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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '880.123', 'ambient_counter': 123, 'ambient_order': 880, 'ambient_tex': 'C_2^3:F_{11}', 'central': True, 'central_factor': False, 'centralizer_order': 880, 'characteristic': False, 'core_order': 2, 'counter': 102, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '880.123.440.a1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '440.a1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '440.11', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 11, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 440, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{22}:C_{10}', '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': '880.123', 'aut_centralizer_order': 1760, 'aut_label': '440.a1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '1.1', 'aut_weyl_index': 3520, 'centralizer': '1.a1.a1', 'complements': ['2.d1.a1', '2.d1.d1', '2.d1.b1', '2.d1.c1'], 'conjugacy_class_count': 1, 'contained_in': ['40.a1.b1', '88.a1.b1', '220.a1.a1', '220.c1.b1', '220.d1.b1'], 'contains': ['880.a1.a1'], 'core': '440.a1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [1791, 5763, 6889, 2612, 2993, 7470, 6844, 4931], 'generators': [460], 'label': '880.123.440.a1.b1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '440.a1.b1', 'normal_contained_in': ['40.a1.b1', '220.a1.a1'], 'normal_contains': ['880.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '440.a1.b1', 'projective_image': '440.11', 'quotient_action_image': '1.1', 'quotient_action_kernel': '440.11', 'quotient_action_kernel_order': 440, 'quotient_fusion': None, 'short_label': '440.a1.b1', '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [2, 2, 2, 10, 44], 'aut_gens': [[1, 2, 40], [441, 442, 840], [1, 442, 840], [21, 22, 40], [1, 2, 280], [1, 62, 60]], 'aut_group': '3520.b', 'aut_hash': 1968696860554166433, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3520, 'aut_permdeg': 48, 'aut_perms': [7193804174127292432771932760465141895504677733267875526748639, 4404908323711177801840459486378867767284885372135740263867779, 99169541223616813582562379856421636462567871915716452894080, 4661933323641448173832125635115639662462662128866509774824239, 527642748481048006916380720515349061796003826419164290261938], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 22, 2, 1], [4, 22, 2, 1], [5, 11, 1, 4], [10, 11, 1, 4], [10, 11, 2, 4], [10, 22, 2, 8], [11, 10, 1, 1], [20, 22, 2, 4], [22, 10, 1, 1], [22, 10, 2, 1], [22, 10, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\wr C_2\\times F_{11}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 22, 2], [4, 22, 2], [5, 11, 4], [10, 11, 12], [10, 22, 16], [11, 10, 1], [20, 22, 8], [22, 10, 7]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '220.7', 'commutator_count': 1, 'commutator_label': '22.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 123, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['440.11', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 22, 1, 2], [4, 22, 1, 2], [5, 11, 4, 1], [10, 11, 4, 3], [10, 22, 4, 4], [11, 10, 1, 1], [20, 22, 4, 2], [22, 10, 1, 3], [22, 10, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 62496, 'exponent': 220, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '440.42', 'hash': 123, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [2, 10, 11], 'inner_gens': [[1, 22, 40], [21, 2, 520], [1, 402, 40]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 220, 'inner_split': False, 'inner_tex': 'C_2\\times F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 10], [10, 8]], 'label': '880.123', 'linC_count': 80, 'linC_degree': 11, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3:F11', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 34, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 58, 'number_divisions': 26, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 64, 'number_subgroup_classes': 108, 'number_subgroups': 680, 'old_label': None, 'order': 880, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 51], [4, 44], [5, 44], [10, 484], [11, 10], [20, 176], [22, 70]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[441, 442, 840], [1, 2, 860], [1, 462, 40], [1, 22, 840]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [6, 289, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 8], [8, 2], [10, 4], [20, 2]], 'representations': {'PC': {'code': 1328162040072869672108102118780982000309, 'gens': [1, 2, 5], 'pres': [6, -2, -2, -2, -5, -2, -11, 265, 31, 50, 7810, 4516, 772, 88, 2891, 2897, 1823]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [35962, 826574, 359380, 36037, 35960, 443772]}, 'Perm': {'d': 17, 'gens': [1313942112127, 289, 55, 2803208814000, 288, 24847381077120]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3:F_{11}', 'transitive_degree': 88, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 110, 'aut_gen_orders': [2, 10, 22], 'aut_gens': [[1, 10, 20], [11, 10, 20], [1, 10, 260], [241, 10, 30]], 'aut_group': '440.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 440, 'aut_permdeg': 24, 'aut_perms': [522570481886452470541130, 17548251277974683845823, 121077927107026838559648], 'aut_phi_ratio': 2.75, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 22, 1, 1], [4, 22, 1, 1], [5, 11, 1, 4], [10, 11, 1, 4], [10, 22, 1, 8], [11, 10, 1, 1], [20, 22, 1, 4], [22, 10, 1, 1], [22, 10, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times F_{11}', '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': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 22, 1], [4, 22, 1], [5, 11, 4], [10, 11, 4], [10, 22, 8], [11, 10, 1], [20, 22, 4], [22, 10, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '220.7', 'commutator_count': 1, 'commutator_label': '22.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 11, '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, 22, 1, 1], [4, 22, 1, 1], [5, 11, 4, 1], [10, 11, 4, 1], [10, 22, 4, 2], [11, 10, 1, 1], [20, 22, 4, 1], [22, 10, 1, 1], [22, 10, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 220, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[10, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '220.7', 'hash': 11, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 1, 22], 'inner_gens': [[1, 10, 270], [1, 10, 20], [211, 10, 20]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 220, 'inner_split': True, 'inner_tex': 'C_2\\times F_{11}', 'inner_used': [1, 3], 'irrC_degree': 10, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 20, 'irrep_stats': [[1, 20], [2, 5], [10, 4]], 'label': '440.11', 'linC_count': 2, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 2, 'linQ_dim': 12, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D22:C10', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 28, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 29, 'number_divisions': 13, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 32, 'number_subgroups': 190, 'old_label': None, 'order': 440, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 25], [4, 22], [5, 44], [10, 220], [11, 10], [20, 88], [22, 30]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[11, 10, 420]], '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': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 4], [8, 1], [10, 2], [20, 1]], 'representations': {'PC': {'code': 81489632502225479303009, 'gens': [1, 3, 4], 'pres': [5, -2, -5, -2, 2, -11, 10, 5403, 3008, 58, 2004, 2009]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [35962, 826574, 36037, 1173952, 371834]}, 'Perm': {'d': 15, 'gens': [6267312007, 5, 13499992920, 16, 105871456080]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{22}:C_{10}', 'transitive_degree': 44, 'wreath_data': None, 'wreath_product': False}
