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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '115200.d', 'ambient_counter': 4, 'ambient_order': 115200, 'ambient_tex': 'S_5^2:D_4', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 1, 'counter': 754, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '115200.d.480.DR', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '480.dl2', 'outer_equivalence': True, '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': 480, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': False, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '240.189', 'subgroup_hash': 189, 'subgroup_order': 240, 'subgroup_tex': 'C_2\\times S_5', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '115200.d', 'aut_centralizer_order': None, 'aut_label': '480.DR', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['160.BG', '160.BI', '240.BE', '240.CG', '240.DH', '240.DJ', '240.DL'], 'contains': [], 'core': '115200.a1', 'coset_action_label': None, 'count': 80, 'diagramx': None, 'generators': [22181903, 1238595856, 766087], 'label': '115200.d.480.DR', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.D', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '20.C', 'old_label': '480.dl2', 'projective_image': '115200.d', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '480.DR', 'subgroup_fusion': None, 'weyl_group': '120.34'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 60, 'aut_gen_orders': [3, 4, 2], 'aut_gens': [[65611, 781878, 63225], [615919, 1175707, 1952169], [624475, 1287808, 63225], [624475, 783753, 63225]], 'aut_group': '240.189', 'aut_hash': 189, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 240, 'aut_permdeg': 7, 'aut_perms': [366, 3048, 2329], 'aut_phi_ratio': 3.75, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 10, 2, 1], [2, 15, 1, 2], [3, 20, 1, 1], [4, 30, 2, 1], [5, 24, 1, 1], [6, 20, 1, 1], [6, 20, 2, 1], [10, 24, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_5', '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': 60, 'autcentquo_group': '120.34', 'autcentquo_hash': 34, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 10, 2], [2, 15, 2], [3, 20, 1], [4, 30, 2], [5, 24, 1], [6, 20, 3], [10, 24, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '120.34', 'commutator_count': 1, 'commutator_label': '60.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '60.5'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 189, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['120.34', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 10, 1, 2], [2, 15, 1, 2], [3, 20, 1, 1], [4, 30, 1, 2], [5, 24, 1, 1], [6, 20, 1, 3], [10, 24, 1, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': 57, 'exponent': 60, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 1, 2], [5, 1, 2], [6, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '240.189', 'hash': 189, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [3, 4, 2], 'inner_gens': [[65611, 1282377, 396361], [18421, 781878, 16275], [693225, 1172502, 63225]], 'inner_hash': 34, 'inner_nilpotent': False, 'inner_order': 120, 'inner_split': True, 'inner_tex': 'S_5', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [4, 4], [5, 4], [6, 2]], 'label': '240.189', 'linC_count': 2, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 2, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C2*S5', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 11, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 14, 'number_divisions': 14, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 45, 'number_subgroup_classes': 57, 'number_subgroups': 535, 'old_label': None, 'order': 240, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 51], [3, 20], [4, 60], [5, 24], [6, 60], [10, 24]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [391251], 'outer_gens': [[65611, 1174377, 63225]], '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': None, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4], [5, 4], [6, 2]], 'representations': {'GLZ': {'b': 3, 'd': 4, 'gens': [35950760, 40477037]}, 'Lie': [{'d': 3, 'q': 5, 'gens': [65611, 63225, 781878], 'family': 'GO'}], 'GLFp': {'d': 3, 'p': 5, 'gens': [1287141, 240669, 1565004]}, 'Perm': {'d': 7, 'gens': [864, 3247]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times S_5', 'transitive_degree': 10, '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': '8.5', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 120, 'aut_gen_orders': [6, 12, 6, 8, 6], 'aut_gens': [[14386070169, 7672776605, 20773152734], [6314288405, 12545301009, 21088812974], [59118635525, 8797829769, 51931812245], [51991683129, 58613522414, 7678677005], [59575270454, 57975568569, 52909039454], [13905684014, 58456500609, 52865131085]], 'aut_group': '460800.bo', 'aut_hash': 1673116337738338913, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 460800, 'aut_permdeg': 320, 'aut_perms': 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20300733636529429203902232606038876829369391396196122912796887295014979709166123619548763778540889732709786474923069968372054118664442327341989969720300749275582618699828561303537626773565565955536550808256726328171114158741841221542267705285896794092998660346636992423965375932925770395040909298159846826965793502542679431179913244668058345645369673341647142022305621696767992790277977614269292296822117336718950331365072216356675611105912322856050285943929444445679339776640640951857442132330493746034373152944656702334426363348535084093606548408342049625134500396819696867314866742423496348213882553668912717860496437570998489546145954310532280454477749206533356], 'aut_phi_ratio': 15.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 20, 4, 1], [2, 30, 1, 2], [2, 30, 2, 1], [2, 100, 2, 1], [2, 200, 1, 1], [2, 225, 1, 2], [2, 240, 1, 1], [2, 300, 4, 1], [2, 450, 1, 1], [3, 40, 1, 1], [3, 400, 1, 1], [4, 60, 4, 1], [4, 240, 1, 1], [4, 600, 2, 2], [4, 900, 2, 1], [4, 900, 4, 1], [4, 1800, 1, 1], [4, 2400, 2, 1], [4, 3600, 1, 2], [5, 48, 1, 1], [5, 576, 1, 1], [6, 40, 1, 1], [6, 40, 2, 1], [6, 40, 4, 1], [6, 400, 1, 1], [6, 400, 2, 3], [6, 400, 4, 1], [6, 600, 1, 2], [6, 600, 2, 1], [6, 600, 4, 1], [6, 800, 1, 2], [6, 800, 4, 1], [6, 4800, 1, 1], [8, 7200, 2, 1], [10, 48, 1, 1], [10, 48, 2, 1], [10, 480, 4, 1], [10, 576, 1, 1], [10, 720, 1, 2], [10, 720, 2, 1], [10, 1152, 1, 1], [10, 5760, 1, 1], [12, 1200, 2, 2], [12, 1200, 4, 1], [12, 4800, 1, 1], [12, 4800, 2, 1], [15, 960, 1, 1], [20, 1440, 4, 1], [20, 5760, 1, 1], [30, 960, 1, 1], [30, 960, 2, 1], [30, 960, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times A_5^2).D_4^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '57600.l', 'autcentquo_hash': 3328040073188890159, 'autcentquo_nilpotent': False, 'autcentquo_order': 57600, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_5^2:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 20, 4], [2, 30, 4], [2, 100, 2], [2, 200, 1], [2, 225, 2], [2, 240, 1], [2, 300, 4], [2, 450, 1], [3, 40, 1], [3, 400, 1], [4, 60, 4], [4, 240, 1], [4, 600, 4], [4, 900, 6], [4, 1800, 1], [4, 2400, 2], [4, 3600, 2], [5, 48, 1], [5, 576, 1], [6, 40, 7], [6, 400, 11], [6, 600, 8], [6, 800, 6], [6, 4800, 1], [8, 7200, 2], [10, 48, 3], [10, 480, 4], [10, 576, 1], [10, 720, 4], [10, 1152, 1], [10, 5760, 1], [12, 1200, 8], [12, 4800, 3], [15, 960, 1], [20, 1440, 4], [20, 5760, 1], [30, 960, 7]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '57600.l', 'commutator_count': 1, 'commutator_label': '14400.ca', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '60.5', '60.5'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 4, '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, 20, 1, 4], [2, 30, 1, 4], [2, 100, 1, 2], [2, 200, 1, 1], [2, 225, 1, 2], [2, 240, 1, 1], [2, 300, 1, 4], [2, 450, 1, 1], [3, 40, 1, 1], [3, 400, 1, 1], [4, 60, 1, 4], [4, 240, 1, 1], [4, 600, 1, 4], [4, 900, 1, 6], [4, 1800, 1, 1], [4, 2400, 1, 2], [4, 3600, 1, 2], [5, 48, 1, 1], [5, 576, 1, 1], [6, 40, 1, 7], [6, 400, 1, 11], [6, 600, 1, 8], [6, 800, 1, 6], [6, 4800, 1, 1], [8, 7200, 1, 2], [10, 48, 1, 3], [10, 480, 1, 4], [10, 576, 1, 1], [10, 720, 1, 4], [10, 1152, 1, 1], [10, 5760, 1, 1], [12, 1200, 1, 8], [12, 4800, 1, 3], [15, 960, 1, 1], [20, 1440, 1, 4], [20, 5760, 1, 1], [30, 960, 1, 7]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 120, 'exponents_of_order': [9, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 1, 8], [10, 1, 8], [12, 1, 4], [32, 1, 4], [40, 1, 8], [48, 1, 4], [50, 1, 4], [60, 1, 4], [72, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '57600.l', 'hash': 8438217488598235417, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 120, 'inner_gen_orders': [4, 8, 4], 'inner_gens': [[14386070169, 15023731814, 58199112845], [6398100138, 7672776605, 14490337694], [58217248218, 52948355885, 20773152734]], 'inner_hash': 3328040073188890159, 'inner_nilpotent': False, 'inner_order': 57600, 'inner_split': True, 'inner_tex': 'S_5^2:C_2^2', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 6], [8, 16], [10, 16], [12, 8], [16, 8], [25, 8], [32, 6], [36, 4], [40, 16], [48, 8], [50, 6], [60, 8], [72, 1]], 'label': '115200.d', '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': False, 'metacyclic': False, 'monomial': False, 'name': 'S5^2:D4', '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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 65, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 119, 'number_divisions': 119, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 4913, 'number_subgroup_classes': 14962, 'number_subgroups': 5371298, 'old_label': None, 'order': 115200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 2943], [3, 440], [4, 22080], [5, 624], [6, 19080], [8, 14400], [10, 12432], [12, 24000], [15, 960], [20, 11520], [30, 6720]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[14386070169, 7672776614, 20773152734], [14386070174, 7672776609, 20773152734]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [8, 16], [10, 16], [12, 8], [16, 8], [25, 8], [32, 6], [36, 4], [40, 16], [48, 8], [50, 6], [60, 8], [72, 1]], 'representations': {'Perm': {'d': 14, 'gens': [14386070169, 7672776605, 20773152734]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 720, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_5^2:D_4', 'transitive_degree': 20, 'wreath_data': ['C_2\times S_5', 'C_2', '2T1'], 'wreath_product': True}
