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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1188.39', 'ambient_counter': 39, 'ambient_order': 1188, 'ambient_tex': 'S_3\\times D_{99}', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 1, 'counter': 60, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1188.39.198.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '198.g1.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': 198, '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': '6.1', 'subgroup_hash': 1, 'subgroup_order': 6, 'subgroup_tex': 'S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1188.39', 'aut_centralizer_order': 30, 'aut_label': '198.g1', 'aut_quo_index': None, 'aut_stab_index': 198, 'aut_weyl_group': '6.1', 'aut_weyl_index': 5940, 'centralizer': '1188.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['18.g1.a1', '66.e1.a1'], 'contains': ['396.c1.a1', '594.c1.a1'], 'core': '1188.a1.a1', 'coset_action_label': None, 'count': 198, 'diagramx': [3842, -1, 3999, -1, 4501, -1, 4499, -1], 'generators': [3, 796], 'label': '1188.39.198.g1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '198.g1.a1', 'old_label': '198.g1.a1', 'projective_image': '1188.39', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '198.g1.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 4], [1, 3], [2, 4]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', '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': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 2, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 3], [2, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 1]], 'label': '6.1', 'linC_count': 1, 'linC_degree': 2, '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': 'S3', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 3, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 6, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 3], [3, 2]], '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': 2, 'perfect': False, 'permutation_degree': 3, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 25, 'gens': [1, 2], 'pres': [2, -2, -3, 17]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [28, 45]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'GL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'SL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGL'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'SO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'SU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PSO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'PSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'GO'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'Omega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PGO'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'POmega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CSO'}, {'d': 2, 'q': 4, 'gens': [20, 194], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'CSOMinus'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'CSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CO'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'COMinus'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGammaL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSigmaL'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGammaU'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGL'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11, 6]}, 'Perm': {'d': 3, 'gens': [1, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3', 'transitive_degree': 3, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 990, 'aut_gen_orders': [30, 10, 90, 30], 'aut_gens': [[1, 2, 12], [9, 478, 192], [9, 658, 636], [9, 646, 84], [5, 1082, 168]], 'aut_group': None, 'aut_hash': 7847050869488114835, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 35640, 'aut_permdeg': 102, 'aut_perms': [406418291063869129739674114442944799911296235623103579778248678581702537548745720033117870759545830397518179946640643286566565974221077945279791067453409664280112, 411329552169017794169720535254644763209185884550668858483245392123768078372456293879446458839938134966263430137141406855762779294818364512475827596015913087504824, 410579076539783953669954928088256751365537129377285892439900762385163418326903942963903836580207199421465523271412307751321811021764922120973951579126463817095319, 193618846551680029775380209837606738985088993792083246083037061844844759868577951900418379618496720483064250300922472428342334207766279455313741751552586268566172], 'aut_phi_ratio': 99.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 99, 1, 1], [2, 297, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [6, 6, 1, 1], [6, 198, 1, 1], [9, 2, 3, 1], [9, 4, 3, 1], [11, 2, 5, 1], [18, 6, 3, 1], [22, 6, 5, 1], [33, 2, 10, 1], [33, 4, 5, 1], [33, 4, 10, 1], [66, 6, 10, 1], [99, 2, 30, 1], [99, 4, 30, 1], [198, 6, 30, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_3\\times C_{99}).C_{30}.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': 990, 'autcentquo_group': None, 'autcentquo_hash': 7847050869488114835, 'autcentquo_nilpotent': False, 'autcentquo_order': 35640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': '(C_3\\times C_{99}).C_{30}.C_2^2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 99, 1], [2, 297, 1], [3, 2, 2], [3, 4, 1], [6, 6, 1], [6, 198, 1], [9, 2, 3], [9, 4, 3], [11, 2, 5], [18, 6, 3], [22, 6, 5], [33, 2, 10], [33, 4, 15], [66, 6, 10], [99, 2, 30], [99, 4, 30], [198, 6, 30]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '1188.39', 'commutator_count': 1, 'commutator_label': '297.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 39, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['198.3', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 99, 1, 1], [2, 297, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [6, 6, 1, 1], [6, 198, 1, 1], [9, 2, 3, 1], [9, 4, 3, 1], [11, 2, 5, 1], [18, 6, 3, 1], [22, 6, 5, 1], [33, 2, 10, 1], [33, 4, 5, 1], [33, 4, 10, 1], [66, 6, 10, 1], [99, 2, 30, 1], [99, 4, 30, 1], [198, 6, 30, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 198, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[4, 1, 30]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '396.22', 'hash': 39, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 198, 'inner_gen_orders': [2, 6, 99], 'inner_gens': [[1, 10, 12], [5, 2, 1176], [1, 26, 12]], 'inner_hash': 39, 'inner_nilpotent': False, 'inner_order': 1188, 'inner_split': True, 'inner_tex': 'S_3\\times D_{99}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 120, 'irrQ_dim': 120, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 100], [4, 49]], 'label': '1188.39', 'linC_count': 150, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 8, 'linQ_dim': 18, 'linQ_dim_count': 8, 'linR_count': 150, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*D99', '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': 21, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 153, 'number_divisions': 21, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 68, 'number_subgroup_classes': 68, 'number_subgroups': 1892, 'old_label': None, 'order': 1188, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 399], [3, 8], [6, 204], [9, 18], [11, 10], [18, 18], [22, 30], [33, 80], [66, 60], [99, 180], [198, 180]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [30], 'outer_gen_pows': [0], 'outer_gens': [[5, 2, 168]], 'outer_group': '30.4', 'outer_hash': 4, 'outer_nilpotent': True, 'outer_order': 30, 'outer_permdeg': 10, 'outer_perms': [373024], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{30}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1], [6, 2], [10, 2], [12, 1], [20, 3], [40, 1], [60, 2], [120, 1]], 'representations': {'PC': {'code': 31579982350277749202110920826001119, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -3, -3, -3, -11, 121, 31, 146, 14121, 93, 17290, 118, 19451]}, 'Perm': {'d': 23, 'gens': [111926654118639728047, 479001600, 1284585462597685248000, 6706022400, 4364893, 2462597285793765120000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times D_{99}', 'transitive_degree': 198, 'wreath_data': None, 'wreath_product': False}
