-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '4608.v', 'ambient_counter': 22, 'ambient_order': 4608, 'ambient_tex': 'D_{24}.C_{96}', 'central': False, 'central_factor': False, 'centralizer_order': 2304, 'characteristic': True, 'core_order': 8, 'counter': 454, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '4608.v.576._.C', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '576.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '576.3880', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3880, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 576, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{12}.C_{24}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '8.1', 'subgroup_hash': 1, 'subgroup_order': 8, 'subgroup_tex': 'C_8', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '4608.v', 'aut_centralizer_order': 3072, 'aut_label': None, 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '4.2', 'aut_weyl_index': 3072, 'centralizer': '2._.F', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['192._.F', '192._.J', '192._.BK', '288._.B', '288._.L', '288._.M', '288._.S'], 'contains': ['1152._.B'], 'core': '576._.C', 'coset_action_label': None, 'count': 1, 'diagramx': [6284, 4289, 5976, 6059, 6019, 4329, 6364, 5728], 'generators': [1078358593, 1380299136, 582313698], 'label': '4608.v.576._.C', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '576._.C', 'normal_contained_in': ['192._.F', '192._.J', '288._.B'], 'normal_contains': ['1152._.B'], 'normalizer': '1.a1.a1', 'old_label': '576.c1.a1', 'projective_image': '1152.31450', 'quotient_action_image': '2.1', 'quotient_action_kernel': '288.327', 'quotient_action_kernel_order': 288, 'quotient_fusion': None, 'short_label': '576._.C', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2], 'aut_gens': [[1], [5], [3]], 'aut_group': '4.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [1, 6], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2], [8, 1, 4]], 'center_label': '8.1', 'center_order': 8, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 8, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 8]], 'label': '8.1', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C8', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 8, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 2], [8, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5], [3]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1]], 'representations': {'PC': {'code': 323, 'gens': [1], 'pres': [3, -2, -2, -2, 6, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26315074]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [77]}, 'Perm': {'d': 8, 'gens': [40176, 16582, 5167]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_8', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '384.5547', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 48, 'aut_gen_orders': [16, 24, 24, 16, 8, 48, 48, 8], 'aut_gens': [[37442, 395398321, 35945290], [6882766, 115025093, 913010366], [481149, 395398321, 158159276], [287377, 992089873, 833930728], [37442, 992089873, 71890580], [343540, 1301219333, 1250896092], [4039104, 395398321, 862686960], [3187202, 395398321, 1049602468], [5622862, 1272463101, 589502756]], 'aut_group': None, 'aut_hash': 2675128990628703738, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 58, 'aut_perms': [1209836495979212660927431096201921996793240735553015761880256400263328622580693, 460404172012635887940176008295548619537204047757590778225083279146513706895115, 174652649647282533900418686626609112483306293175539093391243115546255472235550, 1597639767924934956743718311830340876443831605492568598712822068395741430687228, 1765178233114510148290555356882976171987611967425069352141110052884293477525307, 1074154375750739599809226010366626104094520919021880789240330648703660255467695, 2331923544139373031937616687423484820025328957153845510685169469375855377795222, 460087216947899050909888636323091739061296835940936754624644615139315293135174], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 4, 1], [4, 12, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 3], [6, 2, 4, 1], [6, 12, 4, 1], [8, 1, 4, 1], [8, 2, 2, 3], [8, 12, 4, 1], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 2], [12, 2, 8, 2], [12, 2, 16, 1], [12, 12, 4, 1], [16, 1, 8, 1], [16, 2, 4, 1], [16, 2, 8, 1], [16, 12, 8, 1], [24, 1, 8, 1], [24, 2, 4, 7], [24, 2, 8, 4], [24, 12, 8, 1], [32, 1, 16, 1], [32, 2, 8, 1], [32, 2, 16, 1], [32, 12, 16, 1], [48, 1, 16, 1], [48, 2, 8, 3], [48, 2, 16, 4], [48, 2, 32, 1], [48, 12, 16, 1], [64, 1, 32, 1], [64, 2, 16, 1], [64, 2, 32, 1], [64, 12, 32, 1], [96, 1, 32, 1], [96, 2, 16, 3], [96, 2, 32, 4], [96, 2, 64, 1], [96, 12, 32, 1], [192, 1, 64, 1], [192, 2, 32, 3], [192, 2, 64, 4], [192, 2, 128, 1], [192, 12, 64, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:((C_4\\times C_8).C_2^6.C_2)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '256.55608', 'autcent_hash': 55608, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\times C_{16}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '48.38', 'autcentquo_hash': 38, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 12, 2], [3, 1, 2], [3, 2, 3], [4, 1, 2], [4, 2, 5], [4, 12, 2], [6, 1, 2], [6, 2, 11], [6, 12, 4], [8, 1, 4], [8, 2, 6], [8, 12, 4], [12, 1, 4], [12, 2, 46], [12, 12, 4], [16, 1, 8], [16, 2, 12], [16, 12, 8], [24, 1, 8], [24, 2, 60], [24, 12, 8], [32, 1, 16], [32, 2, 24], [32, 12, 16], [48, 1, 16], [48, 2, 120], [48, 12, 16], [64, 1, 32], [64, 2, 48], [64, 12, 32], [96, 1, 32], [96, 2, 240], [96, 12, 32], [192, 1, 64], [192, 2, 480], [192, 12, 64]], 'center_label': '192.2', 'center_order': 192, 'central_product': True, 'central_quotient': '24.6', 'commutator_count': 1, 'commutator_label': '12.2', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 22, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1536.10843710', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 2, 2], [4, 12, 1, 2], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 5], [6, 12, 2, 2], [8, 1, 4, 1], [8, 2, 2, 3], [8, 12, 2, 2], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 10], [12, 12, 2, 2], [16, 1, 8, 1], [16, 2, 4, 1], [16, 2, 8, 1], [16, 12, 4, 2], [24, 1, 8, 1], [24, 2, 4, 7], [24, 2, 8, 4], [24, 12, 4, 2], [32, 1, 16, 1], [32, 2, 8, 1], [32, 2, 16, 1], [32, 12, 8, 2], [48, 1, 16, 1], [48, 2, 8, 3], [48, 2, 16, 6], [48, 12, 8, 2], [64, 1, 32, 1], [64, 2, 16, 1], [64, 2, 32, 1], [64, 12, 16, 2], [96, 1, 32, 1], [96, 2, 16, 3], [96, 2, 32, 6], [96, 12, 16, 2], [192, 1, 64, 1], [192, 2, 32, 3], [192, 2, 64, 6], [192, 12, 32, 2]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 2236416, 'exponent': 192, 'exponents_of_order': [9, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 256]], 'familial': False, 'frattini_label': '64.50', 'frattini_quotient': '72.48', 'hash': 1280522339947909588, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 12, 1], 'inner_gens': [[37442, 1337164657, 35945290], [1850291, 395398321, 35945290], [37442, 395398321, 35945290]], 'inner_hash': 6, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': False, 'inner_tex': 'D_{12}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 128, 'irrQ_dim': 128, 'irrR_degree': None, 'irrep_stats': [[1, 384], [2, 1056]], 'label': '4608.v', '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': 'D24.C96', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 88, 'number_characteristic_subgroups': 138, 'number_conjugacy_classes': 1440, 'number_divisions': 110, 'number_normal_subgroups': 202, 'number_subgroup_autclasses': 351, 'number_subgroup_classes': 495, 'number_subgroups': 1178, 'old_label': None, 'order': 4608, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 27], [3, 8], [4, 36], [6, 72], [8, 64], [12, 144], [16, 128], [24, 224], [32, 256], [48, 448], [64, 512], [96, 896], [192, 1792]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [16, 8, 16, 8, 4, 8], 'outer_gen_pows': [776418240, 7189058, 7189058, 776418240, 611069953, 7189058], 'outer_gens': [[4039104, 992089873, 366641958], [5622862, 1272463101, 589502756], [4851827, 1301219333, 381020074], [630917, 1272463101, 323507610], [2084593, 86268861, 1200572686], [7188864, 115025093, 1171816454]], 'outer_group': '512.10493039', 'outer_hash': 3046548263559564634, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 512, 'outer_perms': [89245133128511065091163523977343624862408643499109607339967507052980810887062899083300729036366602366927092494572211197883135647892081333522954886275984087880012481432307085931019410429046329658581000071144370198401686744352616214315239204133543135905405889524351461829104777312694166405624571150315351414039259339972763016473062904276677597767785312701205879513910207286274638080187161089226828879757050139078540429755460428692692406475201339931082687823121415630042746197312310089429636983718988130645833496629522807162808911548583956002069237306432506423783910180340195005105847398201863088304992838938163565750452784646666659847142168934277192538965597681051911960148654885022586312175385524365801467902578334515245880323269796984462690672245296527722977555237879653094196819933813519949907173787028298238698304203040517856545388047151700570667401887121714327207926434375467477250021333010581499927742352756925150468010537928769322996306462269261698376973014896709145356590761512649823587074941274999679327935341210554535262023103630007150354061552869826316695724935713631078093405104625001822465381007662352404314753004671439368852020534126902760750838771647309, 277028663836839416629232287075744626575186851688847092247268684824636845602599568546301379067539123187634331466665596749204053483800275832136304868674937470189079576105310334572197553767224537285622501529974573979044523599179961730518695499646008668934458721289605565696677828931098307430740054273550998483930677172796363708067764670156931676693693405985228895629607995839980701714726397306379884014157140060251922504000592474026464175240598887074244480762776028112639872363177319770651120594207334269827640305065967792183429055043724499693287233004781483438597818051758262697134779640796377275828823462518348130357373650736473471193563945139614294508894312244303358485626535369075668346510719546428614960740829014907632118839006972293853071903152309372940512187332857884277042522951480313047830939417233661366756834895062044831070996447121805904435935353420465889405066482164389355330791992931694510514214289999871445662444857418885441905567214420335061995455332115278627052174133595532752930623476721248901110079903388799688164266359243603622996377544481450086103707638616118976023264432559986051981781321178844125732623289475846761854188804220535112364346150661267, 262769844849172437526578055262402768400174677261237054265379536978240920880755421214153144506302692833818104246071718408436318293917752527858944723072157469210144505704470782994378089476002325191740278955283955216213405817557776381551385979849824346131777287516584146942186670891757649389581813221817387384664105285484253173526871043251749924090608586844083709469856215419590543305971879175565471264455277432522832496731373898659053116625116942850024368527294565961150274742302201479354887380755063047069057015263130852296458176304120476443843773629843843575851113581875358406490716176390890557214803653767334017415155680056266489199637449666638794300462763721642790030136534595632037474380738650043524560687164431007002163667026417745927736174702560513414297417937572222794067165972430313497380778663927385586851258452495295936762476569119751110604887316601495340488239214214900105194252510237736664948070278441303390851700527836768707904278231076476917028044378642068063224117627553482422317228587716045762008101195114831184462114360456618538666803056475893509470265613663069596211481160912969489555239921895669957261387262066928409809462855558414557366967223780491, 34122729752962791029292671109264006748516870352735780712645801656820294373757682297179096943041063424864809267962968790532931280778833181906919917762527601306612826290054508040098125568629358678929638813580681919974545048793910338662761666050798964162584904959149745297250792903113114158008395661548461613216575520561923966205945290741874214742175269380340320589015896901128540036429634642196039070042710212224734216593863643528613245100404913193231586580679037044112734299257147185522129792898935251936548262607382266672728949580849759563907835978144535182830806484706312664559146652176161943584329262979515509645740531712612427349750494228904131689065724539009774443367258799960943317048897672050656691994445548076784401226802797567598801447811020666478138980502145872399061427853181149310120283725820682783315573008861448926257320467132794800113902091953336326811007579025510607527920758916034344505706158029873125131622014342251266615835202512543207813246923454395063613483385181544471207156073278388539844033243056594455640876290671552217609526987120842708015839128688786563202723535123069550277988253415749843871568163154459798444259228848140505663933336544502, 109596246932617582673815587043744372956233982585834736936334883807148212396559536440067970705965489868549939610798425396330854105256296000786536883547554801393416468802851133853352064551374344551232492979143970885062273635976893542943812831836843240841411448511599600937563375645294626520234971247775143687499002897925283947980413969297029224066817720341316721190105332834021668832105652982581161943882005569052844681559727649210271539723233567401695173052717491246074529265477434134664040949364495666130281127755440701042493682966918928722701255367938647921572749496901920456347313716991372844434107616204180731350096931630311320560922950592174526060163315110852056315530172852309117343174245857987215656904561940749803931514411246725682373226950645988931643449890979860401595137775580063842545526104803198240880279219391645367359501487806220688114820367351247788342848767384602373618757478220224694224409425263447417482868647258055623315128970091338025420261657955560112630802527627245916034147021286248251788763662334163156138402350399575182367093817945457867495205377578532050991321571606691902749846241199490760332914907068332048206123659900744650638698267902876, 208328131486907050698383471752731415187994464740870099201011205301161655472355356638782530761496986397217620477579955988841747523967002745076353493371048680689017619553063026586572451886039739860104611539449144543883128371666752805422307195864930137495729907032233611222745477676091308875708309187860550307672351299020692576038432658215839083441498438066622648363754355581350673983231340542649148863673500278220649535149623386784489584204940728737130377954919156642288348453350835723256160790057015078675479447632973395928506600154485313502565309846539484468503181059462906440117004213553934902212928114299835284487652167000941186443821578297379663106254673789097124831078395530609957714812689520885261000672753105584231432785958035900295801683594823604903290391549273877339488228330678484225519653228173409617782853824494747639426750537121108069937222917852647511306078218685828749043884558259418773857283405100967560131809832344383631819688146070774605691573385726748837936599744445672633955785788699098144677959250788331889421008106281993247778598002451533496451112947009793403197863474369802968315236253415749843871568163154459798444259228848140505663933336544502], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\times C_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 134, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 32, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [4, 19], [8, 19], [16, 17], [32, 13], [64, 8], [128, 8]], 'representations': {'PC': {'code': '329680630424262361155756325310135021435345469787175764357129', 'gens': [1, 2, 9], 'pres': [11, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 56, 90, 124, 158, 192, 226, 446696, 294, 443529, 328, 371722]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [37442, 395398321, 35945290]}, 'Perm': {'d': 134, 'gens': [15436099512715775457938543278326507815845689574710838830107688810017577763733977450011984164779589723803741620601164288034107911637534346484885657452980049119193810696159407465763620684673469402907159176922987348319091426457781, 30416434948139670563396958516244672379826467384532234219099342505013084076009486586160473368904038175732413567608632246792986883477085494093187563011904349788641140966404643730495870381169235744223786385631570991923106233357116, 45407757218969033080556801194086373207428069848356286463199006820714443992346660515070374613771350849450635387299016895487662001532848635142300672322084731362248726798354717938598556493694381906621611206288096414946036257045493]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 96], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{24}.C_{96}', 'transitive_degree': 384, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '96.176', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 2, 6, 4, 2, 2], 'aut_gens': [[22816850, 73926519, 9506], [40157656, 14602859, 912576], [22816850, 83053259, 577344], [22816850, 14602859, 344738], [22816850, 73926519, 707809], [43808352, 73926519, 912576], [1825348, 73926519, 9506]], 'aut_group': '768.1076129', 'aut_hash': 1076129, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 768, 'aut_permdeg': 22, 'aut_perms': [208012618982370488183, 43159564627920, 1009313067405724682574, 46353646213272442832, 565789213346048622888, 115620738523724313], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 6, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 3], [6, 2, 4, 1], [6, 6, 4, 1], [8, 1, 4, 1], [8, 2, 2, 1], [8, 6, 4, 1], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 2], [12, 6, 4, 1], [16, 1, 8, 1], [16, 2, 4, 1], [16, 6, 8, 1], [24, 1, 8, 1], [24, 2, 4, 3], [24, 2, 8, 2], [24, 6, 8, 1], [48, 1, 16, 1], [48, 2, 8, 3], [48, 2, 16, 2], [48, 6, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times C_{12}:C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.260', 'autcent_hash': 260, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 6, 2], [3, 1, 2], [3, 2, 3], [4, 1, 2], [4, 2, 1], [4, 6, 2], [6, 1, 2], [6, 2, 11], [6, 6, 4], [8, 1, 4], [8, 2, 2], [8, 6, 4], [12, 1, 4], [12, 2, 14], [12, 6, 4], [16, 1, 8], [16, 2, 4], [16, 6, 8], [24, 1, 8], [24, 2, 28], [24, 6, 8], [48, 1, 16], [48, 2, 56], [48, 6, 16]], 'center_label': '48.2', 'center_order': 48, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3880, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['192.460', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 6, 1, 2], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 5], [6, 6, 2, 2], [8, 1, 4, 1], [8, 2, 2, 1], [8, 6, 2, 2], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 2], [12, 6, 2, 2], [16, 1, 8, 1], [16, 2, 4, 1], [16, 6, 4, 2], [24, 1, 8, 1], [24, 2, 4, 3], [24, 2, 8, 2], [24, 6, 4, 2], [48, 1, 16, 1], [48, 2, 8, 3], [48, 2, 16, 2], [48, 6, 8, 2]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 69888, 'exponent': 48, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 32]], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '72.48', 'hash': 3880, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 6, 2], 'inner_gens': [[22816850, 73926519, 9506], [22816850, 73926519, 344738], [22816850, 5476119, 9506]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'D_6', 'inner_used': [2, 3], 'irrC_degree': 2, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 4, 'irrep_stats': [[1, 96], [2, 120]], 'label': '576.3880', 'linC_count': 32, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 36, 'linQ_dim': 20, 'linQ_dim_count': 36, 'linR_count': 16, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D12.C24', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 44, 'number_characteristic_subgroups': 62, 'number_conjugacy_classes': 216, 'number_divisions': 53, 'number_normal_subgroups': 102, 'number_subgroup_autclasses': 137, 'number_subgroup_classes': 182, 'number_subgroups': 332, 'old_label': None, 'order': 576, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 15], [3, 8], [4, 16], [6, 48], [8, 32], [12, 56], [16, 64], [24, 112], [48, 224]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [912674, 68450497, 912674, 912674, 912674], 'outer_gens': [[43808352, 14602859, 9506], [22816850, 73926519, 707809], [22816850, 83053259, 9506], [1825348, 73926519, 9506], [59323810, 73926519, 9506]], 'outer_group': '64.260', 'outer_hash': 260, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 12, 'outer_perms': [5040, 120, 39916800, 362880, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 38, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 8, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 14], [8, 8], [16, 3], [32, 4]], 'representations': {'PC': {'code': 335737604581324773314037571077, 'gens': [1, 2, 7], 'pres': [8, 2, 2, 2, 2, 2, 3, 2, 3, 41, 66, 91, 116, 29574, 166, 24583]}, 'GLFp': {'d': 2, 'p': 97, 'gens': [22816850, 73926519, 9506]}, 'Perm': {'d': 38, 'gens': [13796272150191669027729698877010383278092936, 29761899371701728772487630680010366594605440, 43989186141255759008576375645415539230446000, 325, 58137576012756456081469509965772628716519600, 69649074055620631284864062694384747587022000, 86048084288905699059332487160668418529053440, 435]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 24], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{12}.C_{24}', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
