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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1458.825', 'ambient_counter': 825, 'ambient_order': 1458, 'ambient_tex': 'C_3^4.(C_3\\times S_3)', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 27, 'counter': 51, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1458.825.27.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '27.d1.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': 27, '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': '54.13', 'subgroup_hash': 13, 'subgroup_order': 54, 'subgroup_tex': 'C_3^2:C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1458.825', 'aut_centralizer_order': 81, 'aut_label': '27.d1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '108.17', 'aut_weyl_index': 729, 'centralizer': '162.h1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['9.b1.a1', '9.e1.a1', '9.g1.a1'], 'contains': ['54.b1.a1', '81.b1.a1', '81.d1.a1', '81.d1.b1', '81.d1.c1', '81.h1.a1'], 'core': '54.b1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [8317, -1, 8582, -1, 5507, -1, 8732, -1], 'generators': [111, 54, 96, 486], 'label': '1458.825.27.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.b1.a1', 'old_label': '27.d1.a1', 'projective_image': '486.173', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '27.d1.a1', 'subgroup_fusion': None, 'weyl_group': '18.4'}
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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': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 4, 2, 4, 3, 2, 6, 6], 'aut_gens': [[1, 6, 18], [5, 6, 18], [47, 42, 48], [19, 12, 36], [53, 24, 42], [7, 6, 18], [5, 48, 18], [53, 6, 18], [11, 30, 36]], 'aut_group': '864.4661', 'aut_hash': 4661, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 11, 'aut_perms': [1, 30224881, 7984086, 25453807, 4843488, 2245249, 27748111, 4241545], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 4, 1], [3, 2, 8, 1], [6, 9, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_3^2:\\GL(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': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 1, 2], [3, 2, 12], [6, 9, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 1, 4], [3, 2, 2, 4], [6, 9, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 91, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '54.13', 'hash': 13, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[1, 12, 36], [13, 6, 18], [37, 6, 18]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 12]], 'label': '54.13', 'linC_count': 48, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 24, 'linQ_dim': 6, 'linQ_dim_count': 24, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:C6', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 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': 18, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 32, 'number_subgroups': 72, 'old_label': None, 'order': 54, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 9], [3, 26], [6, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 1, 1], 'outer_gens': [[5, 36, 12], [5, 6, 18], [1, 36, 42], [1, 48, 30], [1, 30, 24]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [143, 127, 576, 126, 121], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [4, 4]], 'representations': {'PC': {'code': 25289495, 'gens': [1, 3, 4], 'pres': [4, -2, -3, -3, -3, 8, 146, 579]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101736062737953, 41624334336939899, 41715841511539063]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [11495158, 22274384, 5504967, 25627796]}, 'Perm': {'d': 9, 'gens': [42001, 97224, 136227, 4]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:C_6', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [3, 3, 9, 3, 6, 3, 6, 3, 3, 3, 3], 'aut_gens': [[1, 6, 18, 54, 162], [1087, 6, 18, 54, 270], [1045, 6, 18, 54, 162], [169, 60, 18, 54, 162], [1, 492, 18, 54, 282], [1, 120, 90, 54, 378], [1, 978, 18, 54, 264], [113, 546, 126, 108, 468], [487, 6, 18, 54, 162], [109, 6, 18, 54, 162], [1, 6, 504, 54, 162], [1, 6, 18, 54, 648]], 'aut_group': None, 'aut_hash': 2044487658610632269, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 78732, 'aut_permdeg': 135, 'aut_perms': [24840618900669467733100903388293537890541259296296902666075765033938299245762144943160664272416488801640195852069129052387589946912180057091011223211789603179899070096664462088211569161560519326293360365594869508269168935743106875, 66261779163770490305712744511541994636830770712334586428139237123545391759870420195949988680161850535644114288431943320318700580911010997903173822932965719467556876601576292097149853900589194734368431326747638908966686612583715314, 58159067687704535825884944545398139572813395501373072281657574720125194324701835289116174121368099817094054065225897263427377831555643728012196317540807334019529804152730215107354949443410080112246007645068514977112117813587577675, 874248503403001989959257516079364027470491799328575938182425376898042363280587101824034038150506787591113144656125496027481580855762914375277366039610731434517892483850741994398574334868705597236319281013850709190626517191003266, 1453389509605303314801168608997865887201486194788079603342785484752899899999019234789342577179948464101453705860045606838957433270062102912815540309222883451590513741971816835840258431788353030391692517389345750113919335040974685, 1256018596403564204240776656463609251414637633383575201665627052321373212484633615620976866132815258275652898521504533572069673340103976333402236704249113797598486765847447291134104967473406914937039088022879220373746964463604290, 1846457112010727301238693476204781670049829150392730277516189667069019129148392798072716198776894998811295654034233515391296183869677071878316587566348549368518332443180617636221451879471228088355882907905256512941658615274561844, 216850781676294106295006085387286879704529784999708505996464041364223363458644149028401462117267771061832400593353964077011371293817793975137663414663888220567879865745287871549144729678946542788315937108025441686970291391820986730, 83028027815117469699049575614103844473037389760800522633144246092780562731615802240112513859620075349785066550376184932606905143561408929445500396987609638169432620464166882736407500218364649160207484547171713274941664043279, 1606365728097929614180378991972351081267945009896286911563085553120163943695456087778709944068024825300300783301643557156441307569981666807526021531022952112325257911194960725638801297992730712399213753317406986554649551347048304, 1612851739706569948569646464387921173673435954831632489550681427634618431645892950083216564809999398323924633773074576916913220193461759525596400026098397301711626137805388482872736099170821322028876499302578182061840804600651570], 'aut_phi_ratio': 162.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 6, 3, 1], [3, 18, 1, 2], [3, 54, 6, 1], [6, 81, 2, 1], [9, 9, 6, 1], [9, 18, 6, 1], [9, 18, 9, 1], [18, 81, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^4.C_3^5.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 3, 'autcent_group': '3.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 3, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': None, 'autcentquo_hash': 8899929293017801025, 'autcentquo_nilpotent': False, 'autcentquo_order': 26244, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^4.C_3^3.D_6', 'cc_stats': [[1, 1, 1], [2, 81, 1], [3, 1, 2], [3, 2, 3], [3, 6, 6], [3, 18, 2], [3, 54, 6], [6, 81, 2], [9, 9, 6], [9, 18, 15], [18, 81, 6]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '486.173', 'commutator_count': 2, 'commutator_label': '243.32', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 825, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 4], [3, 6, 2, 1], [3, 18, 1, 2], [3, 54, 2, 3], [6, 81, 2, 1], [9, 9, 6, 1], [9, 18, 3, 3], [9, 18, 6, 1], [18, 81, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 19656, 'exponent': 18, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[18, 0, 2]], 'familial': False, 'frattini_label': '27.5', 'frattini_quotient': '54.13', 'hash': 825, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 3, 3, 1, 9], 'inner_gens': [[1, 1092, 1062, 54, 1356], [553, 6, 18, 54, 216], [631, 6, 18, 54, 162], [1, 6, 18, 54, 162], [391, 114, 18, 54, 162]], 'inner_hash': 173, 'inner_nilpotent': False, 'inner_order': 486, 'inner_split': True, 'inner_tex': '(C_3^2\\times C_9):C_6', 'inner_used': [1, 2, 3, 5], 'irrC_degree': 18, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 36, 'irrep_stats': [[1, 6], [2, 12], [3, 12], [6, 18], [18, 2]], 'label': '1458.825', 'linC_count': 108, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 36, 'linQ_degree_count': 7, 'linQ_dim': 36, 'linQ_dim_count': 7, 'linR_count': 54, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^4.(C3*S3)', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 50, 'number_divisions': 22, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 115, 'number_subgroup_classes': 176, 'number_subgroups': 2562, 'old_label': None, 'order': 1458, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 81], [3, 404], [6, 162], [9, 324], [18, 486]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 3, 3, 3], 'outer_gen_pows': [0, 0, 111, 111], 'outer_gens': [[113, 12, 36, 108, 228], [1, 492, 18, 54, 282], [1, 606, 90, 54, 942], [1, 120, 90, 54, 378]], 'outer_group': '162.41', 'outer_hash': 41, 'outer_nilpotent': False, 'outer_order': 162, 'outer_permdeg': 12, 'outer_perms': [2309307, 1986024, 288506880, 52014480], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^3:S_3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 54, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [4, 4], [6, 3], [18, 5], [36, 2]], 'representations': {'PC': {'code': 6754232700416793575402707594509535733010879, 'gens': [1, 3, 4, 5, 6], 'pres': [7, 2, 3, 3, 3, 3, 3, 3, 14, 764, 22934, 10278, 29739, 13870, 56957, 25086, 1531, 166, 47634]}, 'Perm': {'d': 54, 'gens': [117642337552012132026437981902971627938309531759870801474617737216000000, 47763910107832494218454531549213300675129201571046124749144900493555984, 86978052431498406760396038791181880126273217563781342123711064716092227, 26142555467905557271450455617090360180902962888139810236108680000744374, 5316610055676179727286798434079463078756123671310400000, 4355572481870163772907365975214962059091431150555334826388371247248884, 8549827396823706354984506769470825596107916453176775172126947119248884]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4.(C_3\\times S_3)', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
