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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1944.3890', 'ambient_counter': 3890, 'ambient_order': 1944, 'ambient_tex': 'C_3^3:F_9', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 34, 'cyclic': True, 'direct': None, 'hall': 2, 'label': '1944.3890.243.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '243.a1', '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': 243, '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': '8.1', 'subgroup_hash': 1, 'subgroup_order': 8, 'subgroup_tex': 'C_8', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '1944.3890', 'aut_centralizer_order': 11520, 'aut_label': '243.a1', 'aut_quo_index': None, 'aut_stab_index': 243, 'aut_weyl_group': '2.1', 'aut_weyl_index': 2799360, 'centralizer': '243.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['27.a1', '81.a1'], 'contains': ['486.a1'], 'core': '1944.a1', 'coset_action_label': None, 'count': 243, 'diagramx': [9014, -1, 9590, -1], 'generators': [1], 'label': '1944.3890.243.a1', 'mobius_quo': None, 'mobius_sub': -9, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '243.a1', 'old_label': '243.a1', 'projective_image': '1944.3890', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '243.a1', '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': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2], 'aut_gens': [[1], [5], [3]], 'aut_group': '4.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [1, 6], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2', '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': 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], [4, 1, 2], [8, 1, 4]], 'center_label': '8.1', 'center_order': 8, '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', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 8, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '4.1', '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 8]], 'label': '8.1', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C8', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 8, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 2], [8, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5], [3]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1]], 'representations': {'PC': {'code': 323, 'gens': [1], 'pres': [3, -2, -2, -2, 6, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26315074]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [77]}, 'Perm': {'d': 8, 'gens': [40176, 16582, 5167]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_8', 'transitive_degree': 8, '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': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 240, 'aut_gen_orders': [48, 10, 6, 8], 'aut_gens': [[1, 8, 24, 72, 216, 648], [1275, 1744, 1112, 72, 1536, 1312], [1777, 1568, 1792, 72, 1328, 1088], [1091, 1320, 232, 72, 688, 456], [1107, 280, 1768, 144, 1120, 672]], 'aut_group': None, 'aut_hash': 6380806534529353498, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5598720, 'aut_permdeg': 486, 'aut_perms': 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'aut_phi_ratio': 8640.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 1, 1], [3, 8, 10, 1], [3, 8, 20, 1], [4, 81, 2, 1], [6, 162, 1, 1], [8, 243, 2, 2], [12, 162, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^5.C_4.C_2^2.A_6.C_2^2', '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': 240, 'autcentquo_group': None, 'autcentquo_hash': 6380806534529353498, 'autcentquo_nilpotent': False, 'autcentquo_order': 5598720, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^5.C_8.C_2.A_6.C_2^2', 'cc_stats': [[1, 1, 1], [2, 81, 1], [3, 2, 1], [3, 8, 30], [4, 81, 2], [6, 162, 1], [8, 243, 4], [12, 162, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1944.3890', 'commutator_count': 1, 'commutator_label': '243.67', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3890, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 1, 1], [3, 8, 1, 10], [3, 8, 2, 10], [4, 81, 2, 1], [6, 162, 1, 1], [8, 243, 4, 1], [12, 162, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 896, 'exponent': 24, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1944.3890', 'hash': 3890, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [8, 3, 3, 3, 3, 3], 'inner_gens': [[1, 24, 56, 144, 648, 1512], [57, 8, 24, 72, 216, 648], [65, 8, 24, 72, 216, 648], [145, 8, 24, 72, 216, 648], [1513, 8, 24, 72, 216, 648], [1729, 8, 24, 72, 216, 648]], 'inner_hash': 3890, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': False, 'inner_tex': 'C_3^3:F_9', 'inner_used': [1, 2, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 4], [8, 30]], 'label': '1944.3890', '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': 'C3^3:F9', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 42, 'number_divisions': 27, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 42, 'number_subgroup_classes': 576, 'number_subgroups': 8600, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 81], [3, 242], [4, 162], [6, 162], [8, 972], [12, 324]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 120, 'outer_gen_orders': [4, 3, 2, 2, 4], 'outer_gen_pows': [4, 7, 2, 4, 6], 'outer_gens': [[1, 432, 1296, 72, 232, 696], [1, 48, 40, 144, 1328, 1088], [1, 928, 232, 72, 896, 224], [3, 440, 928, 144, 448, 896], [1, 1776, 472, 72, 696, 1552]], 'outer_group': '2880.a', 'outer_hash': 5429726546426601382, 'outer_nilpotent': False, 'outer_order': 2880, 'outer_permdeg': 14, 'outer_perms': [14611455480, 56614048560, 51859716487, 1480722616, 19679361982], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'A_6.D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 2], [8, 10], [16, 10]], 'representations': {'PC': {'code': 96986269653408272509132065749860890293778894979559424131871, 'gens': [1, 4, 5, 6, 7, 8], 'pres': [8, -2, -2, -2, -3, 3, -3, -3, 3, 16, 41, 771, 907, 147, 2244, 1292, 500, 6917, 36294, 42350, 6070, 96775, 55311, 20759]}, 'Perm': {'d': 21, 'gens': [129177310999169119, 269696249914313391, 398729704275321620, 2818115596411374011, 5384837059957665106, 5384837059957709125, 7915198737334053978, 43545600]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:F_9', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
