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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '720.441', 'ambient_counter': 441, 'ambient_order': 720, 'ambient_tex': 'D_{10}.S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 180, 'characteristic': False, 'core_order': 1, 'counter': 116, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '720.441.240.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '240.c1.a1', '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': 240, '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': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '720.441', 'aut_centralizer_order': 720, 'aut_label': '240.c1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '2.1', 'aut_weyl_index': 1440, 'centralizer': '4.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['48.c1.a1', '80.a1.a1', '120.c1.a1', '120.f1.a1', '120.i1.a1'], 'contains': ['720.a1.a1'], 'core': '720.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [5413, -1, 6429, -1, 3845, -1, 6235, -1], 'generators': [400], 'label': '720.441.240.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '80.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.c1.a1', 'old_label': '240.c1.a1', 'projective_image': '720.441', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '240.c1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, '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': ['3.1'], '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], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3', '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': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [2, 2, 6, 12, 10], 'aut_gens': [[1, 4, 240], [121, 124, 480], [121, 4, 240], [121, 404, 240], [161, 52, 240], [169, 124, 240]], 'aut_group': '2880.fv', 'aut_hash': 4411530478221040614, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2880, 'aut_permdeg': 36, 'aut_perms': [30704746124937504817866065139415468381355, 30704746124915727699412410827181078092075, 23330977234401596622185670551115415556301, 362857406056026151110148382261934660962879, 339482443439785202480412060609729369244560], 'aut_phi_ratio': 15.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [4, 6, 1, 1], [4, 30, 1, 3], [4, 90, 1, 2], [5, 4, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 10, 1, 4], [6, 20, 1, 2], [10, 4, 1, 1], [12, 6, 2, 1], [12, 30, 2, 3], [15, 4, 2, 1], [15, 8, 1, 1], [15, 8, 2, 1], [20, 12, 2, 1], [30, 4, 2, 1], [30, 8, 1, 1], [30, 8, 2, 1], [60, 12, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_5\\times D_6^2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '720.783', 'autcentquo_hash': 783, 'autcentquo_nilpotent': False, 'autcentquo_order': 720, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5\\times S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 2], [3, 2, 2], [3, 4, 1], [4, 6, 1], [4, 30, 3], [4, 90, 2], [5, 4, 1], [6, 2, 2], [6, 4, 1], [6, 10, 4], [6, 20, 2], [10, 4, 1], [12, 6, 2], [12, 30, 6], [15, 4, 2], [15, 8, 3], [20, 12, 2], [30, 4, 2], [30, 8, 3], [60, 12, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '360.128', 'commutator_count': 1, 'commutator_label': '90.10', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 441, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [4, 6, 1, 1], [4, 30, 1, 1], [4, 30, 2, 1], [4, 90, 2, 1], [5, 4, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 10, 1, 4], [6, 20, 1, 2], [10, 4, 1, 1], [12, 6, 2, 1], [12, 30, 2, 1], [12, 30, 4, 1], [15, 4, 2, 1], [15, 8, 1, 1], [15, 8, 2, 1], [20, 12, 2, 1], [30, 4, 2, 1], [30, 8, 1, 1], [30, 8, 2, 1], [60, 12, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 60, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '360.128', 'hash': 441, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [4, 30, 3], 'inner_gens': [[1, 188, 240], [57, 4, 480], [1, 484, 240]], 'inner_hash': 128, 'inner_nilpotent': False, 'inner_order': 360, 'inner_split': True, 'inner_tex': 'D_5.S_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 18], [4, 16], [8, 6]], 'label': '720.441', 'linC_count': 48, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 10, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D10.S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 36, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 48, 'number_divisions': 33, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 118, 'number_subgroup_classes': 120, 'number_subgroups': 880, 'old_label': None, 'order': 720, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 11], [3, 8], [4, 276], [5, 4], [6, 88], [10, 4], [12, 192], [15, 32], [20, 24], [30, 32], [60, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 44, 480], [1, 4, 480], [1, 164, 240]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 11], [8, 5], [16, 3]], 'representations': {'PC': {'code': 43213287189079580292461484762839009459735838749, 'gens': [1, 3, 7], 'pres': [7, -2, -2, -2, -2, -3, -5, -3, 14, 3950, 2067, 58, 3811, 2138, 80, 1124, 1131, 137, 4037, 4044, 5900]}, 'GLZN': {'d': 2, 'p': 45, 'gens': [152551, 1731376, 2369276, 1205771, 2855941, 91531, 2664709]}, 'Perm': {'d': 15, 'gens': [6745980264, 1465, 13412044800, 2424, 3, 403200, 106817356800]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{10}.S_3^2', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
