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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '7776.cw', 'ambient_counter': 75, 'ambient_order': 7776, 'ambient_tex': 'C_3^4:(S_3\\times \\SD_{16})', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 54, 'counter': 66, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '7776.cw.24._.E', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '24.e1.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': 24, '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': '324.116', 'subgroup_hash': 116, 'subgroup_order': 324, 'subgroup_tex': 'C_3^2:S_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7776.cw', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '7776.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8._.D', '12._.G'], 'contains': ['48._.C', '48._.G', '48._.L', '72._.F', '72._.G', '72._.K'], 'core': '144._.A', 'coset_action_label': None, 'count': 12, 'diagramx': [2734, -1, 2671, -1, 2078, -1, 2004, -1], 'generators': [1, 2592, 864, 48, 2912, 3744], 'label': '7776.cw.24._.E', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2._.A', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '12._.G', 'old_label': '24.e1.a1', 'projective_image': '7776.cw', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24._.E', 'subgroup_fusion': None, 'weyl_group': '648.555'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [3, 6, 3, 3, 3, 2, 2], 'aut_gens': [[1, 6, 18, 108], [7, 6, 18, 108], [41, 48, 18, 216], [79, 6, 126, 108], [49, 6, 18, 108], [79, 42, 18, 108], [37, 12, 90, 216], [1, 12, 90, 108]], 'aut_group': '648.555', 'aut_hash': 555, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 648, 'aut_permdeg': 18, 'aut_perms': [4294835596657382, 5013567591149507, 3384869039994996, 2524925312394339, 3222542592714871, 1813501594690674, 5012074582414896], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 1, 1], [3, 6, 2, 2], [3, 12, 1, 1], [3, 12, 2, 1], [6, 6, 1, 1], [6, 9, 2, 2], [6, 18, 1, 2], [6, 18, 2, 2], [6, 27, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3.S_3^3', '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': 6, 'autcentquo_group': '648.555', 'autcentquo_hash': 555, 'autcentquo_nilpotent': False, 'autcentquo_order': 648, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3.S_3^3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 1], [3, 2, 2], [3, 3, 2], [3, 4, 1], [3, 6, 5], [3, 12, 3], [6, 6, 1], [6, 9, 4], [6, 18, 6], [6, 27, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '324.116', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 116, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['54.5', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 1, 1], [3, 6, 2, 2], [3, 12, 1, 1], [3, 12, 2, 1], [6, 6, 1, 1], [6, 9, 2, 2], [6, 18, 1, 2], [6, 18, 2, 2], [6, 27, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 6, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '108.38', 'hash': 116, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [6, 3, 6, 3], 'inner_gens': [[1, 48, 90, 108], [85, 6, 18, 108], [37, 6, 18, 216], [1, 6, 234, 108]], 'inner_hash': 116, 'inner_nilpotent': False, 'inner_order': 324, 'inner_split': True, 'inner_tex': 'C_3^2:S_3^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 12], [2, 12], [4, 3], [6, 2], [12, 1]], 'label': '324.116', 'linC_count': 12, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:S3^2', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 21, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 30, 'number_divisions': 21, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 100, 'number_subgroup_classes': 100, 'number_subgroups': 676, 'old_label': None, 'order': 324, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 39], [3, 80], [6, 204]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5, 48, 18, 216]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 5], [6, 2], [8, 1], [12, 1]], 'representations': {'PC': {'code': 78354702299892394586715, 'gens': [1, 3, 4, 6], 'pres': [6, -2, -3, -3, -2, -3, -3, 12, 866, 386, 2163, 69, 2164, 455]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [15477644, 9861463, 11495158, 36731395, 32319731, 35376673]}, 'Perm': {'d': 12, 'gens': [1, 1502664, 48285480, 11420784, 3, 95968104]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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:S_3^2', '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [2, 8, 6, 3, 3, 3, 3], 'aut_gens': [[1, 2, 16, 96, 288, 864, 2592], [1097, 6374, 6288, 6080, 5760, 4320, 2592], [7277, 6754, 6512, 1952, 2304, 2592, 6048], [5377, 3778, 6928, 1824, 7488, 1728, 5184], [2689, 7106, 2608, 96, 2880, 864, 2592], [3745, 3746, 7216, 1824, 288, 864, 2592], [6049, 6914, 5200, 96, 288, 864, 2592], [4321, 1730, 4336, 96, 288, 864, 2592]], 'aut_group': '7776.cw', 'aut_hash': 8212187940285761847, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7776, 'aut_permdeg': 225, 'aut_perms': [11231268714318088878867744082404562360510990226514671598596467852112105308964526497545494917507570819614888216266281021383412910480090444199656726890735562817848266391167807026829329218021463950310425797760350239940095047612049778933921736396702327347419707361994649417221333711014234975279434550226154902335999269557510670199033126742173541646154396175709579011930117732152429908326556194085736976454255994395052238842347447493575891, 271488471834983269251046137259492763515915003995221494409717119951333197286100364036583659752021311128779764125120279156946945205540495446662436510353065328138944162389991798972755511793960772457380697811405572817014850016201186111969749740903678315688645256938866510402925918126491118337594428275942698973508801470662403635347663373648639380918765342617885808996318545785064680948853159290733391847567884710391620582840464911809613, 6169312398305143092880081480977234822356588345112915419258815466519754025277236050300712428975881871315929102291914557669783126055010574915921660053914690669158290725252478109249711238085129448289525401518439646129698905163763360333425960129005745833144809500253890917808080045450556360799420199951186199130347478074560110181953470185891524441927750217121885828621329024456898888157515129519824128992464710101683593780426694236976976, 5160807927055650240048602484598268751628988099794258916847138896204896222050753055686540830974092001665916586019406009728228267970377695284140126508330505338947041172202858009481450511200011246328120761669272882912975504080554774825027858221463241842328643456884123844238140995004515578246062746769710124390745648587165454690645734100808029572295824778040877442666243294697006392660416935629001754351139000759969093765713021998559865, 3335599326886604158668422929820389211018687167768085779599077884948032725991418652210822954149586460165405276551920503814319495672060333618660115145504829310208300053620965493027063869834522533680648983956237759341716273355226771624131867878214687790362402925779275296742246745490046015900545973564448959322573525663544776613650913261282832715779482523144891560232835677907843760793064455397557417231875980121274207601354222139212107, 7467909311837943226255482973846087289716225656404315701417180463493461461557530193348685960358533845102208202111532370752658946761862206523300216377506334703130392539196821176810183515569936546097786898467268458968249711619142355858929024767626300291927708105223202590881911752874230215364930662205540228609825692021046729101094542237064340996936660595639838883979038825148739084245879543940344138880279620797018135618508734269662383, 3302816221221346167156239570433023206807198695911642394148280340914675319075148046167865021574676711696534232922260963903526143177151461033136319250086932922362131365492203886266752913381077024331008938579790627142836879708503323697457810436676660229198424190801765952193471637272914891703511418952589999848488134513895951846689129715050975465644604538067691783267654912945124854336555689378797042407906406051827279503106181467985778], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 36, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [3, 8, 1, 1], [3, 18, 1, 1], [3, 24, 1, 1], [3, 48, 1, 1], [3, 144, 1, 1], [4, 162, 1, 1], [4, 324, 1, 1], [4, 486, 1, 1], [4, 972, 1, 1], [6, 72, 1, 1], [6, 162, 1, 1], [6, 216, 1, 6], [6, 432, 1, 2], [8, 162, 1, 2], [8, 486, 1, 2], [12, 324, 1, 1], [12, 648, 1, 1], [24, 324, 1, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4:(S_3\\times \\SD_{16})', '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': 24, 'autcentquo_group': '7776.cw', 'autcentquo_hash': 8212187940285761847, 'autcentquo_nilpotent': False, 'autcentquo_order': 7776, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4:(S_3\\times \\SD_{16})', 'cc_stats': [[1, 1, 1], [2, 27, 2], [2, 36, 1], [2, 81, 1], [2, 108, 1], [3, 8, 1], [3, 18, 1], [3, 24, 1], [3, 48, 1], [3, 144, 1], [4, 162, 1], [4, 324, 1], [4, 486, 1], [4, 972, 1], [6, 72, 1], [6, 162, 1], [6, 216, 6], [6, 432, 2], [8, 162, 2], [8, 486, 2], [12, 324, 1], [12, 648, 1], [24, 324, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '7776.cw', 'commutator_count': 1, 'commutator_label': '972.406', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 75, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 36, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [3, 8, 1, 1], [3, 18, 1, 1], [3, 24, 1, 1], [3, 48, 1, 1], [3, 144, 1, 1], [4, 162, 1, 1], [4, 324, 1, 1], [4, 486, 1, 1], [4, 972, 1, 1], [6, 72, 1, 1], [6, 162, 1, 1], [6, 216, 1, 6], [6, 432, 1, 2], [8, 162, 2, 1], [8, 486, 2, 1], [12, 324, 1, 1], [12, 648, 1, 1], [24, 324, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 17418240, 'exponent': 24, 'exponents_of_order': [5, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 4], [48, 1, 2]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '864.4662', 'hash': 8212187940285761847, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 8, 6, 3, 3, 3, 3], 'inner_gens': [[1, 3046, 16, 1056, 2304, 864, 6912], [6765, 2, 6800, 4544, 576, 2592, 6048], [1, 7426, 16, 1824, 4896, 1728, 5184], [1921, 6306, 880, 96, 2016, 864, 2592], [1441, 578, 6640, 960, 288, 864, 2592], [1, 6050, 1744, 96, 288, 864, 2592], [6049, 6914, 5200, 96, 288, 864, 2592]], 'inner_hash': 8212187940285761847, 'inner_nilpotent': False, 'inner_order': 7776, 'inner_split': True, 'inner_tex': 'C_3^4:(S_3\\times \\SD_{16})', 'inner_used': [1, 2, 3], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 10], [4, 3], [8, 4], [16, 2], [24, 4], [48, 2]], 'label': '7776.cw', 'linC_count': 4, 'linC_degree': 24, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 4, 'linQ_dim': 24, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 24, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': None, 'name': 'C3^4:(S3*SD16)', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 33, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 33, 'number_divisions': 30, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 473, 'number_subgroup_classes': 473, 'number_subgroups': 31930, 'old_label': None, 'order': 7776, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 279], [3, 242], [4, 1944], [6, 2394], [8, 1296], [12, 972], [24, 648]], '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': 7, '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, 6], [4, 3], [8, 5], [16, 2], [24, 4], [48, 2]], 'representations': {'PC': {'code': '58455272015971747385343453536179987028011274495789361867144663252820901170989768250196190849712281178343865739492018193190161126809688095', 'gens': [1, 2, 5, 7, 8, 9, 10], 'pres': [10, -2, -2, -2, -2, -2, -3, 3, -3, -3, 3, 60921, 51, 59882, 82, 38723, 170014, 77424, 5934, 144, 85455, 28825, 515, 73926, 159056, 76186, 1716, 8026, 184327, 23057, 24527, 11577, 1747, 116658, 136108, 19478, 9768, 691209, 302419, 172829, 64839, 32449]}, 'Perm': {'d': 27, 'gens': [9167122780557201012549133103, 6632969369720355530109885969, 3257383389079646387271672220]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:(S_3\\times \\SD_{16})', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
