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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '16934047920.b', 'ambient_counter': 2, 'ambient_order': 16934047920, 'ambient_tex': '\\PGL(3,19)', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 1, 'counter': 37, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '16934047920.b.45720._.L', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '45720.L', 'outer_equivalence': False, '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': 45720, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': None, 'subgroup_hash': None, 'subgroup_order': 370386, 'subgroup_tex': '\\He_{19}:(C_3\\times C_{18})', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '16934047920.b', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 7620, 'diagramx': None, 'generators': [196310485517, 53684935407, 319677897154, 71378628413, 136716455080, 122373361709, 181531096560], 'label': '16934047920.b.45720._.L', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '45720.L', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '45720._.L', 'subgroup_fusion': None, 'weyl_group': None}
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label None does not appear in gps_groups
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[16983693364, 16089815143], [270769875067, 35008046087], [243807198587, 121862180799], [37081001762, 247494246397], [21938407391, 281056621192], [238001375642, 233833939347], [274771737817, 36201431897], [321216475581, 275750437137], [16983693364, 16089811533], [291995051893, 158532055954]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 33868095840, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 137541, 1, 1], [3, 137541, 2, 1], [3, 14815440, 2, 1], [3, 17421860, 1, 1], [4, 47039022, 1, 1], [5, 47039022, 1, 2], [6, 137541, 2, 1], [6, 52265580, 1, 1], [6, 52265580, 2, 1], [8, 47039022, 2, 1], [9, 137541, 2, 3], [9, 52265580, 1, 3], [9, 52265580, 2, 3], [10, 47039022, 1, 2], [12, 47039022, 2, 1], [15, 47039022, 2, 2], [18, 137541, 2, 3], [18, 52265580, 1, 3], [18, 52265580, 2, 15], [19, 137160, 1, 1], [19, 46908720, 1, 1], [20, 47039022, 1, 4], [24, 47039022, 2, 2], [30, 47039022, 2, 2], [36, 47039022, 2, 3], [38, 49514760, 1, 1], [40, 47039022, 2, 4], [45, 47039022, 2, 6], [57, 49514760, 2, 1], [60, 47039022, 2, 4], [72, 47039022, 2, 6], [90, 47039022, 2, 6], [114, 49514760, 2, 1], [120, 47039022, 2, 8], [127, 44446320, 2, 21], [171, 49514760, 2, 3], [180, 47039022, 2, 12], [342, 49514760, 2, 3], [360, 47039022, 2, 24], [381, 44446320, 2, 42]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 137541, 1], [3, 137541, 2], [3, 14815440, 2], [3, 17421860, 1], [4, 47039022, 1], [5, 47039022, 2], [6, 137541, 2], [6, 52265580, 3], [8, 47039022, 2], [9, 137541, 6], [9, 52265580, 9], [10, 47039022, 2], [12, 47039022, 2], [15, 47039022, 4], [18, 137541, 6], [18, 52265580, 33], [19, 137160, 1], [19, 46908720, 1], [20, 47039022, 4], [24, 47039022, 4], [30, 47039022, 4], [36, 47039022, 6], [38, 49514760, 1], [40, 47039022, 8], [45, 47039022, 12], [57, 49514760, 2], [60, 47039022, 8], [72, 47039022, 12], [90, 47039022, 12], [114, 49514760, 2], [120, 47039022, 16], [127, 44446320, 42], [171, 49514760, 6], [180, 47039022, 24], [342, 49514760, 6], [360, 47039022, 48], [381, 44446320, 84]], 'center_label': '1.1', 'center_order': None, 'central_product': None, 'central_quotient': '16934047920.b', 'commutator_count': 1, 'commutator_label': '5644682640.a', 'complements_known': False, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '5644682640.a'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': None, 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 137541, 1, 1], [3, 137541, 2, 1], [3, 14815440, 2, 1], [3, 17421860, 1, 1], [4, 47039022, 1, 1], [5, 47039022, 2, 1], [6, 137541, 2, 1], [6, 52265580, 1, 1], [6, 52265580, 2, 1], [8, 47039022, 2, 1], [9, 137541, 6, 1], [9, 52265580, 3, 1], [9, 52265580, 6, 1], [10, 47039022, 2, 1], [12, 47039022, 2, 1], [15, 47039022, 4, 1], [18, 137541, 6, 1], [18, 52265580, 3, 1], [18, 52265580, 6, 5], [19, 137160, 1, 1], [19, 46908720, 1, 1], [20, 47039022, 4, 1], [24, 47039022, 4, 1], [30, 47039022, 4, 1], [36, 47039022, 6, 1], [38, 49514760, 1, 1], [40, 47039022, 8, 1], [45, 47039022, 12, 1], [57, 49514760, 2, 1], [60, 47039022, 8, 1], [72, 47039022, 12, 1], [90, 47039022, 12, 1], [114, 49514760, 2, 1], [120, 47039022, 16, 1], [127, 44446320, 42, 1], [171, 49514760, 6, 1], [180, 47039022, 24, 1], [342, 49514760, 6, 1], [360, 47039022, 48, 1], [381, 44446320, 84, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': None, 'exponent': 868680, 'exponents_of_order': [5, 4, 3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 19, 127], 'factors_of_order': [2, 3, 5, 19, 127], 'faithful_reps': [[380, 0, 2], [380, 1, 1], [381, 0, 14], [381, 1, 1], [6480, 0, 126], [6858, 0, 162], [6858, 1, 9], [6859, 0, 2], [6859, 1, 1], [7239, 0, 14], [7239, 1, 1], [7620, 0, 38], [7620, 1, 8]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '16934047920.b', 'hash': 4997612786352487109, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': None, 'inner_split': None, 'inner_tex': '\\PGL(3,19)', 'inner_used': None, 'irrC_degree': 380, 'irrQ_degree': 380, 'irrQ_dim': 380, 'irrR_degree': 380, 'irrep_stats': [[1, 3], [380, 3], [381, 15], [6480, 126], [6858, 171], [6859, 3], [7239, 15], [7620, 46]], 'label': '16934047920.b', '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': False, 'metacyclic': False, 'monomial': False, 'name': 'PGL(3,19)', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 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, 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, 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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, 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, 1, 0, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 202, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 382, 'number_divisions': 45, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': None, 'number_subgroup_classes': 719, 'number_subgroups': 9820525962, 'old_label': None, 'order': 16934047920, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 137541], [3, 47327822], [4, 47039022], [5, 94078044], [6, 157071822], [8, 94078044], [9, 471215466], [10, 94078044], [12, 94078044], [15, 188156088], [18, 1725589386], [19, 47045880], [20, 188156088], [24, 188156088], [30, 188156088], [36, 282234132], [38, 49514760], [40, 376312176], [45, 564468264], [57, 99029520], [60, 376312176], [72, 564468264], [90, 564468264], [114, 99029520], [120, 752624352], [127, 1866745440], [171, 297088560], [180, 1128936528], [342, 297088560], [360, 2257873056], [381, 3733490880]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '2.1', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 2, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 381, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [380, 1], [381, 1], [760, 1], [762, 1], [2286, 2], [6858, 1], [6859, 1], [7239, 1], [7620, 2], [13716, 4], [13718, 1], [14478, 1], [15240, 1], [22860, 2], [27432, 4], [41148, 1], [43434, 2], [45720, 6], [54864, 2], [82296, 3], [109728, 1], [164592, 1], [272160, 1], [329184, 1], [544320, 1]], 'representations': {'Lie': [{'d': 3, 'q': 19, 'gens': [16983693364, 16089815143], 'family': 'PGL'}, {'d': 3, 'q': 19, 'family': 'PGammaL'}], 'Perm': {'d': 381, 'gens': 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'subgroup_inclusions_known': False, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\PGL(3,19)', 'transitive_degree': 381, 'wreath_data': None, 'wreath_product': False}
