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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1002.1', 'ambient_counter': 1, 'ambient_order': 1002, 'ambient_tex': 'S_3\\times C_{167}', 'central': False, 'central_factor': True, 'centralizer_order': 167, 'characteristic': True, 'core_order': 6, 'counter': 5, 'cyclic': False, 'direct': True, 'hall': 6, 'label': '1002.1.167.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '167.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '167.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 167, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{167}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, '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': '1002.1', 'aut_centralizer_order': 166, 'aut_label': '167.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '6.1', 'aut_weyl_index': 166, 'centralizer': '6.a1.a1', 'complements': ['6.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['334.a1.a1', '501.a1.a1'], 'core': '167.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7420, 8044, 5933, 8044, 8913, 7880, 5000, 2170], 'generators': [1, 668], 'label': '1002.1.167.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '167.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['334.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '167.a1.a1', 'projective_image': '1002.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '167.1', 'quotient_action_kernel_order': 167, 'quotient_fusion': None, 'short_label': '167.a1.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': True, 'abelian': False, 'abelian_quotient': '334.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 498, 'aut_gen_orders': [2, 498], 'aut_gens': [[1, 2], [1, 670], [669, 728]], 'aut_group': '996.8', 'aut_hash': 8, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 996, 'aut_permdeg': 169, 'aut_perms': [461659132564544618846616966099302292265069832539514738625245486943543327012769915460378707234247225138752687468288583155424963478536293683793087082402081570836221385323530558836848564633600000000000000000000, 21150753763556000227973448005325536733702836250061415371094489430036243645178490197170732379463145274504662368793862780591670211169982601755877887487444464624986083277269471852014127507972285919552471923021572674690803667873664821391528286942422161839747889796410682729992653693093229701622916462087964771], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1], [167, 1, 166, 1], [334, 3, 166, 1], [501, 2, 166, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times C_{166}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 166, 'autcent_group': '166.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 166, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{166}', '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], [167, 1, 166], [334, 3, 166], [501, 2, 166]], 'center_label': '167.1', 'center_order': 167, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '167.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['167.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1], [167, 1, 166, 1], [334, 3, 166, 1], [501, 2, 166, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 504, 'exponent': 1002, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 3, 83], 'factors_of_order': [2, 3, 167], 'faithful_reps': [[2, 0, 166]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1002.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 670], [335, 2]], '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': 332, 'irrQ_dim': 332, 'irrR_degree': 4, 'irrep_stats': [[1, 334], [2, 167]], 'label': '1002.1', 'linC_count': 166, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 168, 'linQ_degree_count': 2, 'linQ_dim': 168, 'linQ_dim_count': 2, 'linR_count': 249, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'S3*C167', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 501, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 12, 'old_label': None, 'order': 1002, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 3], [3, 2], [167, 166], [334, 498], [501, 332]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 166, 'outer_gen_orders': [166], 'outer_gen_pows': [0], 'outer_gens': [[1, 560]], 'outer_group': '166.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 166, 'outer_permdeg': 85, 'outer_perms': [3353220009928858569336442912103262493625458976355624940065453228977352138859978594316152307777012044859965440000000000000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{166}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 170, 'pgroup': 0, 'primary_abelian_invariants': [2, 167], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [166, 2], [332, 1]], 'representations': {'PC': {'code': 40774496914011, 'gens': [1, 2], 'pres': [3, -2, -3, -167, 4021, 22]}, 'Perm': {'d': 170, 'gens': [1, 43195913247082572295709071439924867380446199647133641516885258902299512721861729362348705158067691059345802156887158967072802038554332053088826219201041408332305210555224842687973251851548965605572735965628045335774365608371143361268337876994198392192903108197439150218078266605052213208731613217600866560, 3]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [334], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_{167}', 'transitive_degree': 501, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '167.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 166, 'aut_gen_orders': [166], 'aut_gens': [[1], [67]], 'aut_group': '166.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 166, 'aut_permdeg': 85, 'aut_perms': [3353220009928858569336442912103262493625458976355624940065453228977352138859978594316152307777012044859965440000000000000000000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [167, 1, 166, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{166}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 166, 'autcent_group': '166.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 166, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{166}', '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], [167, 1, 166]], 'center_label': '167.1', 'center_order': 167, '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': ['167.1'], '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], [167, 1, 166, 1]], 'element_repr_type': 'PC', 'elementary': 167, 'eulerian_function': 1, 'exponent': 167, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 83], 'factors_of_order': [167], 'faithful_reps': [[1, 0, 166]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '167.1', 'hash': 1, 'hyperelementary': 167, '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': 166, 'irrQ_dim': 166, 'irrR_degree': 2, 'irrep_stats': [[1, 167]], 'label': '167.1', 'linC_count': 166, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 166, 'linQ_degree_count': 1, 'linQ_dim': 166, 'linQ_dim_count': 1, 'linR_count': 83, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C167', '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': 167, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 167, 'order_factorization_type': 1, 'order_stats': [[1, 1], [167, 166]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 166, 'outer_gen_orders': [166], 'outer_gen_pows': [0], 'outer_gens': [[67]], 'outer_group': '166.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 166, 'outer_permdeg': 85, 'outer_perms': [3353220009928858569336442912103262493625458976355624940065453228977352138859978594316152307777012044859965440000000000000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{166}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 167, 'pgroup': 167, 'primary_abelian_invariants': [167], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [166, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -167]}, 'Lie': [{'d': 1, 'q': 167, 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 167, 'gens': [4657631]}, 'Perm': {'d': 167, 'gens': [1494612823159220602834727446116490698483472390775123871527783668178402860636296923089193082650613568595129745165679275491241374525405329019506357391567179908243866583487687583237455390476159062872416219779074004057956107619164239470402730489650248203178564648960000000000000000000000000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [167], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{167}', 'transitive_degree': 167, 'wreath_data': None, 'wreath_product': False}
