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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '15984.a', 'ambient_counter': 1, 'ambient_order': 15984, 'ambient_tex': 'C_{12}\\times F_{37}', 'central': False, 'central_factor': False, 'centralizer_order': 432, 'characteristic': False, 'core_order': 1, 'counter': 139, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '15984.a.7992.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '7992.b1', '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': 7992, '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': '15984.a', 'aut_centralizer_order': None, 'aut_label': '7992.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '37.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['216.b1', '2664.d1', '2664.e1', '2664.f1', '3996.b1', '3996.d1'], 'contains': ['15984.a1'], 'core': '15984.a1', 'coset_action_label': None, 'count': 74, 'diagramx': [6869, -1, 6132, -1], 'generators': [18], 'label': '15984.a.7992.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '108.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '37.a1', 'old_label': '7992.b1', 'projective_image': '15984.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '7992.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': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '432.200', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1332, 'aut_gen_orders': [6, 18, 36, 18, 18], 'aut_gens': [[1, 36], [12565, 7596], [12061, 8892], [1405, 12780], [7381, 14220], [1873, 4572]], 'aut_group': None, 'aut_hash': 792994677548912317, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 63936, 'aut_permdeg': 259, 'aut_perms': 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1879156675783510685770034401612354623658578968299413828972656133245984566818528333391914555931981458158298920526995513526936638207474229486422367867841602675586614865559010751105449347027549666611686478864155925273061896774643491976294222588772916154432564373815869072993899188218033374062929235936780705814466848475430148462361279275037496026216791095100559916737874793888820240758477235535763428299315624584787298697532300443308221831936598450016368771330614299801680907195903782593833845207334208728537713208889], 'aut_phi_ratio': 12.333333333333334, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 37, 2, 1], [3, 1, 2, 1], [3, 37, 1, 2], [3, 37, 2, 2], [4, 1, 2, 1], [4, 37, 2, 1], [4, 37, 4, 2], [6, 1, 2, 1], [6, 37, 1, 2], [6, 37, 2, 4], [6, 37, 4, 3], [9, 37, 3, 6], [12, 1, 4, 1], [12, 37, 2, 4], [12, 37, 4, 9], [12, 37, 8, 6], [18, 37, 3, 6], [18, 37, 6, 6], [36, 37, 6, 12], [36, 37, 12, 12], [37, 36, 1, 1], [74, 36, 1, 1], [111, 36, 2, 1], [148, 36, 2, 1], [222, 36, 2, 1], [444, 36, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{222}.C_{18}.C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '48.38', 'autcent_hash': 38, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3\\times D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1332, 'autcentquo_group': '1332.31', 'autcentquo_hash': 31, 'autcentquo_nilpotent': False, 'autcentquo_order': 1332, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{37}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 37, 2], [3, 1, 2], [3, 37, 6], [4, 1, 2], [4, 37, 10], [6, 1, 2], [6, 37, 22], [9, 37, 18], [12, 1, 4], [12, 37, 92], [18, 37, 54], [36, 37, 216], [37, 36, 1], [74, 36, 1], [111, 36, 2], [148, 36, 2], [222, 36, 2], [444, 36, 4]], 'center_label': '12.2', 'center_order': 12, 'central_product': True, 'central_quotient': '1332.31', 'commutator_count': 1, 'commutator_label': '37.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '37.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1332.31', 1], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 37, 1, 2], [3, 1, 2, 1], [3, 37, 2, 3], [4, 1, 2, 1], [4, 37, 2, 5], [6, 1, 2, 1], [6, 37, 2, 11], [9, 37, 6, 3], [12, 1, 4, 1], [12, 37, 4, 23], [18, 37, 6, 9], [36, 37, 12, 18], [37, 36, 1, 1], [74, 36, 1, 1], [111, 36, 2, 1], [148, 36, 2, 1], [222, 36, 2, 1], [444, 36, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 864, 'exponent': 1332, 'exponents_of_order': [4, 3, 1], 'factors_of_aut_order': [2, 3, 37], 'factors_of_order': [2, 3, 37], 'faithful_reps': [[36, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '7992.a', 'hash': 1147931464434849270, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 1332, 'inner_gen_orders': [36, 37], 'inner_gens': [[1, 8676], [7345, 36]], 'inner_hash': 31, 'inner_nilpotent': False, 'inner_order': 1332, 'inner_split': True, 'inner_tex': 'F_{37}', 'inner_used': [1, 2], 'irrC_degree': 36, 'irrQ_degree': 144, 'irrQ_dim': 144, 'irrR_degree': 72, 'irrep_stats': [[1, 432], [36, 12]], 'label': '15984.a', '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': True, 'monomial': True, 'name': 'C12*F37', 'ngens': 8, '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 90, 'number_characteristic_subgroups': 41, 'number_conjugacy_classes': 444, 'number_divisions': 86, 'number_normal_subgroups': 156, 'number_subgroup_autclasses': 140, 'number_subgroup_classes': 300, 'number_subgroups': 5484, 'old_label': None, 'order': 15984, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 75], [3, 224], [4, 372], [6, 816], [9, 666], [12, 3408], [18, 1998], [36, 7992], [37, 36], [74, 36], [111, 72], [148, 72], [222, 72], [444, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[5329, 7956], [3997, 8028], [9325, 2628]], 'outer_group': '48.38', 'outer_hash': 38, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 7, 'outer_perms': [745, 1680, 1707], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 44, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 3, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 22], [4, 24], [6, 12], [12, 18], [36, 2], [72, 3], [144, 1]], 'representations': {'PC': {'code': '16524042863852504807657951240713532000254346487170149174608922200597968863130125716179175519619787751934495769927', 'gens': [1, 5], 'pres': [8, -2, -2, -3, -3, -2, -2, -3, -37, 16, 41, 90, 347044, 259932, 82460, 14548, 116, 65669, 240205, 6069, 34877, 141, 153222, 112910, 14134, 6750, 222, 525319, 387087, 48407, 23071]}, 'GLFp': {'d': 2, 'p': 37, 'gens': [50691, 405225, 962409]}, 'Perm': {'d': 44, 'gens': [1473647718507523536228011078992636227327426857028483, 2913759652257673605104566425069958962244530419998948, 61958305861368147964452455736446518711083563906954880]}}, 'schur_multiplier': [12], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [12, 36], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}\\times F_{37}', 'transitive_degree': 444, 'wreath_data': None, 'wreath_product': False}
