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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '2592.ly', 'ambient_counter': 311, 'ambient_order': 2592, 'ambient_tex': 'C_2\\times S_3^4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 18, 'counter': 135, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2592.ly.36.u1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '36.u1', 'outer_equivalence': True, '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': 36, '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': '72.46', 'subgroup_hash': 46, 'subgroup_order': 72, 'subgroup_tex': 'S_3\\times D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2592.ly', 'aut_centralizer_order': 16, 'aut_label': '36.u1', 'aut_quo_index': None, 'aut_stab_index': 108, 'aut_weyl_group': '288.889', 'aut_weyl_index': 1728, 'centralizer': '324.c1', 'complements': None, 'conjugacy_class_count': 12, 'contained_in': ['12.r1', '18.e1', '18.i1'], 'contains': ['72.l1', '72.v1', '72.bt1', '108.p1'], 'core': '144.a1', 'coset_action_label': None, 'count': 108, 'diagramx': None, 'generators': [1, 6706022400, 6230649600, 10080, 5094], 'label': '2592.ly.36.u1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.e1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1', 'old_label': '36.u1', 'projective_image': '1296.3538', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36.u1', 'subgroup_fusion': None, 'weyl_group': '36.10'}
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gps_groups • Show schema
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{'Agroup': True, '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': [2, 2, 2, 2, 2, 3, 3], 'aut_gens': [[1, 2, 12], [37, 46, 12], [6, 25, 40], [1, 38, 12], [1, 10, 60], [37, 38, 12], [9, 2, 12], [1, 26, 12]], 'aut_group': '288.889', 'aut_hash': 889, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 10, 'aut_perms': [726, 374423, 1, 16687, 7, 126000, 782040], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 4, 1], [2, 9, 2, 1], [3, 2, 2, 1], [3, 4, 1, 1], [6, 2, 2, 1], [6, 4, 1, 1], [6, 6, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6\\wr C_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': 12, 'autcentquo_group': '72.40', 'autcentquo_hash': 40, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 9, 2], [3, 2, 2], [3, 4, 1], [6, 2, 2], [6, 4, 1], [6, 6, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '36.10', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 46, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 9, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 336, 'exponent': 6, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '72.46', 'hash': 46, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 3], 'inner_gens': [[1, 10, 12], [5, 2, 60], [1, 26, 12]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'S_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 8], [4, 2]], 'label': '72.46', 'linC_count': 13, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 13, 'linQ_dim': 4, 'linQ_dim_count': 13, 'linR_count': 13, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*D6', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 18, 'number_divisions': 18, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 69, 'number_subgroups': 206, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 31], [3, 8], [6, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 38, 12], [42, 49, 40]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [6, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 2]], 'representations': {'PC': {'code': 1043840168412997, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -3, -2, -3, 101, 26, 122, 608, 58, 609]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [11780359, 26483311, 7115160]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8156, 8101, 13286, 13933]}, 'Perm': {'d': 8, 'gens': [25, 720, 24, 5760, 3]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_6', 'transitive_degree': 12, '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 72, 'aut_gen_orders': [6, 4, 24], 'aut_gens': [[54, 1, 6706022400, 3628800, 3991680, 5040, 5760, 30, 6227020800], [479001601, 1, 3991680, 25, 48, 721, 5760, 6706022400, 7620480], [3628800, 1, 10080, 7, 48, 6227020800, 6706022400, 3991680, 5040], [3628801, 1, 10080, 12933043201, 12454041600, 7, 30, 3991680, 720]], 'aut_group': '497664.g', 'aut_hash': 9178118289268280967, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 497664, 'aut_permdeg': 24, 'aut_perms': [605755605297465525613544, 330087850879264702720568, 396223506198190265978899], 'aut_phi_ratio': 576.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 8, 1], [2, 9, 12, 1], [2, 27, 8, 1], [2, 81, 2, 1], [3, 2, 4, 1], [3, 4, 6, 1], [3, 8, 4, 1], [3, 16, 1, 1], [6, 2, 4, 1], [6, 4, 6, 1], [6, 6, 24, 1], [6, 8, 4, 1], [6, 12, 24, 1], [6, 16, 1, 1], [6, 18, 24, 1], [6, 24, 8, 1], [6, 36, 12, 1], [6, 54, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^4.C_2\\wr S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 72, 'autcentquo_group': '31104.l', 'autcentquo_hash': 3924562976423699589, 'autcentquo_nilpotent': False, 'autcentquo_order': 31104, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\wr S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 8], [2, 9, 12], [2, 27, 8], [2, 81, 2], [3, 2, 4], [3, 4, 6], [3, 8, 4], [3, 16, 1], [6, 2, 4], [6, 4, 6], [6, 6, 24], [6, 8, 4], [6, 12, 24], [6, 16, 1], [6, 18, 24], [6, 24, 8], [6, 36, 12], [6, 54, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '1296.3538', 'commutator_count': 1, 'commutator_label': '81.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 311, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['6.1', 4]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 8], [2, 9, 1, 12], [2, 27, 1, 8], [2, 81, 1, 2], [3, 2, 1, 4], [3, 4, 1, 6], [3, 8, 1, 4], [3, 16, 1, 1], [6, 2, 1, 4], [6, 4, 1, 6], [6, 6, 1, 24], [6, 8, 1, 4], [6, 12, 1, 24], [6, 16, 1, 1], [6, 18, 1, 24], [6, 24, 1, 8], [6, 36, 1, 12], [6, 54, 1, 8]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 66662400000, 'exponent': 6, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '2592.ly', 'hash': 4031814448579710185, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 1, 3, 2, 3, 2, 3, 3, 2], 'inner_gens': [[54, 1, 6706022400, 3628800, 3991680, 5040, 5760, 48, 6227020800], [54, 1, 6706022400, 3628800, 3991680, 5040, 5760, 30, 6227020800], [54, 1, 6706022400, 3628800, 3991680, 5040, 5760, 30, 479001600], [54, 1, 6706022400, 3628800, 7257600, 5040, 5760, 30, 6227020800], [54, 1, 6706022400, 362880, 3991680, 5040, 5760, 30, 6227020800], [54, 1, 6706022400, 3628800, 3991680, 5040, 10080, 30, 6227020800], [54, 1, 6706022400, 3628800, 3991680, 720, 5760, 30, 6227020800], [24, 1, 6706022400, 3628800, 3991680, 5040, 5760, 30, 6227020800], [54, 1, 12454041600, 3628800, 3991680, 5040, 5760, 30, 6227020800]], 'inner_hash': 3538, 'inner_nilpotent': False, 'inner_order': 1296, 'inner_split': True, 'inner_tex': 'S_3^4', 'inner_used': [1, 3, 4, 5, 6, 7, 8, 9], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 32], [2, 64], [4, 48], [8, 16], [16, 2]], 'label': '2592.ly', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*S3^4', '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 162, 'number_divisions': 162, 'number_normal_subgroups': 760, 'number_subgroup_autclasses': 388, 'number_subgroup_classes': 7941, 'number_subgroups': 103944, 'old_label': None, 'order': 2592, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 511], [3, 80], [6, 2000]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 6230649654, 3633840, 5094, 0, 0, 0], 'outer_gens': [[3628801, 1, 6706022400, 55, 30, 5040, 10080, 3991680, 6227020800], [54, 1, 6706022400, 3628800, 7257600, 5041, 10080, 48, 6227020800], [6227020800, 1, 7257600, 55, 30, 5040, 5760, 6706022400, 3628801], [6227020800, 1, 30, 5041, 5760, 3628801, 7257600, 6706022400, 54], [5040, 1, 7257600, 6227020801, 12454041600, 54, 30, 10080, 3628801], [54, 1, 6706022400, 3628801, 3991680, 5040, 10080, 48, 6227020801], [55, 1, 6706022400, 3628801, 3991680, 5040, 10080, 30, 6227020800], [55, 1, 12454041600, 3628801, 3991680, 5041, 5760, 30, 6227020801]], 'outer_group': '384.5602', 'outer_hash': 5602, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 8, 'outer_perms': [5455, 127, 576, 23616, 11536, 7, 126, 5167], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\wr S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 64], [4, 48], [8, 16], [16, 2]], 'representations': {'PC': {'code': '319823301233998242497509901343191028184213475813708885', 'gens': [1, 2, 4, 6, 8], 'pres': [9, -2, -2, -3, -2, -3, -2, -3, -2, -3, 181, 46, 218, 1092, 102, 1093, 1652, 158, 1545, 2212, 214, 1997]}, 'Perm': {'d': 14, 'gens': [54, 1, 6706022400, 3628800, 3991680, 5040, 5760, 30, 6227020800]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times S_3^4', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
