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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '900.147', 'ambient_counter': 147, 'ambient_order': 900, 'ambient_tex': 'C_3^2:C_{10}^2', 'central': True, 'central_factor': False, 'centralizer_order': 900, 'characteristic': True, 'core_order': 2, 'counter': 34, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '900.147.450.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '450.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '450.31', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 31, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 450, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{15}^2:C_2', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '900.147', 'aut_centralizer_order': 414720, 'aut_label': '450.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 414720, 'centralizer': '1.a1', 'complements': ['2.b1'], 'conjugacy_class_count': 1, 'contained_in': ['90.a1', '150.a1', '225.a1'], 'contains': ['900.a1'], 'core': '450.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4805, 3758, 4603, 4502], 'generators': [450], 'label': '900.147.450.a1', 'mobius_quo': -1, 'mobius_sub': -135, 'normal_closure': '450.a1', 'normal_contained_in': ['90.a1', '150.a1'], 'normal_contains': ['900.a1'], 'normalizer': '1.a1', 'old_label': '450.a1', 'projective_image': '450.31', 'quotient_action_image': '1.1', 'quotient_action_kernel': '450.31', 'quotient_action_kernel_order': 450, 'quotient_fusion': None, 'short_label': '450.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '100.16', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [20, 12, 12, 8], 'aut_gens': [[1, 2, 30], [621, 728, 178], [311, 388, 892], [461, 326, 42], [161, 488, 40]], 'aut_group': None, 'aut_hash': 7098974253923069884, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 414720, 'aut_permdeg': 42, 'aut_perms': [75508270995021244508606935280313679319323422629, 480585823280980002129614361855142872431704002280318, 1198095730487683512881658109926709713054745272447712, 782317524639266520031977999916720496772255560932206], 'aut_phi_ratio': 1728.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [3, 2, 4, 1], [5, 1, 24, 1], [6, 2, 4, 1], [10, 1, 24, 1], [10, 9, 48, 1], [15, 2, 96, 1], [30, 2, 96, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times \\GL(2,5)\\times \\AGL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 120, 'autcent_group': '960.5693', 'autcent_hash': 5693, 'autcent_nilpotent': False, 'autcent_order': 960, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times \\GL(2,5)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 2, 4], [5, 1, 24], [6, 2, 4], [10, 1, 24], [10, 9, 48], [15, 2, 96], [30, 2, 96]], 'center_label': '50.5', 'center_order': 50, 'central_product': True, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 147, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['2.1', 1], ['5.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 2, 1, 4], [5, 1, 4, 6], [6, 2, 1, 4], [10, 1, 4, 6], [10, 9, 4, 12], [15, 2, 4, 24], [30, 2, 4, 24]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 651, 'exponent': 30, 'exponents_of_order': [2, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '900.147', 'hash': 147, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[1, 22, 330], [11, 2, 30], [601, 2, 30]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 100], [2, 200]], 'label': '900.147', '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': 'C3^2:C10^2', '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, 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': 10, 'number_conjugacy_classes': 300, 'number_divisions': 84, 'number_normal_subgroups': 120, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 240, 'number_subgroups': 624, 'old_label': None, 'order': 900, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 19], [3, 8], [5, 24], [6, 8], [10, 456], [15, 192], [30, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 120, 'outer_gen_orders': [2, 2, 8, 3, 2, 4, 5, 4, 5], 'outer_gen_pows': [0, 1, 0, 1, 1, 0, 0, 0, 0], 'outer_gens': [[1, 182, 570], [451, 622, 40], [1, 734, 414], [1, 376, 352], [451, 490, 752], [1, 14, 878], [1, 728, 408], [451, 26, 390], [1, 722, 30]], 'outer_group': '23040.k', 'outer_hash': 7481427162996395341, 'outer_nilpotent': False, 'outer_order': 23040, 'outer_permdeg': 30, 'outer_perms': [21647625446779020978355922209200, 6, 144737420471325969673178549761200, 111351886190521302842472056048231, 41250269509990541888656719363841, 107376401474583295007640079008713, 94690903312321211296714256702640, 116361347997971682235198984670767, 4590910108125990234285010940160], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4\\times \\GL(2,5)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 24], [8, 48]], 'representations': {'PC': {'code': 20320830568368769473421767, 'gens': [1, 2, 4], 'pres': [6, -2, -3, -5, -2, -3, -5, 265, 43, 7923, 69, 19804, 118]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [288949, 433456, 7187425, 291149, 18687305, 287521]}, 'Perm': {'d': 18, 'gens': [21010008096000, 87178291200, 444281, 735908, 376610736902400, 711375375110400]}}, 'schur_multiplier': [30], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:C_{10}^2', 'transitive_degree': 450, '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': '50.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [4, 6, 60], 'aut_gens': [[1, 2, 30], [301, 118, 426], [1, 448, 200], [321, 382, 306]], 'aut_group': None, 'aut_hash': 5873696443232014559, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 207360, 'aut_permdeg': 33, 'aut_perms': [4802709757651051228694057872575676972, 210726669210971787274702707252045935, 2675322800603561688327767941570432240], 'aut_phi_ratio': 1728.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 4, 1], [5, 1, 24, 1], [10, 9, 24, 1], [15, 2, 96, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,5)\\times \\AGL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 120, 'autcent_group': '480.218', 'autcent_hash': 218, 'autcent_nilpotent': False, 'autcent_order': 480, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,5)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 4], [5, 1, 24], [10, 9, 24], [15, 2, 96]], 'center_label': '25.2', 'center_order': 25, 'central_product': True, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['5.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 4], [5, 1, 4, 6], [10, 9, 4, 6], [15, 2, 4, 24]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 217, 'exponent': 30, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '450.31', 'hash': 31, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[1, 22, 330], [11, 2, 30], [151, 2, 30]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 50], [2, 100]], 'label': '450.31', 'linC_count': 2880, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 360, 'linQ_dim': 12, 'linQ_dim_count': 360, 'linR_count': 5040, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C15^2:C2', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 150, 'number_divisions': 42, 'number_normal_subgroups': 56, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 96, 'number_subgroups': 224, 'old_label': None, 'order': 450, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 9], [3, 8], [5, 24], [10, 216], [15, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [2, 4, 2, 2, 3, 4], 'outer_gen_pows': [1, 0, 0, 1, 0, 1], 'outer_gens': [[1, 162, 200], [1, 194, 30], [1, 152, 330], [1, 172, 40], [1, 200, 138], [1, 322, 340]], 'outer_group': '11520.r', 'outer_hash': 7046787104755948544, 'outer_nilpotent': False, 'outer_order': 11520, 'outer_permdeg': 28, 'outer_perms': [7, 1245631739646198254262526200, 5, 23, 93435258101123028131282586360, 22], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'S_4\\times \\GL(2,5)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 5, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 4], [4, 12], [8, 24]], 'representations': {'PC': {'code': 12638284614297719, 'gens': [1, 2, 4], 'pres': [5, -2, -3, -5, -3, -5, 221, 36, 6603, 78]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [7187425, 287521, 433456, 12366023, 288949]}, 'Perm': {'d': 16, 'gens': [25, 1401563041200, 1401566019840, 147, 144]}}, 'schur_multiplier': [15], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [5, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^2:C_2', 'transitive_degree': 225, 'wreath_data': None, 'wreath_product': False}
