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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1728.46343', 'ambient_counter': 46343, 'ambient_order': 1728, 'ambient_tex': 'C_2^3.S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 1, 'counter': 1036, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.46343.288.be1.a2', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '288.be1.a2', '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': 288, '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': '1728.46343', 'aut_centralizer_order': 24, 'aut_label': '288.be1', 'aut_quo_index': None, 'aut_stab_index': 48, 'aut_weyl_group': '6.1', 'aut_weyl_index': 1152, 'centralizer': '144.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['72.fd1.a2', '96.y1.a2', '96.z1.a1', '144.y1.a1', '144.by1.a2', '144.by1.b2'], 'contains': ['576.e1.a1', '864.h1.a1'], 'core': '1728.a1.a1', 'coset_action_label': None, 'count': 24, 'diagramx': [9766, -1, 456, -1, 8780, -1, 687, -1], 'generators': [4493461, 4260640], 'label': '1728.46343.288.be1.a2', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.f1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '24.o1.a1', 'old_label': '288.be1.a2', 'projective_image': '1728.46343', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '288.be1.a2', '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': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 6, 2, 6, 2, 12, 12], 'aut_gens': [[31669655, 17190769, 17760224, 4403503, 890233], [10415927, 17190769, 31818860, 4403503, 894937], [31523243, 24300877, 17467400, 32644087, 25487491], [10415927, 17190769, 49518806, 32943967, 894937], [23014478, 24303229, 42657362, 29593381, 25787371], [19374596, 17193121, 11882081, 29593381, 25492195], [31523243, 17188417, 22799207, 4703383, 894937], [19374596, 17193121, 19372013, 29593381, 25492195]], 'aut_group': None, 'aut_hash': 5675430861318798311, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6912, 'aut_permdeg': 24, 'aut_perms': [558200764841319136894680, 187808383078767926841296, 506343450669998798051520, 15146362027394791567997, 333987105201051521714918, 188780635576487580210359, 566550645939695999673982], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 9, 2, 1], [2, 12, 1, 1], [2, 36, 1, 2], [2, 108, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 8, 1, 1], [3, 16, 1, 2], [3, 32, 1, 1], [4, 6, 1, 1], [4, 12, 1, 1], [4, 18, 1, 2], [4, 36, 1, 2], [4, 54, 1, 1], [4, 108, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 1, 4], [6, 6, 2, 1], [6, 8, 1, 1], [6, 12, 1, 2], [6, 12, 2, 1], [6, 16, 1, 2], [6, 18, 2, 1], [6, 24, 1, 1], [6, 24, 2, 2], [6, 32, 1, 1], [6, 36, 2, 1], [6, 48, 2, 1], [6, 72, 1, 1], [12, 12, 1, 1], [12, 12, 2, 1], [12, 24, 1, 1], [12, 24, 2, 2], [12, 36, 1, 1], [12, 36, 2, 1], [12, 48, 2, 1], [12, 72, 1, 1], [12, 72, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_2^2.D_6^2\\times S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '864.4673', 'autcentquo_hash': 4673, 'autcentquo_nilpotent': False, 'autcentquo_order': 864, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 9, 2], [2, 12, 1], [2, 36, 2], [2, 108, 1], [3, 2, 2], [3, 4, 1], [3, 8, 1], [3, 16, 2], [3, 32, 1], [4, 6, 1], [4, 12, 1], [4, 18, 2], [4, 36, 2], [4, 54, 1], [4, 108, 1], [6, 2, 2], [6, 4, 1], [6, 6, 6], [6, 8, 1], [6, 12, 4], [6, 16, 2], [6, 18, 2], [6, 24, 5], [6, 32, 1], [6, 36, 2], [6, 48, 2], [6, 72, 1], [12, 12, 3], [12, 24, 5], [12, 36, 3], [12, 48, 2], [12, 72, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '864.4673', 'commutator_count': 1, 'commutator_label': '216.173', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 46343, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['288.855', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 9, 1, 2], [2, 12, 1, 1], [2, 36, 1, 2], [2, 108, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 8, 1, 1], [3, 16, 1, 2], [3, 32, 1, 1], [4, 6, 1, 1], [4, 12, 1, 1], [4, 18, 1, 2], [4, 36, 1, 2], [4, 54, 1, 1], [4, 108, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 1, 6], [6, 8, 1, 1], [6, 12, 1, 2], [6, 12, 2, 1], [6, 16, 1, 2], [6, 18, 1, 2], [6, 24, 1, 3], [6, 24, 2, 1], [6, 32, 1, 1], [6, 36, 2, 1], [6, 48, 1, 2], [6, 72, 1, 1], [12, 12, 1, 1], [12, 12, 2, 1], [12, 24, 1, 1], [12, 24, 2, 2], [12, 36, 1, 1], [12, 36, 2, 1], [12, 48, 2, 1], [12, 72, 1, 1], [12, 72, 2, 1]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 161280, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4673', 'hash': 46343, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 6, 6, 6], 'inner_gens': [[31669655, 17190769, 11882081, 4403503, 25787371], [31669655, 17190769, 17760224, 4403503, 894937], [47908004, 17190769, 17760224, 7754089, 25787371], [31669655, 17190769, 35460254, 4403503, 890233], [6776045, 17193121, 17467400, 4403503, 890233]], 'inner_hash': 4673, 'inner_nilpotent': False, 'inner_order': 864, 'inner_split': True, 'inner_tex': 'S_4\\times S_3^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 22], [3, 8], [4, 13], [6, 14], [8, 2], [12, 5]], 'label': '1728.46343', 'linC_count': 144, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 448, 'linQ_dim': 9, 'linQ_dim_count': 448, 'linR_count': 1088, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.S3^3', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 57, 'number_characteristic_subgroups': 60, 'number_conjugacy_classes': 72, 'number_divisions': 63, 'number_normal_subgroups': 72, 'number_subgroup_autclasses': 952, 'number_subgroup_classes': 1104, 'number_subgroups': 15012, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 223], [3, 80], [4, 288], [6, 560], [12, 576]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [592705, 592705, 592705], 'outer_gens': [[31669655, 24303229, 17760224, 4403503, 890233], [31669655, 17190769, 17760224, 32943967, 890233], [31669655, 17190769, 5020637, 4403503, 25787371]], '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': 5, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [3, 8], [4, 13], [6, 10], [8, 4], [12, 5], [24, 1]], 'representations': {'PC': {'code': 1367080563112233573322771577047607953141623689965693085336336991082517, 'gens': [1, 2, 3, 6, 8], 'pres': [9, 2, 2, 2, 2, 3, 2, 3, 2, 3, 1190, 74, 1443, 102, 1444, 12983, 6188, 3605, 158, 3048, 31111, 51856, 7801, 11698, 1987, 214, 46673]}, 'GLZN': {'d': 2, 'p': 84, 'gens': [21298627, 7705165, 35310251, 1042546, 892585, 595057, 25782667, 29300557, 598555]}, 'Perm': {'d': 14, 'gens': [368063, 811576, 997925040, 367920, 518918400, 325, 435, 6266937600, 13412044800]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3.S_3^3', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
