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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '17496.nt', 'ambient_counter': 358, 'ambient_order': 17496, 'ambient_tex': 'C_3^4.S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 3, 'counter': 1209, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '17496.nt.972.ck1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '972.ck1', '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': 972, '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': '18.3', 'subgroup_hash': 3, 'subgroup_order': 18, 'subgroup_tex': 'C_3\\times S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '17496.nt', 'aut_centralizer_order': None, 'aut_label': '972.ck1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1458.b1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['324.eg1', '324.el1', '324.em1', '324.en1', '324.eo1', '324.ep1', '324.eq1', '324.er1', '324.es1', '324.et1', '324.eu1', '486.g1', '486.bc1'], 'contains': ['1944.m1', '2916.u1', '2916.w1'], 'core': '5832.b1', 'coset_action_label': None, 'count': 486, 'diagramx': None, 'generators': [9, 1298, 216], 'label': '17496.nt.972.ck1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.d1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '243.a1', 'old_label': '972.ck1', 'projective_image': '17496.nt', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '972.ck1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 6], [5, 12], [17, 6]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', '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': 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, 1, 2], [3, 2, 3], [6, 3, 2]], 'center_label': '3.1', 'center_order': 3, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '18.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 12], [13, 6]], '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': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 6], [2, 3]], 'label': '18.3', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*S3', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 9, 'number_subgroups': 14, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 8], [6, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5, 6]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 5451, 'gens': [1, 3], 'pres': [3, -2, -3, -3, 6, 110]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [11780110, 20974441]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'GU'}, {'d': 2, 'q': 4, 'gens': [20, 130, 194], 'family': 'COPlus'}, {'d': 2, 'q': 2, 'gens': [20, 130, 138], 'family': 'CU'}, {'d': 1, 'q': 9, 'gens': [93882, 62619, 1930], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 687, 1374]}, 'Perm': {'d': 6, 'gens': [450, 147, 243]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times S_3', 'transitive_degree': 6, 'wreath_data': ['C_3', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [6, 12, 18, 6, 6], 'aut_gens': [[1, 6, 36, 648, 1944], [10999, 9420, 4482, 648, 72], [6562, 1299, 8541, 12, 144], [10855, 13944, 10602, 648, 72], [7777, 726, 10548, 648, 9720], [1474, 8451, 8769, 24, 504]], 'aut_group': None, 'aut_hash': 417177536885461405, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 419904, 'aut_permdeg': 117, 'aut_perms': [3768413365915752429657372614894383413818530148804570041223936046020205219658371744571290784131830533048051977567528537474653221972718237754043237720837748134931224306779392651256279090612400883, 139496364879407845035247986555724855103722764111407865983260682575088498641005230925148789445701636682041337446160429486724948114024002962395428692541393445545168530079421047253004537799883015, 3362855752684725456318129790762934306693952121471628426850568031900352605330343080086476146315657000777573220352869884369227825465224717766923185747339134295475279202592989922045992941363051331, 2560043619799424431134653725176522418800116381962811476114426696642122786890225548892045615721746705764280498195775320234794103315045922301805228782713312385484985602885916580619149240127379044, 3455174338008752953093409690563629024317989227409547736911307430148680526220782710141298904539427969586864451387079518683860937739505563149378029276879307076586328820201937126970813864860255694], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 4, 1], [2, 81, 1, 1], [2, 729, 1, 1], [3, 2, 2, 2], [3, 4, 1, 2], [3, 4, 4, 1], [3, 8, 2, 3], [3, 9, 1, 2], [3, 18, 2, 2], [3, 36, 1, 2], [6, 18, 2, 1], [6, 36, 1, 1], [6, 54, 4, 2], [6, 81, 1, 4], [6, 81, 4, 2], [6, 108, 4, 1], [6, 162, 2, 3], [6, 162, 4, 2], [6, 324, 1, 3], [6, 729, 1, 2], [9, 6, 2, 1], [9, 12, 2, 1], [9, 12, 3, 1], [9, 12, 4, 1], [9, 18, 2, 2], [9, 24, 2, 1], [9, 24, 4, 2], [9, 24, 6, 2], [9, 36, 1, 2], [9, 36, 4, 2], [9, 72, 2, 6], [18, 54, 2, 1], [18, 108, 2, 1], [18, 108, 3, 1], [18, 162, 2, 2], [18, 162, 4, 3], [18, 324, 1, 2], [18, 324, 4, 3]], 'aut_supersolvable': False, 'aut_tex': 'C_9^2.C_3^3.(C_3\\times D_4^2)', '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': 36, 'autcentquo_group': None, 'autcentquo_hash': 417177536885461405, 'autcentquo_nilpotent': False, 'autcentquo_order': 419904, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_9^2.C_3^3.(C_3\\times D_4^2)', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 27, 4], [2, 81, 1], [2, 729, 1], [3, 2, 4], [3, 4, 6], [3, 8, 6], [3, 9, 2], [3, 18, 4], [3, 36, 2], [6, 18, 2], [6, 36, 1], [6, 54, 8], [6, 81, 12], [6, 108, 4], [6, 162, 14], [6, 324, 3], [6, 729, 2], [9, 6, 2], [9, 12, 9], [9, 18, 4], [9, 24, 22], [9, 36, 10], [9, 72, 12], [18, 54, 2], [18, 108, 5], [18, 162, 16], [18, 324, 14]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '17496.nt', 'commutator_count': 1, 'commutator_label': '729.415', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 358, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 4], [2, 81, 1, 1], [2, 729, 1, 1], [3, 2, 1, 4], [3, 4, 1, 6], [3, 8, 1, 6], [3, 9, 2, 1], [3, 18, 2, 2], [3, 36, 2, 1], [6, 18, 1, 2], [6, 36, 1, 1], [6, 54, 1, 8], [6, 81, 2, 6], [6, 108, 1, 4], [6, 162, 1, 2], [6, 162, 2, 6], [6, 324, 1, 1], [6, 324, 2, 1], [6, 729, 2, 1], [9, 6, 1, 2], [9, 12, 1, 9], [9, 18, 2, 2], [9, 24, 1, 22], [9, 36, 2, 5], [9, 72, 2, 6], [18, 54, 1, 2], [18, 108, 1, 5], [18, 162, 1, 4], [18, 162, 2, 6], [18, 324, 1, 4], [18, 324, 2, 5]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2515968, 'exponent': 18, 'exponents_of_order': [7, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 6]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '1944.3935', 'hash': 6986637867447006660, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 6, 18, 3, 9], 'inner_gens': [[1, 1110, 2844, 648, 3888], [1525, 6, 1908, 1296, 1944], [17281, 726, 36, 1296, 15552], [1, 1302, 1332, 648, 1944], [15553, 6, 3924, 648, 1944]], 'inner_hash': 6986637867447006660, 'inner_nilpotent': False, 'inner_order': 17496, 'inner_split': False, 'inner_tex': 'C_3^4.S_3^3', 'inner_used': [1, 2, 3], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 24], [2, 48], [4, 36], [6, 8], [8, 18], [12, 18], [24, 22]], 'label': '17496.nt', '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': 'C3^4.S3^3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 74, 'number_characteristic_subgroups': 25, 'number_conjugacy_classes': 174, 'number_divisions': 132, 'number_normal_subgroups': 141, 'number_subgroup_autclasses': 1359, 'number_subgroup_classes': 4290, 'number_subgroups': 187508, 'old_label': None, 'order': 17496, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 927], [3, 242], [6, 6606], [9, 1944], [18, 7776]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 15879, 15879], 'outer_gens': [[1, 30, 36, 648, 1944], [1018, 2607, 14601, 12, 288], [9424, 17205, 2307, 12, 3888]], 'outer_group': '24.14', 'outer_hash': 14, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [7, 136, 1456], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 24], [4, 28], [6, 8], [8, 18], [12, 18], [16, 6], [24, 22]], 'representations': {'PC': {'code': '2332158165460651527911472727197467802570017207282052599261794904398771034370124727206367036052407387533137247', 'gens': [1, 3, 5, 8, 9], 'pres': [10, -2, -3, -2, -3, -2, -3, -3, -3, -3, -3, 20, 33302, 22782, 82, 963, 142204, 189914, 15924, 144, 30245, 8655, 5785, 235, 5066, 17307, 2927, 349928, 349938, 38928, 358, 1166409, 32449]}, 'Perm': {'d': 24, 'gens': [80023424852794295134162, 55088728054715155490169, 26015753540735344915371]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4.S_3^3', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
