354503187579358131808182094082853363451429796693172602175147112715866930902468795451653734666150884271, 259432053499264100735549346284766608670281565380437463919329904132355027935857014261403732858187431547292067923576870454198158932759780410279780327493091007657495847877395379735427449636650837001153426338370697695144703748686152445775487412357820237661433867512408228191682133209608589635834212970304153817603612193687730221760146061713543338144918242767967127758211259064195823697345938], 'aut_phi_ratio': 203.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 203, 1, 1], [3, 1, 2, 1], [6, 203, 2, 1], [7, 2, 3, 1], [21, 2, 6, 1], [29, 2, 14, 1], [87, 2, 28, 1], [203, 2, 84, 1], [609, 2, 168, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{203}.C_{42}.C_2^3', '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': 2436, 'autcentquo_group': None, 'autcentquo_hash': 253864559525441835, 'autcentquo_nilpotent': False, 'autcentquo_order': 34104, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{29}:(C_7^2:(C_2\\times C_{12}))', 'cc_stats': [[1, 1, 1], [2, 203, 1], [3, 1, 2], [6, 203, 2], [7, 2, 3], [21, 2, 6], [29, 2, 14], [87, 2, 28], [203, 2, 84], [609, 2, 168]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '406.5', 'commutator_count': 1, 'commutator_label': '203.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '7.1', '29.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['406.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 203, 1, 1], [3, 1, 2, 1], [6, 203, 2, 1], [7, 2, 3, 1], [21, 2, 6, 1], [29, 2, 14, 1], [87, 2, 28, 1], [203, 2, 84, 1], [609, 2, 168, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 1218, 'exponents_of_order': [1, 1, 1, 1], 'factors_of_aut_order': [2, 3, 7, 29], 'factors_of_order': [2, 3, 7, 29], 'faithful_reps': [[2, 0, 168]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1218.11', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 406, 'inner_gen_orders': [2, 203], 'inner_gens': [[1, 404], [817, 2]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 406, 'inner_split': True, 'inner_tex': 'D_{203}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 336, 'irrQ_dim': 336, 'irrR_degree': 4, 'irrep_stats': [[1, 6], [2, 303]], 'label': '1218.11', 'linC_count': 168, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 36, 'linQ_degree_count': 2, 'linQ_dim': 36, 'linQ_dim_count': 2, 'linR_count': 252, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*D203', 'ngens': 4, '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 309, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 488, 'old_label': None, 'order': 1218, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 203], [3, 2], [6, 406], [7, 6], [21, 12], [29, 28], [87, 56], [203, 168], [609, 336]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [2, 84], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 520], [871, 934]], 'outer_group': '168.39', 'outer_hash': 39, 'outer_nilpotent': True, 'outer_order': 168, 'outer_permdeg': 16, 'outer_perms': [1313941305600, 13452365673], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{84}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 39, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [6, 1], [12, 1], [28, 1], [56, 1], [168, 1], [336, 1]], 'representations': {'PC': {'code': 18799011429973193621943, 'gens': [1, 2], 'pres': [4, -2, -3, -7, -29, 3233, 29, 14546, 94, 18819]}, 'GLFp': {'d': 2, 'p': 811, 'gens': [156551060893, 432063502111]}, 'Perm': {'d': 39, 'gens': [13774094930742041323710367962159238592354664, 3, 444984, 550570807328085194407427960024392768344710400]}}, '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 D_{203}', 'transitive_degree': 609, 'wreath_data': None, 'wreath_product': False}
