Formats: - HTML - YAML - JSON - 2026-08-16T17:22:38.053118
  • gps_subgroup_search • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '512072.a', 'ambient_counter': 1, 'ambient_order': 512072, 'ambient_tex': 'F_{23}\\wr C_2', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': True, 'core_order': 46552, 'counter': 9, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '512072.a.11.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '11.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '11.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 11, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{11}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '46552.a', 'subgroup_hash': 8001351648207747222, 'subgroup_order': 46552, 'subgroup_tex': 'D_{23}^2:D_{11}', 'supersolvable': False, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '512072.a', 'aut_centralizer_order': None, 'aut_label': '11.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '512072.a1.a1', 'complements': ['46552.a1.a1', '46552.f1.a1', '46552.e1.a1', '46552.g1.a1', '46552.d1.a1', '46552.b1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['22.b1.a1', '22.c1.a1', '22.d1.a1', '121.a1.a1', '5819.b1.a1'], 'core': '11.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8751, 6618, 1198, 6701, 8292, 8716, 8297, 8716], 'generators': [80348, 179080, 22264, 209110, 11132, 121721], 'label': '512072.a.11.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '11.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['22.b1.a1', '22.c1.a1', '22.d1.a1'], 'normalizer': '1.a1.a1', 'old_label': '11.a1.a1', 'projective_image': '512072.a', 'quotient_action_image': '11.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '11.a1.a1', 'subgroup_fusion': None, 'weyl_group': '512072.a'}
  • gps_groups • Show schema
    {'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': 3, 'aut_exponent': 1012, 'aut_gen_orders': [22, 22, 506, 23], 'aut_gens': [[1, 2, 44, 2024], [18217, 45296, 30206, 880], [43041, 8362, 43956, 14168], [7191, 29042, 12452, 2024], [41185, 882, 44, 2024]], 'aut_group': '512072.a', 'aut_hash': 7624117046359661246, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 512072, 'aut_permdeg': 552, 'aut_perms': 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'aut_phi_ratio': 25.3, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 46, 1, 1], [2, 506, 1, 1], [2, 529, 1, 1], [4, 11638, 1, 1], [11, 1058, 1, 5], [22, 1058, 1, 15], [23, 44, 1, 1], [23, 44, 11, 1], [46, 1012, 1, 1], [46, 1012, 11, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_{23}\\wr C_2', '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': 1012, 'autcentquo_group': '512072.a', 'autcentquo_hash': 7624117046359661246, 'autcentquo_nilpotent': False, 'autcentquo_order': 512072, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_{23}\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 46, 1], [2, 506, 1], [2, 529, 1], [4, 11638, 1], [11, 1058, 5], [22, 1058, 15], [23, 44, 12], [46, 1012, 12]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '46552.a', 'commutator_count': 1, 'commutator_label': '11638.22', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '11.1', '23.1', '23.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 46, 1, 1], [2, 506, 1, 1], [2, 529, 1, 1], [4, 11638, 1, 1], [11, 1058, 5, 1], [22, 1058, 5, 1], [22, 1058, 10, 1], [23, 44, 1, 1], [23, 44, 11, 1], [46, 1012, 1, 1], [46, 1012, 11, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1440, 'exponent': 1012, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 11, 23], 'factors_of_order': [2, 11, 23], 'faithful_reps': [[44, 1, 24]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '46552.a', 'hash': 8001351648207747222, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 1012, 'inner_gen_orders': [46, 22, 46, 23], 'inner_gens': [[1, 7280, 33814, 1584], [39759, 2, 16940, 8096], [33859, 31682, 44, 44528], [2465, 40482, 4092, 2024]], 'inner_hash': 8001351648207747222, 'inner_nilpotent': False, 'inner_order': 46552, 'inner_split': True, 'inner_tex': 'D_{23}^2:D_{11}', 'inner_used': [1, 2], 'irrC_degree': 44, 'irrQ_degree': 44, 'irrQ_dim': 44, 'irrR_degree': 44, 'irrep_stats': [[1, 4], [2, 21], [44, 24]], 'label': '46552.a', 'linC_count': 24, 'linC_degree': 44, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 44, 'linQ_degree_count': 2, 'linQ_dim': 44, 'linQ_dim_count': 2, 'linR_count': 24, 'linR_degree': 44, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'D23^2:D11', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 29, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 49, 'number_divisions': 12, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 47, 'number_subgroup_classes': 47, 'number_subgroups': 26362, 'old_label': None, 'order': 46552, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1081], [4, 11638], [11, 5290], [22, 15870], [23, 528], [46, 12144]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 11, 'outer_gen_orders': [11], 'outer_gen_pows': [9482], 'outer_gens': [[43301, 28778, 46508, 14168]], 'outer_group': '11.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 11, 'outer_permdeg': 11, 'outer_perms': [4447025], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{11}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 46, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [10, 2], [20, 1], [44, 2], [484, 2]], 'representations': {'PC': {'code': '560400840351995208097766958332331662966330710693498242291647878888145638457712891', 'gens': [1, 2, 4, 6], 'pres': [6, 2, 2, 11, 2, 23, 23, 250800, 87361, 31, 380882, 811539, 203289, 209895, 69, 546484, 22450, 8596, 57029, 145739, 291473, 36455]}, 'Perm': {'d': 46, 'gens': [4611431925783925206182647098209232822158070048417750728423, 3090597174828667965773827252616835498760704556289774904945]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_{23}^2:D_{11}', 'transitive_degree': 46, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '44.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 1012, 'aut_gen_orders': [22, 22, 506, 23], 'aut_gens': [[1, 2, 44, 22264], [462733, 135562, 505342, 9680], [150081, 43562, 347556, 155848], [238131, 67762, 61028, 22264], [283625, 12586, 44, 22264]], 'aut_group': '512072.a', 'aut_hash': 7624117046359661246, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 512072, 'aut_permdeg': 552, 'aut_perms': 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'aut_phi_ratio': 2.3, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 46, 1, 1], [2, 506, 1, 1], [2, 529, 1, 1], [4, 11638, 1, 1], [11, 46, 1, 10], [11, 529, 1, 10], [11, 1058, 1, 45], [22, 46, 1, 10], [22, 529, 1, 10], [22, 1058, 1, 165], [22, 11638, 1, 10], [23, 44, 1, 1], [23, 484, 1, 1], [44, 11638, 1, 10], [46, 1012, 1, 1], [46, 11132, 1, 1], [253, 1012, 1, 10], [506, 1012, 1, 10]], 'aut_supersolvable': False, 'aut_tex': 'F_{23}\\wr C_2', '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': 1012, 'autcentquo_group': '512072.a', 'autcentquo_hash': 7624117046359661246, 'autcentquo_nilpotent': False, 'autcentquo_order': 512072, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_{23}\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 46, 1], [2, 506, 1], [2, 529, 1], [4, 11638, 1], [11, 46, 10], [11, 529, 10], [11, 1058, 45], [22, 46, 10], [22, 529, 10], [22, 1058, 165], [22, 11638, 10], [23, 44, 1], [23, 484, 1], [44, 11638, 10], [46, 1012, 1], [46, 11132, 1], [253, 1012, 10], [506, 1012, 10]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '512072.a', 'commutator_count': 1, 'commutator_label': '11638.22', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '11.1', '11.1', '23.1', '23.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 46, 1, 1], [2, 506, 1, 1], [2, 529, 1, 1], [4, 11638, 1, 1], [11, 46, 10, 1], [11, 529, 10, 1], [11, 1058, 5, 1], [11, 1058, 10, 4], [22, 46, 10, 1], [22, 529, 10, 1], [22, 1058, 5, 1], [22, 1058, 10, 16], [22, 11638, 10, 1], [23, 44, 1, 1], [23, 484, 1, 1], [44, 11638, 10, 1], [46, 1012, 1, 1], [46, 11132, 1, 1], [253, 1012, 10, 1], [506, 1012, 10, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 172800, 'exponent': 1012, 'exponents_of_order': [3, 2, 2], 'factors_of_aut_order': [2, 11, 23], 'factors_of_order': [2, 11, 23], 'faithful_reps': [[44, 0, 20], [44, 1, 2], [484, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '512072.a', 'hash': 7624117046359661246, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 1012, 'inner_gen_orders': [22, 22, 506, 23], 'inner_gens': [[1, 452098, 57158, 1936], [24205, 2, 239140, 222640], [239139, 295242, 44, 155848], [42593, 311698, 378532, 22264]], 'inner_hash': 7624117046359661246, 'inner_nilpotent': False, 'inner_order': 512072, 'inner_split': True, 'inner_tex': 'F_{23}\\wr C_2', 'inner_used': [1, 2, 3], 'irrC_degree': 44, 'irrQ_degree': 44, 'irrQ_dim': 44, 'irrR_degree': 44, 'irrep_stats': [[1, 44], [2, 231], [44, 22], [484, 2]], 'label': '512072.a', 'linC_count': 22, 'linC_degree': 44, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 44, 'linQ_degree_count': 2, 'linQ_dim': 44, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 44, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'F23wrC2', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 299, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 299, 'number_divisions': 39, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 194, 'number_subgroup_classes': 194, 'number_subgroups': 93140, 'old_label': None, 'order': 512072, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1081], [4, 11638], [11, 53360], [22, 296700], [23, 528], [44, 116380], [46, 12144], [253, 10120], [506, 10120]], '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': 4, 'perfect': False, 'permutation_degree': 46, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [10, 6], [20, 22], [44, 2], [440, 2], [484, 2]], 'representations': {'PC': {'code': '2009512658161263764406495818022110616039706956299674565163040514242083078967261616350490708237669655345405470956773691013965559', 'gens': [1, 2, 4, 7], 'pres': [7, 2, 2, 11, 2, 11, 23, 23, 17332, 6329373, 36, 10449434, 1600427, 3347970, 2947885, 80, 4102704, 4473711, 1330578, 417, 1870181, 142308, 30511, 94870, 5454693, 2181892, 173585, 37225]}, 'Perm': {'d': 46, 'gens': [970417196495182273109274329196674182628710082377058649424, 4736529592668538766970483987095405186160907775497953591576]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 22], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'F_{23}\\wr C_2', 'transitive_degree': 46, 'wreath_data': ['F_{23}', 'C_2', '2T1'], 'wreath_product': True}
  • gps_groups • Show schema
    {'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '11.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 10, 'aut_gen_orders': [10], 'aut_gens': [[1], [2]], 'aut_group': '10.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 10, 'aut_permdeg': 7, 'aut_perms': [753], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [11, 1, 10, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{10}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 10, 'autcent_group': '10.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 10, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{10}', '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], [11, 1, 10]], 'center_label': '11.1', 'center_order': 11, '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': ['11.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [11, 1, 10, 1]], 'element_repr_type': 'PC', 'elementary': 11, 'eulerian_function': 1, 'exponent': 11, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [11], 'faithful_reps': [[1, 0, 10]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '11.1', 'hash': 1, 'hyperelementary': 11, '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': 10, 'irrQ_dim': 10, 'irrR_degree': 2, 'irrep_stats': [[1, 11]], 'label': '11.1', 'linC_count': 10, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 5, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C11', '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': 11, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 11, 'order_factorization_type': 1, 'order_stats': [[1, 1], [11, 10]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [10], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '10.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 10, 'outer_permdeg': 7, 'outer_perms': [753], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{10}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 11, 'pgroup': 11, 'primary_abelian_invariants': [11], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [10, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -11]}, 'Lie': [{'d': 1, 'q': 11, 'gens': [4037913], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 11, 'gens': [1343]}, 'Perm': {'d': 11, 'gens': [36288000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [11], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}', 'transitive_degree': 11, 'wreath_data': None, 'wreath_product': False}