Formats: - HTML - YAML - JSON - 2026-07-24T23:50:06.914524
  • gps_subgroup_search • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '311040.j', 'ambient_counter': 10, 'ambient_order': 311040, 'ambient_tex': 'C_3^3:S_3.C_2^4:S_5', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 162, 'counter': 230, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '311040.j.640.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '640.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': 640, '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': '486.146', 'subgroup_hash': 146, 'subgroup_order': 486, 'subgroup_tex': 'C_3^4:C_6', 'supersolvable': True, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '311040.j', 'aut_centralizer_order': None, 'aut_label': '640.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '311040.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['160.l1.a1', '160.m1.a1', '320.a1.a1', '320.b1.a1', '320.b2.a1', '320.c1.a1', '320.d1.a1', '320.e1.a1', '320.f1.a1'], 'contains': ['1280.a1.a1', '1920.a1.a1', '1920.b1.a1'], 'core': '1920.a1.a1', 'coset_action_label': None, 'count': 40, 'diagramx': [2528, -1, 2410, -1, 3060, -1, 2687, -1], 'generators': [4563710714067491935264040177344169864551888572486591018600540525227151423335270770090012696578125793858508085145946273883, 457204064046573758978307144795219438090255661314408443180818087904292671781693752127328358647603419136569127435979989166, 4870863118649996238953491954559592064826773803885751082676587078515901083562009403825362617243342507150687427384123851149, 77005995094495210252915858836731895070683090599749365730144605840758782931379232043737280156581675410091129478750605913, 385935624875196838324085483851290277077168972194299422411085558576842311464152131756432172596584002899647699875185661962, 3345632128488307175833646681330089979655620178094775783321283107311837734822864928974997135331429168616622712467023387249], 'label': '311040.j.640.a1.a1', 'mobius_quo': None, 'mobius_sub': 8, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '40.e1.a1', 'old_label': '640.a1.a1', 'projective_image': '311040.j', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '640.a1.a1', 'subgroup_fusion': None, 'weyl_group': '7776.cw'}
  • gps_groups • Show schema
    {'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': 5, 'aut_exponent': 72, 'aut_gen_orders': [2, 6, 12, 12, 3, 3, 3, 3, 3, 3], 'aut_gens': [[24577758, 24507118, 1305911, 2939606, 36731395, 14159168], [43014227, 42578137, 1305911, 32864349, 36731395, 26018967], [32864349, 11296512, 4514243, 43014227, 11225629, 14201240], [43014227, 42578137, 41439590, 37043725, 14633333, 21377773], [11495158, 42803756, 5504967, 43014227, 11444201, 29044214], [24577758, 24507118, 31289712, 2939606, 36731395, 34559316], [24577758, 24507118, 9861463, 2939606, 36731395, 8261709], [24577758, 32793736, 1305911, 2939606, 32552046, 8188666], [24577758, 11444201, 1305911, 2939606, 3158825, 21271266], [24577758, 24507118, 1305911, 2939606, 36731395, 17072210], [24577758, 24507118, 1305911, 2939606, 36731395, 39300942]], 'aut_group': '629856.le', 'aut_hash': 4095539607023997171, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 629856, 'aut_permdeg': 99, 'aut_perms': [66552576316545305676588122180264224852539928887643423847572569612114416853791789016840753027518966525951262084408516545927091919930891130993657770161870, 114509784116640130390791828025368300770676563414994360087731633874451598011012685586820963115115899922325590636321910651119732408412697430337263060119938847, 301919057001927604551692287087506498878201227169687392859536337989850039274844222330427122339301873226369300793582980589467235891527721513038718173433970730, 658897209693887083221781733704608662856681298162662755402719878325668727850032937574691041968256561897299985039611492767088691662334209338972710589163663090, 430933403696484711511210104516611215061913843999013373724905329266779632925175805475824224812008646904256899466804826913012748875161576396457397901205268306, 688540607637666249858224333031360205313531372761581037298359384967844526814769419481141267979612296084794604767302557847080188348119029826862338952169789795, 13393631877966240740568162804237944635479714579738740631347359503550597412044020337235849805796500597435259556539252179530617226895487323647290975238765670, 758437307748169785768130976425925445954785574440810485479679877701322684201435030749633400587548374696544384011886833063672305592324402232573846627628370790, 13435760249547508828714082045883567944377983566699752521378590501136172868449349004914530102607800848762514203249346147292823706723580555504034008122362846, 758415549878511324383600685924115684583637722257147666496814724301386142308918694689376321079441303802452759473639259077217350215330207060450718113688125950], 'aut_phi_ratio': 3888.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 4, 1], [3, 6, 12, 1], [3, 9, 2, 1], [3, 18, 8, 1], [6, 81, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^5.(S_3\\times C_3^2:\\GL(2,3))', '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': 72, 'autcentquo_group': '629856.le', 'autcentquo_hash': 4095539607023997171, 'autcentquo_nilpotent': False, 'autcentquo_order': 629856, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^5.(S_3\\times C_3^2:\\GL(2,3))', 'cc_stats': [[1, 1, 1], [2, 81, 1], [3, 2, 4], [3, 6, 12], [3, 9, 2], [3, 18, 8], [6, 81, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '486.146', 'commutator_count': 1, 'commutator_label': '81.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 146, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 1, 4], [3, 6, 1, 12], [3, 9, 2, 1], [3, 18, 2, 4], [6, 81, 2, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 91, 'exponent': 6, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '54.13', 'hash': 146, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 3, 3, 3, 2], 'inner_gens': [[24577758, 24507118, 1305911, 2939606, 36731395, 8188666], [24577758, 24507118, 5504967, 2939606, 36731395, 8314856], [24577758, 20327769, 1305911, 2939606, 11537230, 39300942], [24577758, 24507118, 1305911, 2939606, 36731395, 39300942], [24577758, 24507118, 41439590, 2939606, 36731395, 17350487], [20201582, 20254942, 31289712, 32864349, 11225629, 14159168]], 'inner_hash': 146, 'inner_nilpotent': False, 'inner_order': 486, 'inner_split': False, 'inner_tex': 'C_3^4:C_6', 'inner_used': [1, 2, 3, 4, 5, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 12], [6, 12]], 'label': '486.146', '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:C6', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 30, 'number_divisions': 24, 'number_normal_subgroups': 31, 'number_subgroup_autclasses': 38, 'number_subgroup_classes': 258, 'number_subgroups': 3302, 'old_label': None, 'order': 486, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 81], [3, 242], [6, 162]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 36, 'outer_gen_orders': [2, 2, 3, 2, 2, 3, 3, 3], 'outer_gen_pows': [14408200, 14408200, 14408200, 14159168, 14159168, 14159168, 14408200, 14408200], 'outer_gens': [[24577758, 37116768, 4514243, 2939606, 7046280, 14159168], [20201582, 20327769, 8713299, 7138662, 14729278, 14159168], [24577758, 20327769, 5504967, 7138662, 32479219, 14159168], [37043725, 20479832, 5504967, 7138662, 32917739, 14159168], [43014227, 42578137, 35469088, 37043725, 32479219, 14159168], [20201582, 2992993, 31289712, 32864349, 42803756, 14159168], [24577758, 37169915, 1305911, 2939606, 20479832, 14159168], [24577758, 37169915, 1305911, 2939606, 7357854, 14159168]], 'outer_group': '1296.3490', 'outer_hash': 3490, 'outer_nilpotent': False, 'outer_order': 1296, 'outer_permdeg': 9, 'outer_perms': [26869, 15976, 224824, 26149, 26790, 21030, 86400, 81], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_3\\wr S_3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, '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, 12]], 'representations': {'PC': {'code': 1801346832144273154371679, 'gens': [1, 3, 4, 5, 6], 'pres': [6, -2, -3, -3, -3, -3, -3, 12, 866, 386, 867, 12964, 5680, 11669]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 24507118, 1305911, 2939606, 36731395, 14159168]}, 'Perm': {'d': 18, 'gens': [1308157017019, 422641876213640, 85105654963324, 797707230670627, 1171782136762535, 135335]}}, 'schur_multiplier': [3, 3, 3, 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_3^4:C_6', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 120, 'aut_gen_orders': [6, 8], 'aut_gens': [[3938632625685295871943746791959711692177342846391569385563424687439219660033259498372983731343195785497578337558334344355, 5512661209498819967020280990014347681011400779742950735797833264947559131608145782689824052719088350943896887434650941636], [4546827718076786023655309830207872385692149554473375856732050145234214376831883042529265429873627070660882339718929858121, 3559041876905299444537647595027790458300888637494333199111682568957611471079791330646624059617719462449592486946480376470], [179434987654430445210000456310829771807909831560978592961204264964477555123081012567433866574151657732098666864394037488, 1682514694102698159110427440087637326392034045189880765895422001826171333057070931378083979672399706102127499895234226879]], 'aut_group': '311040.j', 'aut_hash': 6830784901558741135, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 311040, 'aut_permdeg': 1080, 'aut_perms': 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'aut_phi_ratio': 3.75, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [2, 90, 1, 1], [2, 1080, 1, 1], [3, 80, 1, 1], [3, 720, 1, 1], [3, 5760, 1, 1], [4, 540, 1, 1], [4, 1620, 1, 1], [4, 3240, 1, 1], [4, 4860, 1, 1], [4, 9720, 1, 1], [5, 31104, 1, 1], [6, 720, 1, 1], [6, 6480, 1, 1], [6, 8640, 1, 2], [6, 17280, 1, 1], [8, 1620, 1, 2], [8, 6480, 1, 2], [8, 9720, 1, 1], [8, 19440, 1, 1], [8, 38880, 1, 1], [10, 31104, 1, 1], [12, 4320, 1, 1], [12, 12960, 1, 1], [12, 25920, 1, 1], [24, 12960, 1, 4]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3:S_3.C_2^4:S_5', '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': 120, 'autcentquo_group': '311040.j', 'autcentquo_hash': 6830784901558741135, 'autcentquo_nilpotent': False, 'autcentquo_order': 311040, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^3:S_3.C_2^4:S_5', 'cc_stats': [[1, 1, 1], [2, 81, 1], [2, 90, 1], [2, 1080, 1], [3, 80, 1], [3, 720, 1], [3, 5760, 1], [4, 540, 1], [4, 1620, 1], [4, 3240, 1], [4, 4860, 1], [4, 9720, 1], [5, 31104, 1], [6, 720, 1], [6, 6480, 1], [6, 8640, 2], [6, 17280, 1], [8, 1620, 2], [8, 6480, 2], [8, 9720, 1], [8, 19440, 1], [8, 38880, 1], [10, 31104, 1], [12, 4320, 1], [12, 12960, 1], [12, 25920, 1], [24, 12960, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '311040.j', 'commutator_count': 1, 'commutator_label': None, 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '60.5'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [2, 90, 1, 1], [2, 1080, 1, 1], [3, 80, 1, 1], [3, 720, 1, 1], [3, 5760, 1, 1], [4, 540, 1, 1], [4, 1620, 1, 1], [4, 3240, 1, 1], [4, 4860, 1, 1], [4, 9720, 1, 1], [5, 31104, 1, 1], [6, 720, 1, 1], [6, 6480, 1, 1], [6, 8640, 1, 2], [6, 17280, 1, 1], [8, 1620, 2, 1], [8, 6480, 2, 1], [8, 9720, 1, 1], [8, 19440, 1, 1], [8, 38880, 1, 1], [10, 31104, 1, 1], [12, 4320, 1, 1], [12, 12960, 1, 1], [12, 25920, 1, 1], [24, 12960, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 136800, 'exponent': 120, 'exponents_of_order': [8, 5, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[80, 1, 2], [160, 0, 2], [160, 1, 1], [240, 1, 2], [320, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '311040.j', 'hash': 6830784901558741135, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 120, 'inner_gen_orders': [6, 8], 'inner_gens': [[3938632625685295871943746791959711692177342846391569385563424687439219660033259498372983731343195785497578337558334344355, 1204688212998658619000334793945242109098409003317537044283164954829151548204437370340377386909595854753169830364257962917], [1369703252167805278818951658925534974015989240540526343408472167481828725179465657638062489175226389158015207098897864557, 5512661209498819967020280990014347681011400779742950735797833264947559131608145782689824052719088350943896887434650941636]], 'inner_hash': 6830784901558741135, 'inner_nilpotent': False, 'inner_order': 311040, 'inner_split': True, 'inner_tex': 'C_3^3:S_3.C_2^4:S_5', 'inner_used': [1, 2], 'irrC_degree': 80, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 80, 'irrep_stats': [[1, 2], [4, 4], [5, 4], [6, 1], [10, 5], [15, 2], [16, 2], [20, 4], [24, 1], [80, 2], [160, 3], [240, 2], [320, 1]], 'label': '311040.j', 'linC_count': 2, 'linC_degree': 80, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 80, 'linQ_degree_count': 2, 'linQ_dim': 80, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 80, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C3^3:S3.C2^4:S5', 'ngens': 2, '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, 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': 33, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 33, 'number_divisions': 29, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 789, 'number_subgroup_classes': 789, 'number_subgroups': 1048248, 'old_label': None, 'order': 311040, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1251], [3, 6560], [4, 19980], [5, 31104], [6, 41760], [8, 84240], [10, 31104], [12, 43200], [24, 51840]], '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': None, 'perfect': False, 'permutation_degree': 81, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [5, 4], [6, 1], [8, 1], [10, 5], [15, 2], [20, 2], [24, 1], [32, 1], [40, 1], [80, 2], [160, 1], [240, 2], [320, 2]], 'representations': {'Perm': {'d': 81, 'gens': [3938632625685295871943746791959711692177342846391569385563424687439219660033259498372983731343195785497578337558334344355, 5512661209498819967020280990014347681011400779742950735797833264947559131608145782689824052719088350943896887434650941636]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3.C_2^4:S_5', 'transitive_degree': 81, 'wreath_data': None, 'wreath_product': False}