Formats: - HTML - YAML - JSON - 2026-07-23T09:25:18.556800
  • gps_subgroup_search • Show schema
    {'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '41472.jn', 'ambient_counter': 248, 'ambient_order': 41472, 'ambient_tex': '(C_3^3\\times C_6).Q_8^2:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 324, 'counter': 207, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '41472.jn.32.cn1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '32.cn1', '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': 32, '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': '1296.3496', 'subgroup_hash': 3496, 'subgroup_order': 1296, 'subgroup_tex': 'C_3^4:C_4^2', 'supersolvable': False, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '41472.jn', 'aut_centralizer_order': None, 'aut_label': '32.cn1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '20736.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['16.t1', '16.cg1', '16.cl1', '16.co1', '16.cq1'], 'contains': ['64.bl1', '64.bm1', '288.dw1'], 'core': '128.c1', 'coset_action_label': None, 'count': 8, 'diagramx': [3821, -1, 5575, -1], 'generators': [21902962535116347720, 980110453081456200, 736694677479767176, 769005177393312205920, 40399713433678516680, 15112469697020332560, 9940908664391742016, 586503093105727268400], 'label': '41472.jn.32.cn1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.b1', 'old_label': '32.cn1', 'projective_image': '41472.jn', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.cn1', 'subgroup_fusion': None, 'weyl_group': None}
  • gps_groups • Show schema
    {'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 48, 'aut_gen_orders': [4, 4, 8], 'aut_gens': [[40647600, 133, 1174320, 23, 56, 333, 7660800, 91854000], [160, 167292720, 314, 203661360, 88265520, 84596400, 311, 56], [40647600, 421, 1174320, 94, 676, 56, 203258160, 88265520], [44352000, 377, 90760320, 687, 333, 78, 214104240, 11249280]], 'aut_group': '41472.m', 'aut_hash': 7917341340725978817, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 41472, 'aut_permdeg': 36, 'aut_perms': [245025168864466323974459964898889443850621, 271798964290846642184411205186584896615008, 129051599181837849020556675003209996555637], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 2, 1], [2, 81, 1, 1], [3, 4, 4, 1], [3, 16, 4, 1], [4, 9, 4, 1], [4, 81, 4, 2], [6, 36, 4, 1], [12, 36, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '\\SOPlus(4,2)^2.D_4', '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': 48, 'autcentquo_group': '41472.m', 'autcentquo_hash': 7917341340725978817, 'autcentquo_nilpotent': False, 'autcentquo_order': 41472, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)^2.D_4', 'cc_stats': [[1, 1, 1], [2, 9, 2], [2, 81, 1], [3, 4, 4], [3, 16, 4], [4, 9, 4], [4, 81, 8], [6, 36, 4], [12, 36, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '1296.3496', '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', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3496, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['36.9', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 81, 1, 1], [3, 4, 1, 4], [3, 16, 1, 4], [4, 9, 2, 2], [4, 81, 2, 4], [6, 36, 1, 4], [12, 36, 2, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 12, 'exponent': 12, 'exponents_of_order': [4, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1296.3496', 'hash': 3496, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 4, 2, 2, 3, 3, 3, 3], 'inner_gens': [[40647600, 133, 1174320, 23, 56, 333, 84596400, 214104240], [40647600, 133, 1174320, 23, 285, 697, 7660800, 91854000], [40647600, 133, 1174320, 23, 56, 333, 11249280, 210551040], [40647600, 133, 1174320, 23, 78, 676, 7660800, 91854000], [40647600, 462, 1174320, 94, 56, 333, 7660800, 91854000], [40647600, 234, 1174320, 687, 56, 333, 7660800, 91854000], [134628480, 133, 12055680, 23, 56, 333, 7660800, 91854000], [55520640, 133, 211241520, 23, 56, 333, 7660800, 91854000]], 'inner_hash': 3496, 'inner_nilpotent': False, 'inner_order': 1296, 'inner_split': False, 'inner_tex': 'C_3^4:C_4^2', 'inner_used': [1, 2, 5, 7], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [4, 16], [16, 4]], 'label': '1296.3496', '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:C4^2', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 36, 'number_divisions': 26, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 65, 'number_subgroup_classes': 196, 'number_subgroups': 4956, 'old_label': None, 'order': 1296, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 99], [3, 80], [4, 684], [6, 144], [12, 288]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [1174320, 40360320, 0, 1174343], 'outer_gens': [[40647600, 130, 1174320, 23, 627, 333, 203258160, 88265520], [40647600, 130, 1174320, 23, 627, 333, 214104240, 11249280], [40360320, 130, 1174320, 23, 627, 333, 203258160, 91854000], [130, 40647600, 23, 1174320, 7660800, 91854000, 333, 78]], 'outer_group': '32.27', 'outer_hash': 27, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [107, 82, 21821, 34938], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\wr C_2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 8], [8, 4], [16, 4]], 'representations': {'PC': {'code': 332479413123530381969022486350160495931449958063248709927, 'gens': [1, 3, 4, 7, 8], 'pres': [8, 2, 2, 3, 2, 2, 3, 3, 3, 16, 1346, 106, 515, 971, 91, 1284, 2412, 116, 2501, 2317, 3390, 710, 3103, 2343]}, 'GLFq': {'d': 3, 'q': 9, 'gens': [129138492, 189999903, 156849001, 46807135, 86218832, 331826950, 344680685, 157244572]}, 'Perm': {'d': 12, 'gens': [40647600, 133, 1174320, 23, 56, 333, 7660800, 91854000]}}, 'schur_multiplier': [3, 12], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:C_4^2', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 48, 'aut_gen_orders': [8, 4, 8, 24, 16, 8], 'aut_gens': [[56469447937013182689, 10380770718769795696, 109758037512010855089], [518906869711399952058, 778833950725398522240, 899094490922231212458], [87608379953761123089, 346627845001394975536, 443345661633832312458], [373908126588359580858, 345761043754427078536, 393043322441358087258], [433879765984520810778, 10380770718769795680, 256009082635146119049], [511950361391708204169, 322162206541605728776, 292354169715654894018], [110644278545480577129, 329339267957661349080, 121693641332869193178]], 'aut_group': None, 'aut_hash': 8218745305919483472, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 331776, 'aut_permdeg': 432, 'aut_perms': 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'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 18, 1, 2], [2, 81, 1, 2], [2, 144, 2, 1], [3, 16, 1, 1], [3, 64, 1, 1], [4, 36, 1, 2], [4, 72, 2, 1], [4, 72, 4, 1], [4, 324, 1, 4], [4, 648, 2, 3], [4, 648, 4, 1], [4, 1296, 2, 1], [6, 16, 1, 1], [6, 64, 1, 1], [6, 144, 1, 2], [6, 576, 2, 2], [8, 324, 2, 2], [8, 432, 2, 2], [8, 648, 2, 1], [8, 1296, 4, 1], [8, 2592, 4, 1], [12, 288, 1, 2], [12, 576, 2, 1], [12, 576, 4, 1], [24, 864, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_4^2.C_2^5.C_2^3', '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': 48, 'autcentquo_group': '41472.m', 'autcentquo_hash': 7917341340725978817, 'autcentquo_nilpotent': False, 'autcentquo_order': 41472, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)^2.D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 18, 2], [2, 81, 2], [2, 144, 2], [3, 16, 1], [3, 64, 1], [4, 36, 2], [4, 72, 6], [4, 324, 4], [4, 648, 10], [4, 1296, 2], [6, 16, 1], [6, 64, 1], [6, 144, 2], [6, 576, 4], [8, 324, 4], [8, 432, 4], [8, 648, 2], [8, 1296, 4], [8, 2592, 4], [12, 288, 2], [12, 576, 6], [24, 864, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '20736.w', 'commutator_count': 1, 'commutator_label': '2592.lu', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 13, 'conjugacy_classes_known': True, 'counter': 248, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 18, 1, 2], [2, 81, 1, 2], [2, 144, 1, 2], [3, 16, 1, 1], [3, 64, 1, 1], [4, 36, 1, 2], [4, 72, 1, 2], [4, 72, 2, 2], [4, 324, 1, 4], [4, 648, 1, 6], [4, 648, 2, 2], [4, 1296, 1, 2], [6, 16, 1, 1], [6, 64, 1, 1], [6, 144, 1, 2], [6, 576, 1, 4], [8, 324, 2, 2], [8, 432, 2, 2], [8, 648, 1, 2], [8, 1296, 2, 2], [8, 2592, 2, 2], [12, 288, 1, 2], [12, 576, 1, 2], [12, 576, 2, 2], [24, 864, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 70533120, 'exponent': 24, 'exponents_of_order': [9, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 1, 4], [32, 0, 2], [32, 1, 1], [64, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '20736.w', 'hash': 1385621723797936428, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [8, 24, 8], 'inner_gens': [[56469447937013182689, 462799966214084496616, 883151761851962830809], [283932961574196101889, 10380770718769795696, 372279316019564178009], [70991887878533220609, 192500330229884563216, 109758037512010855089]], 'inner_hash': 4402493192939309901, 'inner_nilpotent': False, 'inner_order': 20736, 'inner_split': True, 'inner_tex': 'C_3^4:(Q_8^2:C_2^2)', 'inner_used': [1, 2, 3], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 12], [4, 20], [8, 2], [16, 8], [32, 6], [64, 8]], 'label': '41472.jn', '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': False, 'metacyclic': False, 'monomial': True, 'name': '(C3^3*C6).Q8^2:C2^2', 'ngens': 3, '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, 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': 41, 'number_characteristic_subgroups': 28, 'number_conjugacy_classes': 72, 'number_divisions': 56, 'number_normal_subgroups': 64, 'number_subgroup_autclasses': 1297, 'number_subgroup_classes': 2209, 'number_subgroups': 275842, 'old_label': None, 'order': 41472, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 487], [3, 80], [4, 10872], [6, 2672], [8, 19872], [12, 4032], [24, 3456]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[56469447937013182698, 10380770718769795696, 109758037512010855089], [56471121958992738618, 10425227723268029280, 129225877751687593089], [56469447937013182689, 10380770718769795680, 109758037512010855098], [56469447937013182698, 10380770718769795696, 109758037512010855098]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 10], [8, 8], [16, 8], [32, 2], [64, 6], [128, 2]], 'representations': {'PC': {'code': '2092140906442428077210800118079155148750748933497611103381874736368458294795370243754146795070882756463923994046841057310744977392151443658960850080869286216222569627976943730018794086131429053415332613429071205641327982101348622732944651319240748341157160490491370300072706593420483007009656420377848273527933652911720788385821', 'gens': [1, 2, 4, 7, 10, 11, 12], 'pres': [13, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 143312, 415741, 66, 41966, 150100, 1000483, 1064352, 409685, 146, 1132564, 653657, 17450, 186, 102341, 141042, 400015, 417878, 1536827, 650864, 195877, 185334, 11901, 266, 3184903, 1499284, 429345, 437678, 99067, 9640, 306, 269576, 359445, 808738, 209711, 205980, 1945, 1797129, 1830422, 149795, 49968, 31261, 6314, 11007, 1660, 4026890, 1995159, 27492, 13777, 27518, 20667, 13816, 5249, 4073483, 1916952, 828709, 414386, 69951, 168556, 466, 648972, 454297, 10854, 5459, 151488, 146093]}, 'Perm': {'d': 22, 'gens': [56469447937013182689, 10380770718769795696, 109758037512010855089]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_3^3\\times C_6).Q_8^2:C_2^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}