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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '480.945', 'ambient_counter': 945, 'ambient_order': 480, 'ambient_tex': 'A_5:Q_8', 'central': True, 'central_factor': False, 'centralizer_order': 480, 'characteristic': True, 'core_order': 2, 'counter': 71, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '480.945.240.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '240.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '240.189', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 189, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 240, 'quotient_simple': False, 'quotient_solvable': False, 'quotient_supersolvable': False, 'quotient_tex': 'C_2\\times S_5', 'simple': True, 'solvable': True, 'special_labels': ['Z', 'Phi', 'U1', 'C3'], 'split': False, 'standard_generators': False, 'stem': True, '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': '480.945', 'aut_centralizer_order': 960, 'aut_label': '240.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 960, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['48.a1.a1', '80.a1.a1', '120.a1.a1', '120.b1.a1', '120.c1.a1', '120.c1.b1', '120.d1.a1'], 'contains': ['480.a1.a1'], 'core': '240.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5299, 5074, 7484, 6624, 5422, 5905, 4686, 6806], 'generators': [5167], 'label': '480.945.240.a1.a1', 'mobius_quo': -1, 'mobius_sub': -120, 'normal_closure': '240.a1.a1', 'normal_contained_in': ['4.a1.a1', '120.a1.a1'], 'normal_contains': ['480.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '240.a1.a1', 'projective_image': '240.189', 'quotient_action_image': '1.1', 'quotient_action_kernel': '240.189', 'quotient_action_kernel_order': 240, 'quotient_fusion': None, 'short_label': '240.a1.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': 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': 2, 'aut_exponent': 60, 'aut_gen_orders': [20, 4, 2], 'aut_gens': [[1524136312, 386622], [1444286142, 123777239], [2039783639, 479030257], [1085039857, 398039]], 'aut_group': '960.10871', 'aut_hash': 10871, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 960, 'aut_permdeg': 40, 'aut_perms': [617973076486915896897498719060402402688584714653, 158062453747847015690907041072486312109538786906, 220501486763789882328164946655462754172747324296], 'aut_phi_ratio': 7.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 15, 1, 2], [3, 20, 1, 1], [4, 2, 1, 1], [4, 20, 2, 1], [4, 30, 1, 1], [4, 60, 2, 1], [5, 24, 1, 1], [6, 20, 1, 1], [10, 24, 1, 1], [12, 40, 1, 1], [12, 40, 2, 1], [20, 24, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_4\\times 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': 60, 'autcentquo_group': '240.189', 'autcentquo_hash': 189, 'autcentquo_nilpotent': False, 'autcentquo_order': 240, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 15, 2], [3, 20, 1], [4, 2, 1], [4, 20, 2], [4, 30, 1], [4, 60, 2], [5, 24, 1], [6, 20, 1], [10, 24, 1], [12, 40, 3], [20, 24, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '240.189', 'commutator_count': 1, 'commutator_label': '120.35', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '60.5'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 945, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 15, 1, 2], [3, 20, 1, 1], [4, 2, 1, 1], [4, 20, 1, 2], [4, 30, 1, 1], [4, 60, 1, 2], [5, 24, 1, 1], [6, 20, 1, 1], [10, 24, 1, 1], [12, 40, 1, 3], [20, 24, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 57, 'exponent': 60, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 2], [8, -1, 1], [10, -1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '240.189', 'hash': 945, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [6, 2], 'inner_gens': [[1524136312, 479030257], [2003495639, 386622]], 'inner_hash': 189, 'inner_nilpotent': False, 'inner_order': 240, 'inner_split': True, 'inner_tex': 'C_2\\times S_5', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 8, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 1], [4, 4], [5, 4], [6, 4], [8, 1], [10, 1]], 'label': '480.945', 'linC_count': 6, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'A5:Q8', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 15, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 19, 'number_divisions': 18, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 74, 'number_subgroups': 724, 'old_label': None, 'order': 480, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 31], [3, 20], [4, 192], [5, 24], [6, 20], [10, 24], [12, 120], [20, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [11657, 0], 'outer_gens': [[1524124657, 403192], [1524136312, 391537]], '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': None, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 4], [5, 4], [6, 2], [8, 1], [10, 1], [12, 1]], 'representations': {'Perm': {'d': 13, 'gens': [1524136312, 386622]}}, 'schur_multiplier': [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': 'A_5:Q_8', 'transitive_degree': 24, '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': '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}
