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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '7776.cw', 'ambient_counter': 75, 'ambient_order': 7776, 'ambient_tex': 'C_3^4:(S_3\\times \\SD_{16})', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 1, 'counter': 310, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '7776.cw.324._.G', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '324.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': 324, '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': '24.10', 'subgroup_hash': 10, 'subgroup_order': 24, 'subgroup_tex': 'C_3\\times D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7776.cw', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1296._.N', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['36._.H', '162._.A', '162._.D', '162._.E'], 'contains': ['648._.O', '648._.S', '972._.E'], 'core': '7776.a1.a1', 'coset_action_label': None, 'count': 81, 'diagramx': [8084, -1, 7935, -1, 7708, -1, 7728, -1], 'generators': [49, 7052, 2016, 6248], 'label': '7776.cw.324._.G', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4._.B', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '81.a1.a1', 'old_label': '324.g1.a1', 'projective_image': '7776.cw', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '324._.G', 'subgroup_fusion': None, 'weyl_group': '16.11'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2], 'aut_gens': [[1, 2], [7, 14], [13, 22], [1, 10], [13, 2]], 'aut_group': '16.11', 'aut_hash': 11, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 6, 'aut_perms': [24, 126, 1, 414], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [3, 1, 2, 1], [4, 2, 1, 1], [6, 1, 2, 1], [6, 2, 4, 1], [12, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4', '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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [3, 1, 2], [4, 2, 1], [6, 1, 2], [6, 2, 4], [12, 2, 2]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [3, 1, 2, 1], [4, 2, 1, 1], [6, 1, 2, 1], [6, 2, 2, 2], [12, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '12.5', 'hash': 10, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 14], [13, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 12], [2, 3]], 'label': '24.10', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 5, 'linQ_dim': 4, 'linQ_dim_count': 5, 'linR_count': 5, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 15, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 20, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 5], [3, 2], [4, 2], [6, 10], [12, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 2], 'outer_gens': [[1, 10], [7, 2]], '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': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [4, 1]], 'representations': {'PC': {'code': 19095777, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -3, 113, 21, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16563488, 35813297]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 2064, 1717]}, 'Perm': {'d': 7, 'gens': [1680, 24, 4, 744]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times D_4', 'transitive_degree': 12, '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': 24, 'aut_gen_orders': [2, 8, 6, 3, 3, 3, 3], 'aut_gens': [[1, 2, 16, 96, 288, 864, 2592], [1097, 6374, 6288, 6080, 5760, 4320, 2592], [7277, 6754, 6512, 1952, 2304, 2592, 6048], [5377, 3778, 6928, 1824, 7488, 1728, 5184], [2689, 7106, 2608, 96, 2880, 864, 2592], [3745, 3746, 7216, 1824, 288, 864, 2592], [6049, 6914, 5200, 96, 288, 864, 2592], [4321, 1730, 4336, 96, 288, 864, 2592]], 'aut_group': '7776.cw', 'aut_hash': 8212187940285761847, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7776, 'aut_permdeg': 225, 'aut_perms': 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3302816221221346167156239570433023206807198695911642394148280340914675319075148046167865021574676711696534232922260963903526143177151461033136319250086932922362131365492203886266752913381077024331008938579790627142836879708503323697457810436676660229198424190801765952193471637272914891703511418952589999848488134513895951846689129715050975465644604538067691783267654912945124854336555689378797042407906406051827279503106181467985778], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 36, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [3, 8, 1, 1], [3, 18, 1, 1], [3, 24, 1, 1], [3, 48, 1, 1], [3, 144, 1, 1], [4, 162, 1, 1], [4, 324, 1, 1], [4, 486, 1, 1], [4, 972, 1, 1], [6, 72, 1, 1], [6, 162, 1, 1], [6, 216, 1, 6], [6, 432, 1, 2], [8, 162, 1, 2], [8, 486, 1, 2], [12, 324, 1, 1], [12, 648, 1, 1], [24, 324, 1, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4:(S_3\\times \\SD_{16})', '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': 24, 'autcentquo_group': '7776.cw', 'autcentquo_hash': 8212187940285761847, 'autcentquo_nilpotent': False, 'autcentquo_order': 7776, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4:(S_3\\times \\SD_{16})', 'cc_stats': [[1, 1, 1], [2, 27, 2], [2, 36, 1], [2, 81, 1], [2, 108, 1], [3, 8, 1], [3, 18, 1], [3, 24, 1], [3, 48, 1], [3, 144, 1], [4, 162, 1], [4, 324, 1], [4, 486, 1], [4, 972, 1], [6, 72, 1], [6, 162, 1], [6, 216, 6], [6, 432, 2], [8, 162, 2], [8, 486, 2], [12, 324, 1], [12, 648, 1], [24, 324, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '7776.cw', 'commutator_count': 1, 'commutator_label': '972.406', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 75, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 36, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [3, 8, 1, 1], [3, 18, 1, 1], [3, 24, 1, 1], [3, 48, 1, 1], [3, 144, 1, 1], [4, 162, 1, 1], [4, 324, 1, 1], [4, 486, 1, 1], [4, 972, 1, 1], [6, 72, 1, 1], [6, 162, 1, 1], [6, 216, 1, 6], [6, 432, 1, 2], [8, 162, 2, 1], [8, 486, 2, 1], [12, 324, 1, 1], [12, 648, 1, 1], [24, 324, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 17418240, 'exponent': 24, 'exponents_of_order': [5, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 4], [48, 1, 2]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '864.4662', 'hash': 8212187940285761847, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 8, 6, 3, 3, 3, 3], 'inner_gens': [[1, 3046, 16, 1056, 2304, 864, 6912], [6765, 2, 6800, 4544, 576, 2592, 6048], [1, 7426, 16, 1824, 4896, 1728, 5184], [1921, 6306, 880, 96, 2016, 864, 2592], [1441, 578, 6640, 960, 288, 864, 2592], [1, 6050, 1744, 96, 288, 864, 2592], [6049, 6914, 5200, 96, 288, 864, 2592]], 'inner_hash': 8212187940285761847, 'inner_nilpotent': False, 'inner_order': 7776, 'inner_split': True, 'inner_tex': 'C_3^4:(S_3\\times \\SD_{16})', 'inner_used': [1, 2, 3], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 10], [4, 3], [8, 4], [16, 2], [24, 4], [48, 2]], 'label': '7776.cw', 'linC_count': 4, 'linC_degree': 24, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 4, 'linQ_dim': 24, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 24, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': None, 'name': 'C3^4:(S3*SD16)', 'ngens': 10, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 33, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 33, 'number_divisions': 30, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 473, 'number_subgroup_classes': 473, 'number_subgroups': 31930, 'old_label': None, 'order': 7776, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 279], [3, 242], [4, 1944], [6, 2394], [8, 1296], [12, 972], [24, 648]], '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': 7, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 3], [8, 5], [16, 2], [24, 4], [48, 2]], 'representations': {'PC': {'code': '58455272015971747385343453536179987028011274495789361867144663252820901170989768250196190849712281178343865739492018193190161126809688095', 'gens': [1, 2, 5, 7, 8, 9, 10], 'pres': [10, -2, -2, -2, -2, -2, -3, 3, -3, -3, 3, 60921, 51, 59882, 82, 38723, 170014, 77424, 5934, 144, 85455, 28825, 515, 73926, 159056, 76186, 1716, 8026, 184327, 23057, 24527, 11577, 1747, 116658, 136108, 19478, 9768, 691209, 302419, 172829, 64839, 32449]}, 'Perm': {'d': 27, 'gens': [9167122780557201012549133103, 6632969369720355530109885969, 3257383389079646387271672220]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:(S_3\\times \\SD_{16})', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
