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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '500.46', 'ambient_counter': 46, 'ambient_order': 500, 'ambient_tex': 'C_5^3:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 125, 'characteristic': False, 'core_order': 5, 'counter': 32, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '500.46.100.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '100.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '100.10', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 10, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 100, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5:F_5', 'simple': True, 'solvable': True, 'special_labels': ['C4'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '5.1', 'subgroup_hash': 1, 'subgroup_order': 5, 'subgroup_tex': 'C_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '500.46', 'aut_centralizer_order': 2000, 'aut_label': '100.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '4.1', 'aut_weyl_index': 4000, 'centralizer': '4.a1.a1', 'complements': ['5.a1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['20.a1.a1', '20.b1.a1', '20.d1.a1', '50.b1.a1'], 'contains': ['500.a1.a1'], 'core': '100.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4920, 3842, 6187, 5201, 4142, 3333, 5518, 5149], 'generators': [100], 'label': '500.46.100.b1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '100.b1.a1', 'normal_contained_in': ['20.a1.a1', '20.b1.a1'], 'normal_contains': ['500.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '100.b1.a1', 'projective_image': '500.46', 'quotient_action_image': '4.1', 'quotient_action_kernel': '25.2', 'quotient_action_kernel_order': 25, 'quotient_fusion': None, 'short_label': '100.b1.a1', 'subgroup_fusion': None, 'weyl_group': '4.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '5.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [2]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', '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]], 'center_label': '5.1', 'center_order': 5, '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': ['5.1'], '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], [5, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '5.1', 'hash': 1, 'hyperelementary': 5, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 5]], 'label': '5.1', 'linC_count': 4, 'linC_degree': 1, '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': 'C5', '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': 5, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 5, 'order_factorization_type': 1, 'order_stats': [[1, 1], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 5, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -5]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26329372]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131]}, 'Perm': {'d': 5, 'gens': [96]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [5], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5', 'transitive_degree': 5, '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': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 40, 'aut_gen_orders': [8, 4, 20, 20, 5], 'aut_gens': [[1, 4, 20, 100], [3, 300, 60, 4], [405, 12, 40, 400], [121, 12, 20, 300], [313, 12, 20, 100], [9, 4, 20, 100]], 'aut_group': '16000.cf', 'aut_hash': 1154461776096652454, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16000, 'aut_permdeg': 250, 'aut_perms': [179966185513871411574681212092234500764364639136491661753169905228703326991439231446304339603784611266329621492338118954043236728796746463325620195321182419505265947977282459112104667373417840061554561058664462201895120143415415336365174328805399807475232143524921958125910194659566276140572354319010624261782756907505986200119379187911920631499112510999082262819811707426985337057646875995681497290829182936818216843062920809353988612932260309948125785538390554820521975245069852542102916840, 1349243054504321523397685388216327823730149047197376084202316869894141662595719911456685893788424609764729434654457744404589808861265539951701798458844739207611007516019233240502100130381035188665742852367542097844044659866339474656488560204771087599708436691929974585669563121557594955182744295900814318434899897324488184270033754619394102784699445977033874179713788157719499291524124784288333527499528447290234393578427963932136234708700136754500232150958800408031224246268469095595646807853, 2520480377931600786632948240043902654714499962835101646057120833006195059856110476341477815031462577725393271995317167940068208448848644770785127018313194497915562145992992704465171405802651491847060852209197880171295810204186286999105474249475908366426230357709921462035088268223451711722896151131348102658474851052906441916405511792604270420328779363731884243782978560659519278387133957793520299569624645442952496808854657183064444935309220315598964437302240300885062877545597690258112088453, 3201131023436489347443134670504285329873522170499538582074070530525824864550917292094769751157789470142419630474527237150933840280868038885986425122594421889639856543245782022752994313278286219317836472224704768913094187901239295376079604697536328571347813699235231388421692696152783596589984549408221799024908887200035120606791206195733781926966397116306018684012177516155155550642097205032034705332311283121415159604480589088945550960358898077760349268472337825114190919249124846902556615262, 226039248161456279201392456231276392991321611590213177246367172133317986027615796535598943496891926293291918275634578657202484517525381366450671113640804697362525123048463710706794779083203478368703850663566298094661173787947562853348063233239621332893760780282190244716046122337448361403263482576306471354666315788468791251784028502094374369339947544226757934910806837830711381849995360871768354449041397683614709540242948498808346352420051784274259475631079857139320255370880561282186731704], 'aut_phi_ratio': 80.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [4, 125, 2, 1], [5, 2, 2, 1], [5, 4, 2, 1], [5, 4, 4, 1], [5, 4, 8, 1], [5, 4, 16, 1], [10, 50, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5^3:C_2', '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': 40, 'autcentquo_group': '16000.cf', 'autcentquo_hash': 1154461776096652454, 'autcentquo_nilpotent': False, 'autcentquo_order': 16000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5^3:C_2', 'cc_stats': [[1, 1, 1], [2, 25, 1], [4, 125, 2], [5, 2, 2], [5, 4, 30], [10, 50, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '500.46', 'commutator_count': 1, 'commutator_label': '125.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 46, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [4, 125, 2, 1], [5, 2, 2, 1], [5, 4, 1, 2], [5, 4, 2, 2], [5, 4, 4, 6], [10, 50, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 20, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 0, 16]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '500.46', 'hash': 46, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 5, 5, 5], 'inner_gens': [[1, 8, 80, 300], [17, 4, 20, 100], [41, 4, 20, 100], [301, 4, 20, 100]], 'inner_hash': 46, 'inner_nilpotent': False, 'inner_order': 500, 'inner_split': False, 'inner_tex': 'C_5^3:C_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 4], [4, 30]], 'label': '500.46', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 6, 'linQ_dim': 12, 'linQ_dim_count': 3, 'linR_count': 8, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5^3:C4', 'ngens': 5, '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': 9, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 38, 'number_divisions': 15, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 44, 'number_subgroups': 392, 'old_label': None, 'order': 500, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 25], [4, 250], [5, 124], [10, 100]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4], 'outer_gen_pows': [2, 0, 0], 'outer_gens': [[3, 100, 80, 16], [1, 8, 20, 100], [1, 4, 40, 100]], 'outer_group': '32.24', 'outer_hash': 24, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 12, 'outer_perms': [287175000, 11291173, 95212080], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2:C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 4], [8, 2], [16, 6]], 'representations': {'PC': {'code': 1651511457430373893183, 'gens': [1, 3, 4, 5], 'pres': [5, -2, -2, -5, -5, -5, 10, 122, 127, 1603, 7504, 5009]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [10358659849, 101789761111, 3916019267, 99619144034, 14631557504]}, 'Perm': {'d': 15, 'gens': [88706873527, 2091640320, 37, 194174920800, 289013317200]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5^3:C_4', 'transitive_degree': 20, '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': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 20, 5], 'aut_gens': [[1, 4, 20], [1, 12, 20], [13, 4, 60], [21, 4, 20]], 'aut_group': '400.205', 'aut_hash': 205, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 400, 'aut_permdeg': 10, 'aut_perms': [18, 50515, 454320], 'aut_phi_ratio': 10.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 25, 1, 2], [5, 2, 2, 1], [5, 4, 1, 1], [5, 4, 4, 1], [10, 10, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_5^2', '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': '400.205', 'autcentquo_hash': 205, 'autcentquo_nilpotent': False, 'autcentquo_order': 400, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5^2', 'cc_stats': [[1, 1, 1], [2, 5, 1], [4, 25, 2], [5, 2, 2], [5, 4, 5], [10, 10, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '100.10', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 25, 2, 1], [5, 2, 2, 1], [5, 4, 1, 1], [5, 4, 4, 1], [10, 10, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 20, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '100.10', 'hash': 10, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 5, 5], 'inner_gens': [[1, 8, 80], [17, 4, 20], [41, 4, 20]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 100, 'inner_split': True, 'inner_tex': 'C_5:F_5', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 4], [4, 5]], 'label': '100.10', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5:F5', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 13, 'number_divisions': 7, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 13, 'number_subgroup_classes': 13, 'number_subgroups': 56, 'old_label': None, 'order': 100, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 5], [4, 50], [5, 24], [10, 20]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[1, 4, 60]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 3], [16, 1]], 'representations': {'PC': {'code': 1970081001743, 'gens': [1, 3, 4], 'pres': [4, -2, -2, -5, -5, 8, 98, 102, 1283]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [118869, 517634, 1786579, 851419]}, 'Perm': {'d': 10, 'gens': [46087, 90720, 37, 493920]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5:F_5', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
