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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1000.136', 'ambient_counter': 136, 'ambient_order': 1000, 'ambient_tex': 'C_{10}.D_5^2', 'central': False, 'central_factor': True, 'centralizer_order': 100, 'characteristic': True, 'core_order': 20, 'counter': 28, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1000.136.50.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '50.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '50.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 50, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5:D_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '20.1', 'subgroup_hash': 1, 'subgroup_order': 20, 'subgroup_tex': 'C_5:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1000.136', 'aut_centralizer_order': 24000, 'aut_label': '50.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '40.12', 'aut_weyl_index': 24000, 'centralizer': '10.b1', 'complements': ['20.f1', '20.e1'], 'conjugacy_class_count': 1, 'contained_in': ['10.a1', '25.b1'], 'contains': ['100.b1', '250.b1'], 'core': '50.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8078, 9136, 7931, 1301], 'generators': [50, 200, 100], 'label': '1000.136.50.a1', 'mobius_quo': 0, 'mobius_sub': -125, 'normal_closure': '50.a1', 'normal_contained_in': ['10.a1'], 'normal_contains': ['100.b1'], 'normalizer': '1.a1', 'old_label': '50.a1', 'projective_image': '500.51', 'quotient_action_image': '2.1', 'quotient_action_kernel': '25.2', 'quotient_action_kernel_order': 25, 'quotient_fusion': None, 'short_label': '50.a1', 'subgroup_fusion': None, 'weyl_group': '10.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, '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, 10], 'aut_gens': [[1, 4], [1, 12], [11, 4]], 'aut_group': '40.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 40, 'aut_permdeg': 7, 'aut_perms': [361, 1855], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 5, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_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': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 5, 2], [5, 2, 2], [10, 2, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 5, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, -1, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '10.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 16], [9, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 4]], 'label': '20.1', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5:C4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 8, 'number_divisions': 5, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 10, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 10], [5, 4], [10, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [1, 0], 'outer_gens': [[1, 8], [3, 4]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2]], 'representations': {'PC': {'code': 133699, 'gens': [1, 3], 'pres': [3, -2, -2, -5, 6, 146]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [91655872471905395, 125101739906136769]}, 'Lie': [{'d': 2, 'q': 11, 'gens': [440, 6664, 7988, 13320], 'family': 'CSOPlus'}, {'d': 2, 'q': 9, 'gens': [1460, 2020, 6241], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 377]}, 'Perm': {'d': 9, 'gens': [5169, 16, 50520]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5: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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 120, 'aut_gen_orders': [10, 24, 20, 20, 10], 'aut_gens': [[1, 2, 10, 200], [187, 44, 552, 200], [83, 166, 298, 400], [165, 44, 510, 400], [63, 124, 416, 600], [141, 8, 494, 800]], 'aut_group': None, 'aut_hash': 4457645683847538779, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 960000, 'aut_permdeg': 60, 'aut_perms': [7459239018985203215341435830508200412550572462921633387201318108987687188910411132, 1423631093714486909583379934461808143930063991946818655864547707546416305184368391, 3362192420429427954907288040144789883216273302614808386748527257972658585947534751, 3998454176408219004169737186657714475827796440975676502306582207570955672239569393, 2750280248791739684569544376515706692617507805423577056995610841013502150350461496], 'aut_phi_ratio': 2400.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 125, 2, 1], [4, 5, 2, 1], [4, 25, 2, 1], [5, 2, 2, 1], [5, 2, 12, 1], [5, 4, 24, 1], [10, 2, 2, 1], [10, 2, 12, 1], [10, 4, 24, 1], [20, 10, 24, 1], [20, 50, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(D_5\\times C_5:D_5).C_2^4.S_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': None, 'autcentquo_hash': 1096677362456611443, 'autcentquo_nilpotent': False, 'autcentquo_order': 240000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^3.C_4^2.S_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 125, 2], [4, 5, 2], [4, 25, 2], [5, 2, 14], [5, 4, 24], [10, 2, 14], [10, 4, 24], [20, 10, 24], [20, 50, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '500.51', 'commutator_count': 1, 'commutator_label': '125.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 136, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 125, 1, 2], [4, 5, 2, 1], [4, 25, 2, 1], [5, 2, 2, 7], [5, 4, 2, 12], [10, 2, 2, 7], [10, 4, 2, 12], [20, 10, 4, 6], [20, 50, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 504, 'exponent': 20, 'exponents_of_order': [3, 3], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '500.51', 'hash': 136, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5, 10, 5], 'inner_gens': [[1, 8, 90, 200], [5, 2, 10, 200], [121, 2, 10, 800], [1, 2, 410, 200]], 'inner_hash': 51, 'inner_nilpotent': False, 'inner_order': 500, 'inner_split': False, 'inner_tex': 'C_5:D_5^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 56], [4, 48]], 'label': '1000.136', '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': 'C10.D5^2', 'ngens': 6, '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': 13, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 112, 'number_divisions': 51, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 50, 'number_subgroup_classes': 248, 'number_subgroups': 3792, 'old_label': None, 'order': 1000, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 251], [4, 60], [5, 124], [10, 124], [20, 440]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 60, 'outer_gen_orders': [2, 4, 10, 2], 'outer_gen_pows': [0, 0, 0, 50], 'outer_gens': [[101, 166, 174, 800], [101, 120, 12, 600], [101, 86, 156, 200], [1, 48, 10, 600]], 'outer_group': '1920.240589', 'outer_hash': 4048372991184717723, 'outer_nilpotent': False, 'outer_order': 1920, 'outer_permdeg': 15, 'outer_perms': [88176211223, 7185432367, 100631059216, 6266937720], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^4.S_5', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 14], [8, 31]], 'representations': {'PC': {'code': 12949412341024104731161256710927, 'gens': [1, 2, 3, 6], 'pres': [6, -2, -5, -2, -2, -5, -5, 600, 97, 1622, 50, 4323, 69, 4804, 2897]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 61175388599, 44573825876, 54120021252, 3457043888, 90859387996]}, 'Perm': {'d': 19, 'gens': [5170, 356995585094413, 23, 7115056236288000, 7115063424940800, 50520]}}, 'schur_multiplier': [10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}.D_5^2', 'transitive_degree': 100, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 120, 'aut_gen_orders': [3, 4, 5], 'aut_gens': [[1, 2, 10], [41, 36, 18], [1, 4, 14], [3, 2, 10]], 'aut_group': '12000.a', 'aut_hash': 5912173750379521722, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12000, 'aut_permdeg': 25, 'aut_perms': [9442776828922870746177219, 1890682562858195671719856, 701386956536875213774337], 'aut_phi_ratio': 600.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [5, 2, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^2.\\GL(2,5)', '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': 120, 'autcentquo_group': '12000.a', 'autcentquo_hash': 5912173750379521722, 'autcentquo_nilpotent': False, 'autcentquo_order': 12000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^2.\\GL(2,5)', 'cc_stats': [[1, 1, 1], [2, 25, 1], [5, 2, 12]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '50.4', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '5.1'], 'composition_length': 3, '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, 25, 1, 1], [5, 2, 2, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 7, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '50.4', 'hash': 4, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5, 5], 'inner_gens': [[1, 8, 40], [5, 2, 10], [21, 2, 10]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 50, 'inner_split': False, 'inner_tex': 'C_5:D_5', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 12]], 'label': '50.4', 'linC_count': 60, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 15, 'linQ_dim': 8, 'linQ_dim_count': 15, 'linR_count': 60, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5:D5', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 14, 'number_divisions': 8, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 16, 'number_subgroups': 64, 'old_label': None, 'order': 50, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 25], [5, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [4, 12, 2], 'outer_gen_pows': [1, 1, 1], 'outer_gens': [[1, 26, 28], [1, 14, 4], [1, 6, 30]], 'outer_group': '240.91', 'outer_hash': 91, 'outer_nilpotent': False, 'outer_order': 240, 'outer_permdeg': 9, 'outer_perms': [1569, 131889, 16], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'A_5:C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 6]], 'representations': {'PC': {'code': 338703, 'gens': [1, 2, 3], 'pres': [3, -2, -5, -5, 49, 362]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [25977, 848025, 1594376]}, 'Perm': {'d': 10, 'gens': [41047, 37, 444240]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5:D_5', 'transitive_degree': 25, 'wreath_data': None, 'wreath_product': False}
