Formats: - HTML - YAML - JSON - 2026-07-25T03:09:53.328022
  • gps_subgroup_search • Show schema
    {'Agroup': True, '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': 157, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '311040.j.240.x1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '240.x1.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': 240, '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': '1296.3506', 'subgroup_hash': 3506, 'subgroup_order': 1296, 'subgroup_tex': 'C_3^2:(C_2\\times F_9)', 'supersolvable': False, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '311040.j', 'aut_centralizer_order': None, 'aut_label': '240.x1', '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': ['80.h1.a1', '120.e1.a1', '120.y1.a1', '120.bh1.a1'], 'contains': ['480.h1.a1', '480.n1.a1', '480.p1.a1', '2160.v1.a1'], 'core': '1920.a1.a1', 'coset_action_label': None, 'count': 60, 'diagramx': [1681, -1, 1210, -1, 2002, -1, 2385, -1], 'generators': [1470178075756205444310996234758515644761071745510428406883456259212552201138311022944891752990084434471159923897789030174, 4563710714067491935264040177344169864551888572486591018600540525227151423335270770090012696578125793858508085145946273883, 3938632625685295871943746791959711692177342846391569385563424687439219660033259498372983731343195785497578337558334344355, 457204064046573758978307144795219438090255661314408443180818087904292671781693752127328358647603419136569127435979989166, 3623136856969618423096610467161572174743516431721993124033073209273972701610267345987062058640201436624985258555172182340, 77005995094495210252915858836731895070683090599749365730144605840758782931379232043737280156581675410091129478750605913, 385935624875196838324085483851290277077168972194299422411085558576842311464152131756432172596584002899647699875185661962, 3345632128488307175833646681330089979655620178094775783321283107311837734822864928974997135331429168616622712467023387249], 'label': '311040.j.240.x1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '60.n1.a1', 'old_label': '240.x1.a1', 'projective_image': '311040.j', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '240.x1.a1', 'subgroup_fusion': None, 'weyl_group': '5184.cc'}
  • gps_groups • Show schema
    {'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 48, 'aut_gen_orders': [8, 6, 12], 'aut_gens': [[24577758, 24507118, 2939606, 36731395, 17205882, 32649813, 7355200, 29111455], [14729278, 11225629, 14357270, 42578137, 25872955, 11221654, 11448105, 28897601], [7138662, 6973453, 2939606, 11225629, 21313965, 24950601, 37173792, 21310395], [11495158, 11391081, 43014227, 24783208, 29114980, 37009130, 6970070, 39281013]], 'aut_group': '20736.x', 'aut_hash': 5863683207372618949, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20736, 'aut_permdeg': 180, 'aut_perms': [837376639342417379023611334889478164536685508888703592664907172553696744738774607546475913020875071390519481465119213938474910888188777695571605568810486058836349293597208736222116861028812501444809694405030156304947087074458229520590656478497785213294306124208386661818740660417834409499449739914762640314143111128981427859789, 4557350792559871712720534213464916280737934917033419672721096245580734562231215459400256624804903402711703565200010840604496233769673255553995350002331649301141128743882892228205166316385481794913101678427330854838757201282503777201197225320168863554939597326887509467656280090927515883062025783702527052631642629479614493381274, 172434950814924844425773895289422043858541932599759705933472306006077183421231946943110269692384708017485185176946552359223653278310336629757626672422676491790977564272152864896147420462596430918523095867601793298239885625762118244999547580432115254287471053971317764284971278065104343144162515153757591628925432876652441569733795], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 2, 1], [2, 81, 1, 1], [3, 8, 2, 1], [3, 16, 4, 1], [4, 81, 2, 2], [6, 72, 2, 1], [8, 81, 2, 2], [8, 81, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_9\\wr C_2:C_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': 48, 'autcentquo_group': '20736.x', 'autcentquo_hash': 5863683207372618949, 'autcentquo_nilpotent': False, 'autcentquo_order': 20736, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9\\wr C_2:C_2', 'cc_stats': [[1, 1, 1], [2, 9, 2], [2, 81, 1], [3, 8, 2], [3, 16, 4], [4, 81, 4], [6, 72, 2], [8, 81, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1296.3506', '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': 3506, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 81, 1, 1], [3, 8, 1, 2], [3, 16, 1, 4], [4, 81, 2, 2], [6, 72, 1, 2], [8, 81, 4, 2]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 24, 'exponent': 24, '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': '1296.3506', 'hash': 3506, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [3, 3, 3, 3, 8, 4, 2, 2], 'inner_gens': [[24577758, 24507118, 2939606, 36731395, 38903140, 7042278, 11534549, 39281013], [24577758, 24507118, 2939606, 36731395, 39069872, 7144047, 11607619, 39281013], [24577758, 24507118, 2939606, 36731395, 34704108, 37006309, 20477175, 17111339], [24577758, 24507118, 2939606, 36731395, 28857609, 20425536, 32870443, 8168737], [37043725, 37010401, 20201582, 7411244, 17205882, 20260201, 32870443, 3989361], [7138662, 6973453, 11495158, 2721277, 8170251, 32649813, 7195929, 3989361], [20201582, 20254942, 32864349, 11225629, 13981311, 32796943, 7355200, 21310395], [20201582, 20327769, 32864349, 11537230, 4083907, 2666102, 42522093, 29111455]], 'inner_hash': 3506, 'inner_nilpotent': False, 'inner_order': 1296, 'inner_split': False, 'inner_tex': 'C_3^2:(C_2\\times F_9)', 'inner_used': [1, 2, 3, 4, 5, 8], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [8, 4], [16, 4]], 'label': '1296.3506', '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': 'C3^2:(C2*F9)', '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': 11, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 24, 'number_divisions': 16, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 146, 'number_subgroups': 4228, 'old_label': None, 'order': 1296, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 99], [3, 80], [4, 324], [6, 144], [8, 648]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [14408200, 14480438, 28750608, 28855529], 'outer_gens': [[24577758, 24507118, 37043725, 2721277, 8389996, 11173191, 42522093, 29111455], [14633333, 42648777, 14853281, 32793736, 17370790, 11170697, 14725868, 28855529], [14633333, 36678278, 14357270, 20646633, 38993165, 43015059, 14414294, 28855529], [24577758, 24732251, 2939606, 36678278, 17329885, 32922890, 7355200, 29111455]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [24, 127, 1, 414], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 2], [8, 4], [16, 4]], 'representations': {'PC': {'code': 1215595967756470772268752302947114495665880253430961740766499438864503575, 'gens': [1, 4, 5, 7, 8], 'pres': [8, 2, 2, 2, 3, 2, 3, 3, 3, 16, 41, 3331, 779, 147, 1604, 1612, 1220, 116, 2693, 397, 1173, 48390, 16142, 4054, 710, 73735, 32271, 13847, 2343]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 24507118, 2939606, 36731395, 17205882, 32649813, 7355200, 29111455]}, 'Perm': {'d': 18, 'gens': [107684389640120, 3402106385166720, 71057056916073, 21210673483915, 2644211021173174, 1510738997195764, 2267507354388480, 711549292688884]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:(C_2\\times F_9)', 'transitive_degree': 18, '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}