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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '126000.a', 'ambient_counter': 1, 'ambient_order': 126000, 'ambient_tex': '\\PSU(3,5)', 'central': False, 'central_factor': False, 'centralizer_order': 5, 'characteristic': False, 'core_order': 1, 'counter': 33, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '126000.a.2520.a1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '2520.a1.c1', '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': 2520, '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': '50.3', 'subgroup_hash': 3, 'subgroup_order': 50, 'subgroup_tex': 'C_5\\times D_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '126000.a', 'aut_centralizer_order': None, 'aut_label': '2520.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '25200.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['504.a1.a1', '1260.a1.c1'], 'contains': ['5040.a1.c1', '12600.a1.a1', '12600.b1.c1'], 'core': '126000.a1.a1', 'coset_action_label': None, 'count': 1260, 'diagramx': [687, -1, 9067, -1, 1087, -1, 1746, -1], 'generators': [1850837234387, 3449475634554, 1723001285964], 'label': '126000.a.2520.a1.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '1260.a1.c1', 'old_label': '2520.a1.c1', 'projective_image': '126000.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2520.a1.c1', 'subgroup_fusion': None, 'weyl_group': '20.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '10.2', '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], 'aut_gens': [[1, 10], [1, 30], [27, 10]], 'aut_group': '80.30', 'aut_hash': 30, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 80, 'aut_permdeg': 9, 'aut_perms': [6496, 131169], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [10, 5, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times F_5', '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': 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, 5, 1], [5, 1, 4], [5, 2, 10], [10, 5, 4]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', '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': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [10, 5, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 0, 8]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '50.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 40], [21, 10]], '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': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 10], [2, 10]], 'label': '50.3', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 20, 'number_divisions': 7, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 10, 'number_subgroups': 20, 'old_label': None, 'order': 50, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 5], [5, 24], [10, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [1, 0], 'outer_gens': [[9, 30], [7, 10]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 3], [8, 2]], 'representations': {'PC': {'code': 318031, 'gens': [1, 3], 'pres': [3, -2, -5, -5, 6, 362]}, 'Lie': [{'d': 1, 'q': 25, 'gens': [1818624673360232724641098, 4935872799604142007576, 800006261860070844758196], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 11, 'gens': [132, 3997, 5325]}, 'Perm': {'d': 10, 'gens': [2045400, 408993, 1451553]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_5', 'transitive_degree': 10, 'wreath_data': ['C_5', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 840, 'aut_gen_orders': [24, 5, 2], 'aut_gens': [[2288823046880, 245766370], [3082871762819, 1329953871835], [3420146851493, 696061564992], [382659606435, 2242348482782]], 'aut_group': '756000.a', 'aut_hash': 6064165764739422449, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 756000, 'aut_permdeg': 504, 'aut_perms': 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'aut_phi_ratio': 26.25, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 525, 1, 1], [3, 3500, 1, 1], [4, 15750, 1, 1], [5, 504, 1, 1], [5, 5040, 3, 1], [6, 10500, 1, 1], [7, 18000, 2, 1], [8, 15750, 2, 1], [10, 12600, 1, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PGammaU(3,5)', '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': 840, 'autcentquo_group': '756000.a', 'autcentquo_hash': 6064165764739422449, 'autcentquo_nilpotent': False, 'autcentquo_order': 756000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PGammaU(3,5)', 'cc_stats': [[1, 1, 1], [2, 525, 1], [3, 3500, 1], [4, 15750, 1], [5, 504, 1], [5, 5040, 3], [6, 10500, 1], [7, 18000, 2], [8, 15750, 2], [10, 12600, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '126000.a', 'commutator_count': 1, 'commutator_label': '126000.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['126000.a'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 525, 1, 1], [3, 3500, 1, 1], [4, 15750, 1, 1], [5, 504, 1, 1], [5, 5040, 1, 3], [6, 10500, 1, 1], [7, 18000, 2, 1], [8, 15750, 2, 1], [10, 12600, 1, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': 19483, 'exponent': 840, 'exponents_of_order': [4, 3, 2, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3, 5, 7], 'faithful_reps': [[20, -1, 1], [21, 1, 1], [28, 1, 3], [84, 1, 1], [105, 1, 1], [125, 1, 1], [126, 0, 2], [126, 1, 1], [144, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '126000.a', 'hash': 2459828284312693258, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 840, 'inner_gen_orders': [8, 5], 'inner_gens': [[2288823046880, 490095820], [2526612984398, 245766370]], 'inner_hash': 2459828284312693258, 'inner_nilpotent': False, 'inner_order': 126000, 'inner_split': True, 'inner_tex': '\\PSU(3,5)', 'inner_used': [1, 2], 'irrC_degree': 20, 'irrQ_degree': 20, 'irrQ_dim': 21, 'irrR_degree': 21, 'irrep_stats': [[1, 1], [20, 1], [21, 1], [28, 3], [84, 1], [105, 1], [125, 1], [126, 3], [144, 2]], 'label': '126000.a', 'linC_count': 1, 'linC_degree': 20, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 1, 'linQ_dim': 21, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 21, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'PSU(3,5)', 'ngens': 2, '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, 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': 10, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 14, 'number_divisions': 12, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 50, 'number_subgroup_classes': 80, 'number_subgroups': 179308, 'old_label': None, 'order': 126000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 525], [3, 3500], [4, 15750], [5, 15624], [6, 10500], [7, 36000], [8, 31500], [10, 12600]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [273550032880, 2741221610660], 'outer_gens': [[2817869890364, 1361024809107], [2679028197433, 288297607691]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': None, 'perfect': True, 'permutation_degree': 50, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': True, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [20, 1], [21, 1], [28, 3], [84, 1], [105, 1], [125, 1], [126, 1], [252, 1], [288, 1]], 'representations': {'Lie': [{'d': 3, 'q': 5, 'gens': [245766370, 2288823046880], 'family': 'PSU'}], 'Perm': {'d': 50, 'gens': [1254868217757642304475528878034949162654523002517600559981109231, 633908189927600454010912420558966103240499253035585062840780050]}}, 'schur_multiplier': [3], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\PSU(3,5)', 'transitive_degree': 50, 'wreath_data': None, 'wreath_product': False}
