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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '419904.gm', 'ambient_counter': 169, 'ambient_order': 419904, 'ambient_tex': 'C_3^4.D_6^2:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 69984, 'counter': 37, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '419904.gm.6.A', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '6.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '6.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 6, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'S_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '69984.fe', 'subgroup_hash': 6506385555687228630, 'subgroup_order': 69984, 'subgroup_tex': 'C_3^4.D_6^2:S_3', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '419904.gm', 'aut_centralizer_order': None, 'aut_label': '6.A', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 2, 'contained_in': None, 'contains': None, 'core': '6.A', 'coset_action_label': None, 'count': 2, 'diagramx': [719, 509, 880, 3054], 'generators': [74552088921494400, 497125975455214583, 1482411000, 273868160173941327240, 4103670016102320, 300820665600, 22056933293259420360, 879340827866723043253, 16, 1482411023, 321217635742324272840, 2512139796684120], 'label': '419904.gm.6.A', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '6.A', 'normal_contained_in': [], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '6.a1', 'projective_image': '209952.eg', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.A', 'subgroup_fusion': None, 'weyl_group': '209952.eg'}
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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': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [12, 12, 12, 12, 18, 36], 'aut_gens': [[56348826589715068207, 56470409094976537680, 109495301021054299562], [671969488780430316840, 77001517697232845536, 272652449890101902053], [56347800207063418103, 56350896638040345736, 338363567591272766650], [484846220222188812007, 671971282500091474103, 786827305393236540861], [818602649703866368223, 818624734737966389527, 772748025127233524650], [193366407034316124007, 857920450806341411063, 628980388521126129381], [75940487817749750903, 467605657438960048687, 261714356211970413370]], 'aut_group': None, 'aut_hash': 2015464893847043307, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5038848, 'aut_permdeg': 540, 'aut_perms': 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'aut_phi_ratio': 216.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 54, 4, 1], [2, 324, 1, 1], [2, 729, 2, 1], [2, 1458, 1, 1], [3, 4, 1, 2], [3, 12, 1, 1], [3, 12, 3, 2], [3, 12, 9, 1], [3, 24, 1, 1], [3, 24, 3, 1], [3, 24, 9, 2], [3, 162, 1, 1], [4, 324, 1, 1], [4, 8748, 2, 1], [6, 4, 1, 2], [6, 4, 2, 1], [6, 8, 1, 1], [6, 12, 1, 1], [6, 12, 2, 1], [6, 12, 3, 2], [6, 12, 6, 1], [6, 12, 9, 1], [6, 24, 1, 1], [6, 24, 2, 1], [6, 24, 3, 2], [6, 24, 6, 1], [6, 24, 9, 3], [6, 24, 18, 2], [6, 108, 4, 1], [6, 162, 1, 1], [6, 162, 2, 1], [6, 324, 4, 1], [6, 324, 12, 1], [6, 486, 8, 1], [6, 648, 1, 1], [6, 648, 3, 1], [6, 648, 9, 1], [6, 1458, 2, 2], [9, 324, 2, 1], [9, 648, 1, 1], [12, 648, 1, 1], [12, 648, 3, 1], [12, 648, 9, 1], [18, 324, 2, 1], [18, 324, 4, 1], [18, 648, 1, 1], [18, 648, 2, 1], [18, 972, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^6.C_6^3.C_2^5', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 1960495289627526838, 'autcentquo_nilpotent': False, 'autcentquo_order': 629856, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^6.C_3.C_6^2.C_2^3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 54, 4], [2, 324, 1], [2, 729, 2], [2, 1458, 1], [3, 4, 2], [3, 12, 16], [3, 24, 22], [3, 162, 1], [4, 324, 1], [4, 8748, 2], [6, 4, 4], [6, 8, 1], [6, 12, 24], [6, 24, 78], [6, 108, 4], [6, 162, 3], [6, 324, 16], [6, 486, 8], [6, 648, 13], [6, 1458, 4], [9, 324, 2], [9, 648, 1], [12, 648, 13], [18, 324, 6], [18, 648, 3], [18, 972, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '34992.fm', 'commutator_count': 1, 'commutator_label': None, 'complements_known': False, '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', '3.1', '3.1', '3.1'], 'composition_length': 12, 'conjugacy_classes_known': True, 'counter': 135, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 54, 1, 4], [2, 324, 1, 1], [2, 729, 1, 2], [2, 1458, 1, 1], [3, 4, 1, 2], [3, 12, 1, 16], [3, 24, 1, 22], [3, 162, 1, 1], [4, 324, 1, 1], [4, 8748, 1, 2], [6, 4, 1, 4], [6, 8, 1, 1], [6, 12, 1, 24], [6, 24, 1, 78], [6, 108, 1, 4], [6, 162, 1, 1], [6, 162, 2, 1], [6, 324, 1, 16], [6, 486, 2, 4], [6, 648, 1, 13], [6, 1458, 1, 2], [6, 1458, 2, 1], [9, 324, 2, 1], [9, 648, 1, 1], [12, 648, 1, 13], [18, 324, 2, 3], [18, 648, 1, 1], [18, 648, 2, 1], [18, 972, 2, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 36, 'exponents_of_order': [7, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 12], [24, 1, 51]], 'familial': False, 'frattini_label': '162.55', 'frattini_quotient': '432.755', 'hash': 6506385555687228630, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [18, 18, 4], 'inner_gens': [[56348826589715068207, 672124946530191130800, 626521739710383095901], [467636638859946224767, 56470409094976537680, 525914934293212103642], [601485438048787288936, 153581046180931369440, 109495301021054299562]], 'inner_hash': 6895329934315343326, 'inner_nilpotent': False, 'inner_order': 34992, 'inner_split': True, 'inner_tex': 'C_3^6.C_6:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 22], [4, 28], [8, 5], [12, 80], [24, 100]], 'label': '69984.fe', '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': None, 'name': 'C3^4.D6^2:S3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 61, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 243, 'number_divisions': 228, 'number_normal_subgroups': 53, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 69984, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 3459], [3, 890], [4, 17820], [6, 26430], [9, 1296], [12, 8424], [18, 11664]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 6, 12], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[818797452882877980127, 818797669292963001720, 725032691499755093162], [498015588400808734927, 496920766098529795823, 129023294142635400242], [602216019994802790360, 193304862932321147056, 124840368088737518890]], 'outer_group': '144.162', 'outer_hash': 162, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 10, 'outer_perms': [1992360, 458647, 453733], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:D_6', 'pc_rank': None, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 18], [8, 11], [12, 80], [16, 1], [24, 100]], 'representations': {'PC': {'code': '9372022037964788007992841084580977707540283743512582399914230666635285847955852396762758984721732870679238824940079432179634343666380802390256771895386664948845561979477731183279428490982750468205813473984381745494403756719', 'gens': [1, 2, 4, 6, 8, 10, 11, 12], 'pres': [12, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 176328, 1435057, 61, 2371106, 1743843, 1538223, 705771, 135, 164164, 1607776, 925228, 523589, 852785, 937469, 254489, 209, 314502, 1971666, 545358, 430458, 481543, 2891539, 595615, 275371, 283, 749096, 2303012, 1268168, 615644, 852489, 2911701, 994353, 398925, 4561930, 3025462, 575026, 760366, 8377355, 385367, 1872323, 551279]}, 'Perm': {'d': 22, 'gens': [56348826589715068207, 56470409094976537680, 109495301021054299562]}}, 'schur_multiplier': [2, 2, 2, 6], 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 54, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4.D_6^2:S_3', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [6, 4, 12, 6, 6, 12, 12], 'aut_gens': [[5393457543661705567, 59018593227322993333, 107580278241953401703, 163749987579054967221], [27424444062770905696, 188213897708719516930, 590364892492071753023, 515951904674640687141], [821360693276132370616, 461095454043691165450, 843302926060969621680, 445602041598254567061], [579558990260979843856, 545906116061861311581, 694007803813265596807, 163761725253178524261], [333558146813363499007, 433488142617382189573, 17156251658634647183, 54552038512223427842], [141108863704931364616, 59028198147811751882, 222496789026236986927, 643460645721110168301], [208395139727459755456, 619016027253672872890, 703739329114287968640, 725397023075214834842], [326298274000758023767, 475709583097670359101, 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'aut_phi_ratio': 144.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 54, 4, 1], [2, 81, 2, 1], [2, 162, 1, 1], [2, 324, 2, 1], [2, 486, 4, 1], [2, 729, 2, 2], [2, 1458, 1, 2], [3, 4, 1, 2], [3, 12, 1, 1], [3, 12, 3, 1], [3, 24, 1, 1], [3, 24, 3, 1], [3, 36, 1, 2], [3, 36, 3, 1], [3, 72, 1, 1], [3, 72, 3, 2], [3, 162, 2, 1], [3, 216, 1, 1], [3, 216, 3, 1], [4, 324, 2, 1], [4, 8748, 4, 1], [6, 4, 1, 2], [6, 4, 2, 1], [6, 8, 1, 1], [6, 12, 1, 1], [6, 12, 2, 1], [6, 12, 3, 1], [6, 12, 6, 1], [6, 24, 1, 1], [6, 24, 2, 1], [6, 24, 3, 1], [6, 24, 6, 1], [6, 36, 1, 2], [6, 36, 2, 1], [6, 36, 3, 1], [6, 72, 1, 2], [6, 72, 2, 1], [6, 72, 3, 3], [6, 72, 6, 2], [6, 108, 4, 2], [6, 162, 2, 1], [6, 216, 1, 1], [6, 216, 2, 1], [6, 216, 3, 1], [6, 216, 4, 1], [6, 216, 6, 1], [6, 324, 2, 4], [6, 324, 4, 1], [6, 324, 6, 3], [6, 324, 12, 1], [6, 648, 1, 1], [6, 648, 2, 1], [6, 648, 3, 1], [6, 648, 4, 1], [6, 648, 6, 4], [6, 648, 12, 1], [6, 972, 4, 3], [6, 972, 8, 1], [6, 972, 12, 1], [6, 1458, 4, 1], [6, 1944, 2, 1], [6, 1944, 4, 1], [6, 1944, 6, 1], [6, 1944, 12, 1], [6, 2916, 2, 6], [6, 5832, 1, 1], [6, 5832, 2, 1], [9, 72, 1, 1], [9, 144, 1, 1], [9, 432, 1, 1], [9, 432, 3, 1], [9, 648, 2, 2], [12, 648, 2, 1], [12, 1944, 2, 1], [12, 1944, 6, 1], [12, 5832, 2, 1], [12, 17496, 4, 1], [18, 72, 1, 1], [18, 72, 2, 1], [18, 144, 1, 1], [18, 144, 2, 1], [18, 432, 1, 1], [18, 432, 2, 1], [18, 432, 3, 1], [18, 432, 6, 1], [18, 648, 2, 2], [18, 648, 4, 1], [18, 1296, 2, 1], [18, 1944, 4, 1], [18, 1944, 8, 1], [18, 11664, 2, 1], [36, 11664, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^6.C_3^2.C_2^4.C_6.C_2^5', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 7482597521743094414, 'autcentquo_nilpotent': False, 'autcentquo_order': 1259712, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_3^4.C_2.C_6.C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 54, 4], [2, 81, 2], [2, 162, 1], [2, 324, 2], [2, 486, 4], [2, 729, 4], [2, 1458, 2], [3, 4, 2], [3, 12, 4], [3, 24, 4], [3, 36, 5], [3, 72, 7], [3, 162, 2], [3, 216, 4], [4, 324, 2], [4, 8748, 4], [6, 4, 4], [6, 8, 1], [6, 12, 12], [6, 24, 12], [6, 36, 7], [6, 72, 25], [6, 108, 8], [6, 162, 2], [6, 216, 16], [6, 324, 42], [6, 648, 46], [6, 972, 32], [6, 1458, 4], [6, 1944, 24], [6, 2916, 12], [6, 5832, 3], [9, 72, 1], [9, 144, 1], [9, 432, 4], [9, 648, 4], [12, 648, 2], [12, 1944, 8], [12, 5832, 2], [12, 17496, 4], [18, 72, 3], [18, 144, 3], [18, 432, 12], [18, 648, 8], [18, 1296, 2], [18, 1944, 12], [18, 11664, 2], [36, 11664, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '209952.eg', 'commutator_count': 1, 'commutator_label': None, 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 14, 'conjugacy_classes_known': True, 'counter': 169, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 54, 1, 4], [2, 81, 1, 2], [2, 162, 1, 1], [2, 324, 1, 2], [2, 486, 1, 4], [2, 729, 1, 4], [2, 1458, 1, 2], [3, 4, 1, 2], [3, 12, 1, 4], [3, 24, 1, 4], [3, 36, 1, 5], [3, 72, 1, 7], [3, 162, 1, 2], [3, 216, 1, 4], [4, 324, 1, 2], [4, 8748, 1, 4], [6, 4, 1, 4], [6, 8, 1, 1], [6, 12, 1, 12], [6, 24, 1, 12], [6, 36, 1, 7], [6, 72, 1, 25], [6, 108, 1, 8], [6, 162, 1, 2], [6, 216, 1, 16], [6, 324, 1, 42], [6, 648, 1, 46], [6, 972, 1, 32], [6, 1458, 1, 4], [6, 1944, 1, 24], [6, 2916, 1, 12], [6, 5832, 1, 3], [9, 72, 1, 1], [9, 144, 1, 1], [9, 432, 1, 4], [9, 648, 1, 4], [12, 648, 1, 2], [12, 1944, 1, 8], [12, 5832, 1, 2], [12, 17496, 1, 4], [18, 72, 1, 3], [18, 144, 1, 3], [18, 432, 1, 12], [18, 648, 1, 8], [18, 1296, 1, 2], [18, 1944, 1, 12], [18, 11664, 1, 2], [36, 11664, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 36, 'exponents_of_order': [8, 6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 24], [24, 1, 24], [48, 1, 6], [72, 1, 30]], 'familial': False, 'frattini_label': '162.55', 'frattini_quotient': '2592.fh', 'hash': 5861061776370282502, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 12, 6, 6], 'inner_gens': [[5393457543661705567, 166957216531083822250, 309018245953667459903, 495783346533573779402], [564837757490524866496, 59018593227322993333, 321445546314025085423, 164492008917904645922], [333603319193850586447, 749726880812446578613, 107580278241953401703, 358663255587204842301], [586839495066246223216, 367512788132453427250, 840865401935624581943, 163749987579054967221]], 'inner_hash': 6564233059903904080, 'inner_nilpotent': False, 'inner_order': 209952, 'inner_split': None, 'inner_tex': 'C_3^5:S_3.D_6^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 16], [2, 28], [4, 52], [8, 42], [12, 64], [16, 6], [24, 64], [36, 32], [48, 16], [72, 56]], 'label': '419904.gm', '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': None, 'name': 'C3^4.D6^2:S3^2', 'ngens': 4, '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, 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': 124, 'number_characteristic_subgroups': 47, 'number_conjugacy_classes': 376, 'number_divisions': 376, 'number_normal_subgroups': 195, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 419904, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 8967], [3, 2024], [4, 35640], [6, 186648], [9, 4536], [12, 98496], [18, 60264], [36, 23328]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [497084124609198000, 583051434130679678656, 690267889663496812200, 0], 'outer_gens': [[821360693276132370616, 461095454043691165450, 843302926060969621680, 445602041598254567061], [564837757490524866496, 382095825041505576970, 601460068728639945000, 498164817606285771501], [128998106827168729696, 884210670604511122322, 288603055825435704247, 188187013539694028901], [32304098586040938847, 424868468137781835002, 22078758743915344687, 497676817867987311962]], 'outer_group': '96.221', 'outer_hash': 221, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 48, 'outer_perms': [561294661562426644742398195844942356686751452768723659482015, 11494616539392570028685562161422277666689534141478727364087949, 11533108311790620895780788272872148098439619213691553146575869, 12402887702626976533440839407023515213993959105187538129888269], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:C_2^3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': False, 'ratrep_stats': [[1, 16], [2, 28], [4, 52], [8, 42], [12, 64], [16, 6], [24, 64], [36, 32], [48, 16], [72, 56]], 'representations': {'PC': {'code': '538110532104426739453645389170495643900569175734521104062965797400009094849251086591355468455167274303851282499985496692741020434944198864372903671485128163589325916597033946747826140701472992831809516994502381469796066181301957223687799423618119700210276773206541663985048328871290427585673422543671010682541576287756703585173258669542182367756063135945672824640471541698240922377767599', 'gens': [1, 2, 4, 6, 8, 10, 12, 13, 14], 'pres': [14, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 824712, 1154777, 71, 6086138, 3352512, 395811, 2033825, 5818375, 157, 25766164, 8957778, 2936252, 15262133, 7268203, 7077705, 2009495, 875257, 243, 37679046, 11219060, 4978042, 3358704, 1659398, 43948807, 10612245, 4540067, 926065, 491295, 315245, 329, 28340936, 23850310, 10133460, 609386, 272980, 709206, 35017929, 17508983, 453637, 151251, 415, 55085194, 27542616, 399206, 133108, 58689803, 10898521, 8134599, 524213, 1931395, 461745, 37005708, 36966410, 9199048, 3066390, 1179428, 235954, 55206157, 27603099, 16567529, 1978087, 1200765, 738611]}, 'Perm': {'d': 22, 'gens': [5393457543661705567, 59018593227322993333, 107580278241953401703, 163749987579054967221]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 36, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4.D_6^2:S_3^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 4], [1, 3], [2, 4]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', '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': 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, 2, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 3], [2, 4]], '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 1]], 'label': '6.1', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'S3', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 3, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 6, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 3], [3, 2]], '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': 2, 'perfect': False, 'permutation_degree': 3, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 25, 'gens': [1, 2], 'pres': [2, -2, -3, 17]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [28, 45]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'GL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'SL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGL'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'SO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'SU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PSO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'PSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'GO'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'Omega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PGO'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'POmega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CSO'}, {'d': 2, 'q': 4, 'gens': [20, 194], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'CSOMinus'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'CSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CO'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'COMinus'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGammaL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSigmaL'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGammaU'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGL'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11, 6]}, 'Perm': {'d': 3, 'gens': [1, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
