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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1980.70', 'ambient_counter': 70, 'ambient_order': 1980, 'ambient_tex': 'C_{33}:(S_3\\times C_{10})', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 330, 'counter': 5, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1980.70.3.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '3.a1.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': 3, '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': '660.18', 'subgroup_hash': 18, 'subgroup_order': 660, 'subgroup_tex': 'C_{66}:C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1980.70', 'aut_centralizer_order': 6, 'aut_label': '3.a1', 'aut_quo_index': None, 'aut_stab_index': 12, 'aut_weyl_group': '1320.142', 'aut_weyl_index': 72, 'centralizer': '990.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '6.b1.a1', '6.b1.e1', '9.a1.a1', '15.a1.a1', '33.a1.a1'], 'core': '6.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [3813, -1, 5456, -1, 3574, -1, 6176, -1], 'generators': [5, 2, 1320, 990, 180], 'label': '1980.70.3.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '3.a1.a1', 'projective_image': '990.16', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3.a1.a1', 'subgroup_fusion': None, 'weyl_group': '330.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 330, 'aut_gen_orders': [2, 10, 66], 'aut_gens': [[1, 10], [331, 230], [331, 610], [11, 10]], 'aut_group': '1320.142', 'aut_hash': 142, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1320, 'aut_permdeg': 17, 'aut_perms': [792835169070, 121264337885270, 27226629342999], 'aut_phi_ratio': 8.25, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [5, 11, 1, 4], [6, 2, 1, 1], [10, 11, 1, 4], [10, 33, 2, 4], [11, 5, 2, 1], [15, 22, 1, 4], [22, 5, 2, 1], [22, 15, 4, 1], [30, 22, 1, 4], [33, 10, 2, 1], [66, 10, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6\\times F_{11}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '660.15', 'autcentquo_hash': 15, 'autcentquo_nilpotent': False, 'autcentquo_order': 660, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 2, 1], [5, 11, 4], [6, 2, 1], [10, 11, 4], [10, 33, 8], [11, 5, 2], [15, 22, 4], [22, 5, 2], [22, 15, 4], [30, 22, 4], [33, 10, 2], [66, 10, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '330.2', 'commutator_count': 1, 'commutator_label': '33.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 18, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['55.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [5, 11, 4, 1], [6, 2, 1, 1], [10, 11, 4, 1], [10, 33, 4, 2], [11, 5, 2, 1], [15, 22, 4, 1], [22, 5, 2, 1], [22, 15, 2, 2], [30, 22, 4, 1], [33, 10, 2, 1], [66, 10, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 330, 'exponents_of_order': [2, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[10, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '660.18', 'hash': 18, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [10, 33], 'inner_gens': [[1, 590], [81, 10]], 'inner_hash': 2, 'inner_nilpotent': False, 'inner_order': 330, 'inner_split': True, 'inner_tex': 'C_{33}:C_{10}', 'inner_used': [1, 2], 'irrC_degree': 10, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 20, 'irrep_stats': [[1, 20], [2, 10], [5, 8], [10, 4]], 'label': '660.18', 'linC_count': 60, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 6, 'linQ_dim': 12, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C66:C10', 'ngens': 5, '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': 30, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 42, 'number_divisions': 18, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 40, 'number_subgroups': 224, 'old_label': None, 'order': 660, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 7], [3, 2], [5, 44], [6, 2], [10, 308], [11, 10], [15, 88], [22, 70], [30, 88], [33, 20], [66, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 110], 'outer_gens': [[331, 430], [441, 430]], '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': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 4], [8, 2], [10, 4], [20, 2]], 'representations': {'PC': {'code': 236404182472859997731104561303019, 'gens': [1, 3], 'pres': [5, -2, -5, -2, -3, -11, 10, 8852, 3682, 42, 10403, 3208, 78, 6004, 3759]}, 'GLZN': {'d': 2, 'p': 22, 'gens': [10693, 133343, 10891, 10653, 223629]}, 'Perm': {'d': 16, 'gens': [7, 1, 100724642640, 30, 1496093004000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{66}:C_{10}', 'transitive_degree': 66, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 1320, 'aut_gen_orders': [20, 10, 66, 6], 'aut_gens': [[1, 10, 30], [371, 680, 220], [781, 670, 510], [1151, 10, 690], [931, 20, 1310]], 'aut_group': None, 'aut_hash': 721779419800622324, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 95040, 'aut_permdeg': 29, 'aut_perms': [5212760139801825153349556333766, 1134048229849192730728562972194, 6696655330182053663976843842340, 5348096140114981417068958741122], 'aut_phi_ratio': 198.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [3, 2, 4, 1], [5, 11, 1, 4], [6, 2, 4, 1], [10, 11, 1, 4], [10, 99, 2, 4], [11, 5, 2, 1], [15, 22, 4, 4], [22, 5, 2, 1], [22, 45, 4, 1], [30, 22, 4, 4], [33, 10, 8, 1], [66, 10, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSU(3,2).C_{33}.C_5.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1320, 'autcentquo_group': '47520.b', 'autcentquo_hash': 4147157100757437592, 'autcentquo_nilpotent': False, 'autcentquo_order': 47520, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_{11}\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 2, 4], [5, 11, 4], [6, 2, 4], [10, 11, 4], [10, 99, 8], [11, 5, 2], [15, 22, 16], [22, 5, 2], [22, 45, 4], [30, 22, 16], [33, 10, 8], [66, 10, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '990.16', 'commutator_count': 1, 'commutator_label': '99.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '5.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 70, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['2.1', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 2, 1, 4], [5, 11, 4, 1], [6, 2, 1, 4], [10, 11, 4, 1], [10, 99, 4, 2], [11, 5, 2, 1], [15, 22, 4, 4], [22, 5, 2, 1], [22, 45, 2, 2], [30, 22, 4, 4], [33, 10, 2, 4], [66, 10, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 31248, 'exponent': 330, 'exponents_of_order': [2, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1980.70', 'hash': 70, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [10, 3, 33], 'inner_gens': [[1, 20, 1410], [21, 10, 30], [601, 10, 30]], 'inner_hash': 16, 'inner_nilpotent': False, 'inner_order': 990, 'inner_split': False, 'inner_tex': '(C_3\\times C_{33}):C_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 20], [2, 40], [5, 8], [10, 16]], 'label': '1980.70', 'linC_count': 4200, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 84, 'linQ_dim': 14, 'linQ_dim_count': 84, 'linR_count': 84, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C33:(S3*C10)', '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, 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': 30, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 84, 'number_divisions': 36, 'number_normal_subgroups': 45, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 120, 'number_subgroups': 1092, 'old_label': None, 'order': 1980, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 19], [3, 8], [5, 44], [6, 8], [10, 836], [11, 10], [15, 352], [22, 190], [30, 352], [33, 80], [66, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2], 'outer_gen_pows': [10, 0, 0, 5, 1335], 'outer_gens': [[11, 10, 1290], [1, 660, 1360], [1651, 660, 1970], [1, 1340, 50], [661, 660, 1370]], 'outer_group': '96.226', 'outer_hash': 226, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [7, 120, 1456, 11520, 5160], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 4], [8, 8], [10, 4], [20, 8]], 'representations': {'PC': {'code': 3926004301827688511632572892449881049031019, 'gens': [1, 3, 4], 'pres': [6, -2, -5, -3, -2, -3, -11, 12, 362, 33843, 11169, 69, 25204, 27910, 118, 19445, 29171]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [287521, 433456, 6613135, 287893, 288949, 12362371]}, 'Perm': {'d': 19, 'gens': [745, 24, 401719050391680, 5763, 4, 7178193517380480]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{33}:(S_3\\times C_{10})', 'transitive_degree': 198, 'wreath_data': None, 'wreath_product': False}
