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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '22400.a', 'ambient_counter': 1, 'ambient_order': 22400, 'ambient_tex': 'C_{40}.D_{280}', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 700, 'counter': 51, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '22400.a.16._.M', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '16.M', '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': 16, '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': '1400.100', 'subgroup_hash': None, 'subgroup_order': 1400, 'subgroup_tex': 'D_{70}:C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '22400.a', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 4, 'diagramx': None, 'generators': [4016035480, 1908171527, 5147625598, 2840069249, 79242, 4237915959], 'label': '22400.a.16._.M', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '16.M', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16._.M', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [6, 12, 12, 20, 4, 28], 'aut_gens': [[5010755, 5112, 21832578], [20400997, 189925, 4652854], [16463963, 12638, 15032284], [357957, 273989, 15032284], [19327208, 287692, 12526954], [19327264, 296638, 15748105], [5010808, 302034, 24695894]], 'aut_group': None, 'aut_hash': 6942332147878176941, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 13440, 'aut_permdeg': 76, 'aut_perms': [750569999405823540116164799845413446257191919712678265080051438506932148019669248683515024430149211097890940355, 185486445920418840796227371224883251958330151361288864372475960187573686064240774838279697161112781740779905607, 587389293961444674040200210292755132053132057550624702679553353090140933105846367792410821962201178571052414748, 782277786705423827180006220694060171767928993180300351515813856865902353984521033014588579776696900620770060041, 1604935142115896769234947992223735493801461840844411083532324430553616722236263659682486290510123146952512853230, 470788719355117287703135413140751215812829390024999444609949904098417283213095622810952423592610398880047868286], 'aut_phi_ratio': 28.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 70, 1, 1], [4, 70, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [7, 2, 3, 1], [10, 1, 4, 1], [10, 2, 2, 1], [10, 2, 4, 2], [10, 2, 8, 1], [10, 2, 16, 1], [10, 70, 4, 1], [14, 2, 3, 1], [14, 2, 6, 1], [20, 70, 4, 1], [35, 2, 12, 2], [35, 2, 48, 1], [70, 2, 12, 2], [70, 2, 24, 2], [70, 2, 48, 1], [70, 2, 96, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_4\\times C_7:(C_2\\times C_6\\times F_5)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 420, 'autcentquo_group': '840.139', 'autcentquo_hash': 139, 'autcentquo_nilpotent': False, 'autcentquo_order': 840, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 70, 1], [4, 70, 1], [5, 1, 4], [5, 2, 10], [7, 2, 3], [10, 1, 4], [10, 2, 34], [10, 70, 4], [14, 2, 9], [20, 70, 4], [35, 2, 72], [70, 2, 216]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '140.10', 'commutator_count': 1, 'commutator_label': '70.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1', '5.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 100, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['280.28', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 70, 1, 1], [4, 70, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [7, 2, 3, 1], [10, 1, 4, 1], [10, 2, 2, 1], [10, 2, 4, 8], [10, 70, 4, 1], [14, 2, 3, 1], [14, 2, 6, 1], [20, 70, 4, 1], [35, 2, 12, 2], [35, 2, 24, 2], [70, 2, 12, 2], [70, 2, 24, 8]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 36, 'exponent': 140, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': [[2, 0, 96]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '700.33', 'hash': 100, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [10, 2, 35], 'inner_gens': [[5010755, 333700, 21832578], [357925, 5112, 2505438], [5010755, 249068, 21832578]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 140, 'inner_split': True, 'inner_tex': 'D_{70}', 'inner_used': [1, 2, 3], 'irrC_degree': 2, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 4, 'irrep_stats': [[1, 20], [2, 345]], 'label': '1400.100', 'linC_count': 96, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 8, 'linQ_dim': 14, 'linQ_dim_count': 8, 'linR_count': 48, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D70:C10', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 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': 28, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 365, 'number_divisions': 38, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 84, 'number_subgroups': 560, 'old_label': None, 'order': 1400, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 73], [4, 70], [5, 24], [7, 6], [10, 352], [14, 18], [20, 280], [35, 144], [70, 432]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 4, 12], 'outer_gen_pows': [357912, 357912, 357912], 'outer_gens': [[5010755, 5112, 12526954], [23622180, 5112, 15748105], [23622151, 5112, 20043068]], 'outer_group': '96.161', 'outer_hash': 161, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 13, 'outer_perms': [479002344, 2400, 4032003], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4\\times C_{12}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 6], [6, 2], [8, 10], [12, 1], [24, 4], [48, 10]], 'representations': {'PC': {'code': 1320725744473228972928296223, 'gens': [1, 2, 4], 'pres': [6, 2, 2, 5, 2, 5, 7, 31, 33363, 69, 40804, 178, 43205]}, 'GLFp': {'d': 2, 'p': 71, 'gens': [5010755, 5112, 21832578]}, 'Perm': {'d': 21, 'gens': [122002101778608796, 997920000, 862393, 13412044800, 1212985, 2676904891524864000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{70}:C_{10}', 'transitive_degree': 140, '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': '80.23', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[4881369021, 2085675857, 79242], [4881369021, 2085675857, 15222613], [22188261, 66564217, 79242], [4881369021, 2085675857, 4249001], [4881369021, 2085675857, 22187760], [4881369021, 2085675857, 79242], [4881369021, 2085675857, 18018001], [4881369021, 2085675857, 22187760], [4881369021, 2085675857, 79242], [4881369021, 2085675857, 7149483], [4881369021, 2085675857, 8653114], [4881369021, 1841607640, 79242], [2063488045, 465949035, 79242], [1331282651, 2706941207, 79242], [4260103879, 3527898595, 79242], [3971659492, 2706941207, 79242], [4171351709, 3527898595, 79242], [2862257290, 6168275585, 79242]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 860160, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 112.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 280, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 4, 1], [4, 280, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [7, 2, 3, 1], [8, 1, 4, 1], [8, 2, 2, 3], [8, 2, 16, 1], [8, 280, 2, 1], [10, 1, 4, 1], [10, 2, 2, 1], [10, 2, 4, 2], [10, 2, 8, 1], [10, 2, 16, 1], [10, 280, 4, 1], [14, 2, 3, 1], [14, 2, 6, 1], [16, 280, 4, 1], [20, 1, 8, 1], [20, 2, 4, 3], [20, 2, 16, 4], [20, 2, 64, 1], [20, 280, 4, 1], [28, 2, 6, 2], [28, 2, 24, 1], [35, 2, 12, 2], [35, 2, 48, 1], [40, 1, 16, 1], [40, 2, 8, 7], [40, 2, 32, 4], [40, 2, 64, 2], [40, 2, 256, 1], [40, 280, 8, 1], [56, 2, 12, 4], [56, 2, 96, 1], [70, 2, 12, 2], [70, 2, 24, 2], [70, 2, 48, 1], [70, 2, 96, 1], [80, 280, 16, 1], [140, 2, 24, 4], [140, 2, 96, 4], [140, 2, 384, 1], [280, 2, 48, 8], [280, 2, 192, 4], [280, 2, 384, 2], [280, 2, 1536, 1]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 280, 1], [4, 1, 2], [4, 2, 5], [4, 280, 1], [5, 1, 4], [5, 2, 10], [7, 2, 3], [8, 1, 4], [8, 2, 22], [8, 280, 2], [10, 1, 4], [10, 2, 34], [10, 280, 4], [14, 2, 9], [16, 280, 4], [20, 1, 8], [20, 2, 140], [20, 280, 4], [28, 2, 36], [35, 2, 72], [40, 1, 16], [40, 2, 568], [40, 280, 8], [56, 2, 144], [70, 2, 216], [80, 280, 16], [140, 2, 864], [280, 2, 3456]], 'center_label': '40.2', 'center_order': 40, 'central_product': None, 'central_quotient': '560.67', 'commutator_count': None, 'commutator_label': '280.4', 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1', '7.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': None, 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 280, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 2, 2], [4, 280, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [7, 2, 3, 1], [8, 1, 4, 1], [8, 2, 2, 3], [8, 2, 4, 4], [8, 280, 2, 1], [10, 1, 4, 1], [10, 2, 2, 1], [10, 2, 4, 8], [10, 280, 4, 1], [14, 2, 3, 1], [14, 2, 6, 1], [16, 280, 4, 1], [20, 1, 8, 1], [20, 2, 4, 3], [20, 2, 8, 16], [20, 280, 4, 1], [28, 2, 6, 2], [28, 2, 12, 2], [35, 2, 12, 2], [35, 2, 24, 2], [40, 1, 16, 1], [40, 2, 8, 7], [40, 2, 16, 32], [40, 280, 8, 1], [56, 2, 12, 4], [56, 2, 24, 4], [70, 2, 12, 2], [70, 2, 24, 8], [80, 280, 16, 1], [140, 2, 24, 4], [140, 2, 48, 16], [280, 2, 48, 8], [280, 2, 96, 32]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': None, 'exponent': 560, 'exponents_of_order': [7, 2, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': None, 'familial': False, 'frattini_label': '32.3', 'frattini_quotient': '700.33', 'hash': 1111805358260240536, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [40, 140, 2], 'inner_gens': [[4881369021, 2085675857, 12063892], [4881369021, 2085675857, 745774], [22188261, 66564217, 79242]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 560, 'inner_split': None, 'inner_tex': 'D_{280}', 'inner_used': None, 'irrC_degree': None, 'irrQ_degree': None, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': None, 'label': '22400.a', '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': False, 'monomial': True, 'name': 'C40.D280', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 1, 3, 2, 1, 5, 2, 1, 3, 6, 1, 3, 1, 2, 10, 1, 5, 1, 2, 6, 3, 3, 0, 6, 2, 1, 5, 1, 10, 2, 1, 3, 1, 8, 0, 3, 1, 0, 4, 5, 1, 4, 5, 0, 1, 3, 3, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 96, 'number_characteristic_subgroups': 134, 'number_conjugacy_classes': 5660, 'number_divisions': 186, 'number_normal_subgroups': 134, 'number_subgroup_autclasses': None, 'number_subgroup_classes': 548, 'number_subgroups': 6640, 'old_label': None, 'order': 22400, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 283], [4, 292], [5, 24], [7, 6], [8, 608], [10, 1192], [14, 18], [16, 1120], [20, 1408], [28, 72], [35, 144], [40, 3392], [56, 288], [70, 432], [80, 4480], [140, 1728], [280, 6912]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '1536.7532393', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 1536, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': 3, 'perfect': False, 'permutation_degree': 33, 'pgroup': 0, 'primary_abelian_invariants': [2, 8, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': None, 'representations': {'PC': {'code': '155237108204583153947767911863838321769484110376797119514060715938832362611', 'gens': [1, 2, 7], 'pres': [10, -2, -2, -2, -2, -5, -7, -2, -2, -2, -5, 223961, 51, 655082, 82, 828483, 113, 923204, 294, 28805, 206, 237, 268]}, 'GLFp': {'d': 2, 'p': 281, 'gens': [4881369021, 2085675857, 79242]}, 'Perm': {'d': 33, 'gens': [280664389747619095505423194857407575, 550965601643868530543787759304055856, 823389696669084561367747535394697217]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 40], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{40}.D_{280}', 'transitive_degree': 560, 'wreath_data': None, 'wreath_product': False}
