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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '53240.bd', 'ambient_counter': 30, 'ambient_order': 53240, 'ambient_tex': 'D_{11}\\times C_{22}:F_{11}', 'central': False, 'central_factor': False, 'centralizer_order': 440, 'characteristic': False, 'core_order': 1, 'counter': 154, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '53240.bd.26620.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '26620.c1', 'outer_equivalence': True, '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': 26620, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, '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': '53240.bd', 'aut_centralizer_order': None, 'aut_label': '26620.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '121.b1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['2420.i1', '2420.j1', '2420.k1', '5324.c1', '13310.b1', '13310.d1'], 'contains': ['53240.a1'], 'core': '53240.a1', 'coset_action_label': None, 'count': 242, 'diagramx': [5026, -1, 6031, -1], 'generators': [2530], 'label': '53240.bd.26620.c1', 'mobius_quo': None, 'mobius_sub': 22, 'normal_closure': '110.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '121.b1', 'old_label': '26620.c1', 'projective_image': '53240.bd', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '26620.c1', '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 220, 'aut_gen_orders': [110, 20, 10, 110, 55, 10], 'aut_gens': [[1, 10, 220, 2420], [5081, 27870, 11880, 2420], [22309, 44350, 5940, 42900], [47821, 50030, 45320, 41140], [3021, 41110, 24420, 41140], [14701, 9770, 34980, 2420], [32589, 32170, 40920, 3520]], 'aut_group': None, 'aut_hash': 906806283152441688, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10648000, 'aut_permdeg': 506, 'aut_perms': 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'aut_phi_ratio': 550.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 2, 1], [2, 121, 2, 1], [2, 1331, 2, 1], [5, 121, 2, 2], [10, 121, 2, 2], [10, 121, 4, 2], [10, 1331, 4, 4], [11, 2, 5, 1], [11, 10, 2, 1], [11, 10, 10, 1], [11, 20, 10, 1], [11, 20, 50, 1], [22, 2, 5, 1], [22, 10, 2, 1], [22, 10, 10, 1], [22, 20, 10, 1], [22, 20, 50, 1], [22, 110, 4, 1], [22, 110, 20, 1], [22, 242, 10, 1], [55, 242, 10, 2], [110, 242, 10, 2], [110, 242, 20, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_{11}^3.C_5.C_{10}^2.C_2^4', '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': 220, 'autcentquo_group': None, 'autcentquo_hash': 3149364867277236831, 'autcentquo_nilpotent': False, 'autcentquo_order': 2662000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{11}^3.C_{10}.C_{10}^2.C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 11, 2], [2, 121, 2], [2, 1331, 2], [5, 121, 4], [10, 121, 12], [10, 1331, 16], [11, 2, 5], [11, 10, 12], [11, 20, 60], [22, 2, 5], [22, 10, 12], [22, 20, 60], [22, 110, 24], [22, 242, 10], [55, 242, 20], [110, 242, 60]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '26620.bb', 'commutator_count': 1, 'commutator_label': '1331.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1', '11.1', '11.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 30, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1210.10', 1], ['2.1', 1], ['22.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 1, 2], [2, 121, 1, 2], [2, 1331, 1, 2], [5, 121, 4, 1], [10, 121, 4, 3], [10, 1331, 4, 4], [11, 2, 5, 1], [11, 10, 1, 2], [11, 10, 5, 2], [11, 20, 5, 12], [22, 2, 5, 1], [22, 10, 1, 2], [22, 10, 5, 2], [22, 20, 5, 12], [22, 110, 1, 4], [22, 110, 5, 4], [22, 242, 5, 2], [55, 242, 20, 1], [110, 242, 20, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4499712, 'exponent': 110, 'exponents_of_order': [3, 3, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[20, 1, 50]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '53240.bd', 'hash': 3100417070322456923, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 22, 11, 11], 'inner_gens': [[1, 48610, 20460, 21780], [4861, 10, 2200, 50820], [35421, 450, 220, 2420], [33881, 4850, 220, 2420]], 'inner_hash': 738845080847280697, 'inner_nilpotent': False, 'inner_order': 26620, 'inner_split': False, 'inner_tex': 'D_{11}\\times C_{11}:F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': 20, 'irrQ_degree': 100, 'irrQ_dim': 100, 'irrR_degree': 20, 'irrep_stats': [[1, 40], [2, 100], [10, 48], [20, 120]], 'label': '53240.bd', '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': 'D11*C22:F11', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 34, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 308, 'number_divisions': 64, 'number_normal_subgroups': 63, 'number_subgroup_autclasses': 156, 'number_subgroup_classes': 440, 'number_subgroups': 94916, 'old_label': None, 'order': 53240, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 2927], [5, 484], [10, 22748], [11, 1330], [22, 6390], [55, 4840], [110, 14520]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 10, 10], 'outer_gen_pows': [0, 8, 0, 0], 'outer_gens': [[9, 45750, 20460, 42900], [1, 50830, 35860, 12100], [26621, 31530, 26400, 36300], [1, 19370, 5060, 31460]], 'outer_group': '400.221', 'outer_hash': 221, 'outer_nilpotent': True, 'outer_order': 400, 'outer_permdeg': 18, 'outer_perms': [355687428096000, 1307714284800, 6227475162, 39916842], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{10}^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 35, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 8], [10, 12], [40, 4], [50, 8], [100, 24]], 'representations': {'PC': {'code': '17758002571555044466105400900421978247758065417477149017511461742040987647090309909', 'gens': [1, 3, 5, 6], 'pres': [7, -2, -5, -2, -11, -11, -2, -11, 14, 1020812, 50934, 58, 5603, 716104, 96261, 7718, 914765, 762312, 213463, 124, 2134446, 474333, 237180]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [32906036359581576, 18721992341619836, 11300846666707172, 1754382938723032, 36037487804644665, 32182222080390795, 41772741070013040]}, 'Perm': {'d': 35, 'gens': [340007756491608160105911900865578977880, 608909717286147822420363932861514282487, 938901755970236865128710790579787429294, 1242035941944727800399071423394049536000, 62659640997738622771558431091307779806, 1547473788333472666977512918876160000000, 1]}}, 'schur_multiplier': [2, 2, 22], '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': 'D_{11}\\times C_{22}:F_{11}', 'transitive_degree': 220, 'wreath_data': None, 'wreath_product': False}
