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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '65536.p', 'ambient_counter': 16, 'ambient_order': 65536, 'ambient_tex': 'C_{256}.D_{128}', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 512, 'counter': 173, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '65536.p.64._.BO', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '64.BO', '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': 64, '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': '1024.dco', 'subgroup_hash': None, 'subgroup_order': 1024, 'subgroup_tex': 'C_8.D_{64}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '65536.p', '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': 2, 'diagramx': None, 'generators': [1374942198, 1052424966, 4345495809, 1086374205, 4226673689, 15726601, 2308544888, 4090876929, 4090876914, 4345496064], 'label': '65536.p.64._.BO', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '64.BO', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '64._.BO', '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': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 64, 'aut_gen_orders': [32, 32, 32, 8, 32, 32], 'aut_gens': [[3513740760, 66306, 2800807926], [3870207213, 13069992, 1188221756], [4141800776, 4292157, 2868706290], [1765357840, 12555221, 1612586381], [1918129216, 4485421, 4175749948], [2681985925, 2294239, 594110990], [2698960502, 9278471, 373441268]], 'aut_group': None, 'aut_hash': 7664937467520673312, 'aut_nilpotency_class': 6, 'aut_nilpotent': True, 'aut_order': 16384, 'aut_permdeg': 320, 'aut_perms': 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2147521789919108124910685979286036235096377048258498891125879486863432037047436470531425359398641991480399072101557583339062540110578889860527953544994374130221028799287915569389360313602980824296183548831712882140383521959161830060589650180954632111395875482552444572146457824648806188276369359959339816539592192616159937980280352581232955627455538732203671664528762866524269673149078051677938092394511119407962024838908705490456474309533662299040641109376052755097265207515610077765075269274968198865377604801117175768087075025724321620050798791236805425238176732392622204016999679603907452852599874370163603104192181062702142797453190491286924224164580919036638], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 64, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 4, 1], [4, 64, 1, 1], [8, 1, 4, 1], [8, 2, 2, 3], [8, 64, 2, 1], [16, 2, 4, 2], [16, 2, 8, 1], [16, 64, 4, 1], [32, 2, 8, 2], [32, 2, 16, 1], [64, 2, 16, 2], [64, 2, 32, 1], [128, 2, 128, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{32}.C_{32}.C_2^4', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 32, 'autcentquo_group': '1024.dng', 'autcentquo_hash': 7288729358377021530, 'autcentquo_nilpotent': True, 'autcentquo_order': 1024, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{32}:C_{16}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 64, 1], [4, 1, 2], [4, 2, 5], [4, 64, 1], [8, 1, 4], [8, 2, 6], [8, 64, 2], [16, 2, 16], [16, 64, 4], [32, 2, 32], [64, 2, 64], [128, 2, 128]], 'center_label': '8.1', 'center_order': 8, 'central_product': False, 'central_quotient': '128.161', 'commutator_count': 1, 'commutator_label': '64.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 2095, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 64, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 2, 2], [4, 64, 1, 1], [8, 1, 4, 1], [8, 2, 2, 3], [8, 64, 2, 1], [16, 2, 4, 2], [16, 2, 8, 1], [16, 64, 4, 1], [32, 2, 8, 2], [32, 2, 16, 1], [64, 2, 16, 2], [64, 2, 32, 1], [128, 2, 64, 2]], 'element_repr_type': 'GLFp', 'elementary': 2, 'eulerian_function': 24, 'exponent': 128, 'exponents_of_order': [10], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 128]], 'familial': False, 'frattini_label': '256.497', 'frattini_quotient': '4.2', 'hash': 1615446465600477466, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 64, 'inner_gen_orders': [64, 2, 32], 'inner_gens': [[3513740760, 12555221, 2800807926], [152771544, 66306, 1374942198], [3513740760, 9044344, 2800807926]], 'inner_hash': 161, 'inner_nilpotent': True, 'inner_order': 128, 'inner_split': True, 'inner_tex': 'D_{64}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 128, 'irrQ_dim': 128, 'irrR_degree': 4, 'irrep_stats': [[1, 16], [2, 252]], 'label': '1024.dco', 'linC_count': 128, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 128, 'linQ_degree_count': 2, 'linQ_dim': 128, 'linQ_dim_count': 2, 'linR_count': 64, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C8.D64', 'ngens': 3, 'nilpotency_class': 7, 'nilpotent': True, 'normal_counts': [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': 24, 'number_characteristic_subgroups': 37, 'number_conjugacy_classes': 268, 'number_divisions': 26, 'number_normal_subgroups': 37, 'number_subgroup_autclasses': 86, 'number_subgroup_classes': 88, 'number_subgroups': 617, 'old_label': None, 'order': 1024, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 67], [4, 76], [8, 144], [16, 288], [32, 64], [64, 128], [128, 256]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 32, 'outer_gen_orders': [2, 2, 32], 'outer_gen_pows': [16974594, 16974594, 16974594], 'outer_gens': [[611085405, 66306, 1374942198], [3513740999, 66306, 1561662732], [3021477750, 66306, 2376443143]], 'outer_group': '128.988', 'outer_hash': 988, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 36, 'outer_perms': [10333147966386144929666651337523200000000, 8744435920973300526071438768797780081, 27175422691743435819738647695393008], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{32}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 256, 'pgroup': 2, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 7], [8, 2], [16, 3], [32, 3], [64, 1], [128, 2]], 'representations': {'PC': {'code': '100727962972253419728838842804624023322781710289203118702462', 'gens': [1, 2, 9], 'pres': [10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7001, 51, 21002, 82, 4803, 113, 12004, 144, 13445, 175, 13446, 206, 40328, 268, 64009]}, 'GLFp': {'d': 2, 'p': 257, 'gens': [3513740760, 66306, 2800807926]}, 'Perm': {'d': 256, 'gens': 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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '512.60895', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 256, 'aut_gen_orders': [64, 256, 128, 256, 64, 128], 'aut_gens': [[1459815001, 152771338, 66306], [3734410585, 611085545, 10918902], [2902655657, 1425865907, 7130208], [4022978708, 4073902459, 6573546], [729907505, 831755194, 14427466], [695958476, 594110994, 15043752], [2240646313, 1969052924, 10236310]], 'aut_group': None, 'aut_hash': 3734133122171447582, 'aut_nilpotency_class': 8, 'aut_nilpotent': True, 'aut_order': 4194304, 'aut_permdeg': 384, 'aut_perms': 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2, 1], [8, 1, 4, 1], [8, 2, 2, 3], [8, 2, 16, 1], [8, 128, 4, 1], [16, 1, 8, 1], [16, 2, 4, 3], [16, 2, 8, 2], [16, 2, 64, 1], [16, 128, 8, 1], [32, 1, 16, 1], [32, 2, 8, 3], [32, 2, 16, 2], [32, 2, 32, 2], [32, 2, 256, 1], [32, 128, 16, 1], [64, 1, 32, 1], [64, 2, 16, 3], [64, 2, 32, 2], [64, 2, 64, 2], [64, 2, 128, 2], [64, 2, 1024, 1], [64, 128, 32, 1], [128, 1, 64, 1], [128, 2, 32, 3], [128, 2, 64, 2], [128, 2, 128, 2], [128, 2, 256, 2], [128, 2, 512, 2], [128, 2, 4096, 1], [128, 128, 64, 1], [256, 1, 128, 1], [256, 2, 64, 3], [256, 2, 128, 2], [256, 2, 256, 2], [256, 2, 512, 2], [256, 2, 1024, 2], [256, 2, 2048, 2], [256, 128, 128, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{32}.C_4^3.C_8^2.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 64, 'autcent_group': '512.420501', 'autcent_hash': 1709335650994231389, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\times C_{64}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 128, 'autcentquo_group': None, 'autcentquo_hash': 1312969192172081052, 'autcentquo_nilpotent': True, 'autcentquo_order': 8192, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{64}.C_{32}.C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 128, 2], [4, 1, 2], [4, 2, 5], [4, 128, 2], [8, 1, 4], [8, 2, 22], [8, 128, 4], [16, 1, 8], [16, 2, 92], [16, 128, 8], [32, 1, 16], [32, 2, 376], [32, 128, 16], [64, 1, 32], [64, 2, 1520], [64, 128, 32], [128, 1, 64], [128, 2, 6112], [128, 128, 64], [256, 1, 128], [256, 2, 8128], [256, 128, 128]], 'center_label': '256.1', 'center_order': 256, 'central_product': None, 'central_quotient': '256.539', 'commutator_count': None, 'commutator_label': '128.1', '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', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 16, 'conjugacy_classes_known': True, 'counter': 16, '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, 128, 1, 2], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 2, 2], [4, 128, 1, 2], [8, 1, 4, 1], [8, 2, 2, 3], [8, 2, 4, 4], [8, 128, 2, 2], [16, 1, 8, 1], [16, 2, 4, 3], [16, 2, 8, 10], [16, 128, 4, 2], [32, 1, 16, 1], [32, 2, 8, 3], [32, 2, 16, 22], [32, 128, 8, 2], [64, 1, 32, 1], [64, 2, 16, 3], [64, 2, 32, 46], [64, 128, 16, 2], [128, 1, 64, 1], [128, 2, 32, 3], [128, 2, 64, 94], [128, 128, 32, 2], [256, 1, 128, 1], [256, 2, 64, 3], [256, 2, 128, 62], [256, 128, 64, 2]], 'element_repr_type': 'GLFp', 'elementary': 2, 'eulerian_function': None, 'exponent': 256, 'exponents_of_order': [16], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': None, 'familial': False, 'frattini_label': '8192.vg', 'frattini_quotient': '8.5', 'hash': 7308071633351697028, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 128, 'inner_gen_orders': [128, 128, 2], 'inner_gens': [[1459815001, 152771338, 645841], [1459815001, 152771338, 13212113], [50923865, 16974602, 66306]], 'inner_hash': 539, 'inner_nilpotent': True, 'inner_order': 256, 'inner_split': False, 'inner_tex': 'D_{128}', 'inner_used': [1, 3], 'irrC_degree': None, 'irrQ_degree': None, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': None, 'label': '65536.p', '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': 'C256.D128', 'ngens': 3, 'nilpotency_class': 8, 'nilpotent': True, 'normal_counts': [1, 1, 3, 5, 7, 9, 11, 13, 19, 21, 19, 17, 15, 13, 11, 7, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 73, 'number_characteristic_subgroups': 129, 'number_conjugacy_classes': 16768, 'number_divisions': 285, 'number_normal_subgroups': 173, 'number_subgroup_autclasses': None, 'number_subgroup_classes': 884, 'number_subgroups': 6863, 'old_label': None, 'order': 65536, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 259], [4, 268], [8, 560], [16, 1216], [32, 2816], [64, 7168], [128, 20480], [256, 32768]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 64, 'outer_gen_orders': [64, 32, 64, 64, 64], 'outer_gen_pows': [16974594, 237644358, 2851731763, 16974594, 16974594], 'outer_gens': [[1256119948, 2087875147, 66306], [1357967591, 441339539, 5322727], [1578637301, 356466640, 6181621], [2138798769, 2868706438, 66306], [1408891315, 4175749914, 66306]], 'outer_group': None, 'outer_hash': 1709335650928231063, 'outer_nilpotent': True, 'outer_order': 16384, 'outer_permdeg': 132, 'outer_perms': [154249900121750978389357575958032494645042937946875580787750681721394876158418863902306579117876121171126676569818774791817718726904465498726750646627672598395901601864062426027771339620791879895479940690137066798725, 847158139380657876769287550670158282197148382034890233578194885468717571194507897228041276787733927479435948272897987145655760395070716678653021809429473563632484171169959913627704645709752866243274172511891484081041362537, 847207950105543122566557463972451320877777454173045484010134841593667729738511728825817555380977960060481426577767122131246604604074753152866063380880056887197508543448652670570072332033980382570635941242894085571592788576, 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'schur_multiplier': [2, 2], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 2, 128], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{256}.D_{128}', 'transitive_degree': 512, 'wreath_data': None, 'wreath_product': False}
