-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.2722', 'ambient_counter': 2722, 'ambient_order': 1344, 'ambient_tex': '(C_2\\times C_{24}).D_{14}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 224, 'counter': 9, 'cyclic': False, 'direct': None, 'hall': 14, 'label': '1344.2722.3.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '3.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': 3, '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': '448.383', 'subgroup_hash': 383, 'subgroup_order': 448, 'subgroup_tex': 'C_{28}:Q_{16}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1344.2722', 'aut_centralizer_order': 2, 'aut_label': '3.a1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': None, 'aut_weyl_index': 6, 'centralizer': '336.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '6.b1.a1', '6.c1.a1', '6.d1.a1', '6.e1.a1', '6.f1.a1', '6.g1.a1', '21.a1.a1'], 'core': '6.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [6732, -1, 4220, -1, 2578, -1, 5391, -1], 'generators': [1, 56, 112, 84, 14, 16, 224], 'label': '1344.2722.3.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '3.a1.a1', 'projective_image': '336.149', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3.a1.a1', 'subgroup_fusion': None, 'weyl_group': '112.29'}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [2, 12, 6, 12, 28, 28, 6, 6], 'aut_gens': [[1, 2, 112], [77, 54, 112], [61, 234, 168], [333, 318, 336], [273, 202, 392], [229, 2, 336], [301, 338, 168], [101, 430, 336], [309, 38, 336]], 'aut_group': None, 'aut_hash': 8021432174918964053, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21504, 'aut_permdeg': 72, 'aut_perms': [25022353084953788813136683588875973545896918133801228043743893765444862732373300249457620279224516767415, 1283234917074326590736672579396960778012420499256835905900379753332098989445985123222259758696835261735, 32899606088484743447415532755077416568882758571149497497949110390374879927478753592161989542261368829687, 33373238023354467664558208145680759337235292071144697215654075443768624773346419645934756888604129466783, 55259897525991951444409903910317661408613166226456637032995863200718008580744459762302587408310668714672, 41473434726035824099718310697539357176154487572988928285430552924331410013132892643046284101613493386304, 11998679793514262574984351180060189036033283277051351368171329936126757544122657117007663148977817739040, 2479706379394499387824564879256589943649290150861638273802395543852078076591478794914818193662009666613], 'aut_phi_ratio': 112.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 1], [4, 28, 2, 2], [4, 56, 2, 1], [7, 2, 3, 1], [8, 4, 4, 1], [14, 2, 3, 3], [28, 2, 6, 2], [28, 4, 6, 2], [56, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_7.(C_2^4\\times C_6).C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '336.216', 'autcentquo_hash': 216, 'autcentquo_nilpotent': False, 'autcentquo_order': 336, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 4], [4, 4, 1], [4, 28, 4], [4, 56, 2], [7, 2, 3], [8, 4, 4], [14, 2, 9], [28, 2, 12], [28, 4, 12], [56, 4, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '112.29', 'commutator_count': 1, 'commutator_label': '56.8', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 383, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 4], [4, 4, 1, 1], [4, 28, 1, 2], [4, 28, 2, 1], [4, 56, 1, 2], [7, 2, 3, 1], [8, 4, 2, 2], [14, 2, 3, 3], [28, 2, 6, 2], [28, 4, 3, 2], [28, 4, 6, 1], [56, 4, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 56, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '56.12', 'hash': 383, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 28, 2], 'inner_gens': [[1, 110, 112], [5, 2, 336], [1, 226, 112]], 'inner_hash': 29, 'inner_nilpotent': False, 'inner_order': 112, 'inner_split': True, 'inner_tex': 'C_2\\times D_{28}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 58], [4, 13]], 'label': '448.383', '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': 'C28:Q16', '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': 21, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 79, 'number_divisions': 27, 'number_normal_subgroups': 45, 'number_subgroup_autclasses': 78, 'number_subgroup_classes': 108, 'number_subgroups': 548, 'old_label': None, 'order': 448, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 236], [7, 6], [8, 16], [14, 18], [28, 72], [56, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[225, 30, 112], [253, 30, 112], [225, 2, 112], [225, 30, 168], [1, 122, 168]], 'outer_group': '192.1531', 'outer_hash': 1531, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 13, 'outer_perms': [1520507520, 1520467944, 482631144, 1680, 2403], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 4], [6, 4], [12, 4], [24, 3]], 'representations': {'PC': {'code': 28535701444609954457558841933716874036, 'gens': [1, 2, 6], 'pres': [7, -2, -2, -2, -2, -7, -2, -2, 392, 1541, 36, 2270, 58, 2915, 80, 3364, 7068, 124]}, 'Perm': {'d': 27, 'gens': [453786483447377770016315647, 8023680, 873160106454638555546568960, 1292478586697946400166937600, 3669120, 1711998581393250427602124800, 973]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{28}:Q_{16}', 'transitive_degree': 448, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [12, 6, 6, 2, 6, 6, 6, 6, 12, 2], 'aut_gens': [[1, 2, 112, 448], [317, 298, 1288, 896], [297, 90, 616, 896], [293, 46, 1064, 448], [261, 334, 112, 896], [281, 22, 840, 448], [237, 62, 784, 896], [297, 246, 1232, 896], [65, 18, 1064, 448], [69, 18, 560, 896], [253, 254, 336, 896]], 'aut_group': None, 'aut_hash': 5255589997166976132, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 64512, 'aut_permdeg': 76, 'aut_perms': [393931110280199932917311391338184363768694857921397639052917804329866039385740704271865869764513475222334746745, 1112319804506979170335766418277107917747041009573451678905284984941631762891381717728919254250718629647144908146, 92423402978473631460685422593472039770885196249473889839391447709723284710181204028815142382492819496172226502, 594318473756100801679915564002109042461029977794332575872363364543915744752643731763655324317314840885342853513, 559294929173538899317106964969718829765766332811948250373549557865644142038798980147130515431748598929988178423, 719265891147976239517613999739339952829711132040624042672493050042781829348720411268653892554864590388982919152, 1878574354867370338935555671479155408254225969935022144486515029466186840940571576673000775218560702336382993278, 776372830800013271490598890438353338255615385475448065984508340969203979920940188637939317923311080427489370752, 166397139471215798676390393618531555680038195810932166602768490637074728569536083257024537199001633052742756825, 1141768674237862080742153253426881443997273626522869157578260527428134970520606080211494705479857556080004440876], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 2, 1, 2], [4, 6, 2, 1], [4, 12, 1, 1], [4, 28, 2, 1], [4, 56, 1, 1], [4, 84, 2, 1], [4, 168, 1, 1], [6, 2, 1, 3], [7, 2, 3, 1], [8, 4, 2, 1], [8, 12, 2, 1], [12, 4, 1, 2], [12, 56, 2, 2], [14, 2, 3, 3], [21, 4, 3, 1], [24, 4, 4, 1], [28, 2, 6, 2], [28, 12, 6, 2], [42, 4, 3, 3], [56, 4, 12, 1], [56, 12, 12, 1], [84, 4, 6, 2], [168, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '1008.906', 'autcentquo_hash': 906, 'autcentquo_nilpotent': False, 'autcentquo_order': 1008, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 1], [4, 2, 2], [4, 6, 2], [4, 12, 1], [4, 28, 2], [4, 56, 1], [4, 84, 2], [4, 168, 1], [6, 2, 3], [7, 2, 3], [8, 4, 2], [8, 12, 2], [12, 4, 2], [12, 56, 4], [14, 2, 9], [21, 4, 3], [24, 4, 4], [28, 2, 12], [28, 12, 12], [42, 4, 9], [56, 4, 12], [56, 12, 12], [84, 4, 12], [168, 4, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '336.149', 'commutator_count': 1, 'commutator_label': '168.39', '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', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2722, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 2, 1, 2], [4, 6, 1, 2], [4, 12, 1, 1], [4, 28, 1, 2], [4, 56, 1, 1], [4, 84, 2, 1], [4, 168, 1, 1], [6, 2, 1, 3], [7, 2, 3, 1], [8, 4, 2, 1], [8, 12, 2, 1], [12, 4, 1, 2], [12, 56, 1, 2], [12, 56, 2, 1], [14, 2, 3, 3], [21, 4, 3, 1], [24, 4, 4, 1], [28, 2, 6, 2], [28, 12, 3, 2], [28, 12, 6, 1], [42, 4, 3, 3], [56, 4, 12, 1], [56, 12, 12, 1], [84, 4, 6, 2], [168, 4, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 10752, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '168.50', 'hash': 2722, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 28, 2, 3], 'inner_gens': [[1, 110, 112, 448], [5, 2, 336, 448], [1, 226, 112, 896], [1, 2, 1008, 448]], 'inner_hash': 149, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': None, 'inner_tex': 'S_3\\times D_{28}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 66], [4, 67]], 'label': '1344.2722', 'linC_count': 96, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 132, 'linQ_dim': 16, 'linQ_dim_count': 8, 'linR_count': 240, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*C24).D14', '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, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 40, 'number_characteristic_subgroups': 60, 'number_conjugacy_classes': 141, 'number_divisions': 44, 'number_normal_subgroups': 68, 'number_subgroup_autclasses': 196, 'number_subgroup_classes': 216, 'number_subgroups': 1716, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 476], [6, 6], [7, 6], [8, 32], [12, 232], [14, 18], [21, 12], [24, 16], [28, 168], [42, 36], [56, 192], [84, 48], [168, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 2, 6], 'outer_gen_pows': [84, 96, 928, 0, 84, 0], 'outer_gens': [[309, 282, 1288, 896], [237, 86, 392, 896], [285, 30, 1064, 448], [261, 334, 112, 896], [269, 26, 168, 448], [249, 326, 1232, 896]], 'outer_group': '192.1543', 'outer_hash': 1543, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 192, 'outer_perms': [113059758369372306527443777582306753938330667108089765130649718401774375870252207934735299345276368930107706501707776089476409041404256362586090243982330208675614388775942158631027967746527916346308339776060452471908075409613981384904874296711934504899848000004029479623730169764817281509200298411810882296379002402474767321124038620792091265312018192630067, 108381699444940428301897281363122981495254965463988171298035072041308966273122712866974615173235779640655788838176344685541485184169317513730284986628770351464353974930347725674088356782018501286102041284745678865288769901330980875460805146631417990622191852048754616099491534167023819014807135704752837295240316301684567786074168511188964433529970677250192, 170440786812010173235648578665798686066148566496646373100088642104679004050272856018144459941042186991074991747639341050458944486424335371483388560469524350712977528198574603813109222663614395139746079247039805588464783826482773596707434884668301255380930613089898379706002593017646161971738136277177426590939559966113425632346167181199437741816133843499303, 7561717069511715701283261292875255187466202006790353238632507626453288627628015750683075811127007185377209140979144288171154420806069583389431482633721852045355564418452172609978394759383702286456298039833579065520455999873367392073924216733259096345049682152984132391523400526002416456203340622374265627080895766655333032967748783809710880758237338253245, 59284674151066013795670693418848471573368518242785705857045441576282225491377580186728917755126239751482462283267380235436580710390013317200548587620275963692104390460999817050096321775871343410618056885375877056055718415405821013447491614357612159385560301051246574549100839588363326508151363960950066791068161186824796558116480117047487899885238106991777, 176412994596974717075814881072608860048712432215214704807547262051095124490797267461548320904889951749816803465270788741498575252227181873481291040267806762402365230493258736942711742532760237726677465072582944133746197103064690988435489515650347925492855766721080874877963903312417192909404580296889723373128654140319118255728355488925087680426021703178346], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 8], [6, 4], [8, 2], [12, 6], [24, 6], [48, 2]], 'representations': {'PC': {'code': 1112334152330650844897500934591146716306158129977, 'gens': [1, 2, 6, 8], 'pres': [8, -2, -2, -2, -2, -7, -2, -2, -3, 448, 1761, 41, 2594, 66, 3331, 91, 3844, 8077, 141, 559]}, 'Perm': {'d': 30, 'gens': [305292256521392544629513581560, 51369540337068307327, 404838313573811045, 16, 535000309689702360, 663578103817364400, 1175091669949317120000, 9452345886333784620057231360000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times C_{24}).D_{14}', 'transitive_degree': 1344, 'wreath_data': None, 'wreath_product': False}
