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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '17496.ig', 'ambient_counter': 215, 'ambient_order': 17496, 'ambient_tex': 'C_3^4:S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': True, 'core_order': 2916, 'counter': 17, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '17496.ig.6.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '6.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '6.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 6, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '2916.lt', 'subgroup_hash': 6948860401795433662, 'subgroup_order': 2916, 'subgroup_tex': 'C_3^5:D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '17496.ig', 'aut_centralizer_order': None, 'aut_label': '6.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1944.b1', 'complements': ['2916.be1', '2916.bg1', '2916.bi1', '2916.bk1', '2916.cf1', '2916.cj1', '2916.ch1', '2916.cl1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1', '3.a1'], 'contains': ['12.c1', '12.f1', '12.k1', '18.a1', '18.k1', '18.m1', '18.ba1'], 'core': '6.b1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [96005, 488211443649364, 361031173626964, 3, 2971211522227200, 2866516142567764, 2866516142515204, 2866871122617600], 'label': '17496.ig.6.b1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '6.b1', 'normal_contained_in': ['2.a1', '3.a1'], 'normal_contains': ['12.c1', '12.f1', '12.k1', '18.a1'], 'normalizer': '1.a1', 'old_label': '6.b1', 'projective_image': '17496.ig', 'quotient_action_image': '2.1', 'quotient_action_kernel': '3.1', 'quotient_action_kernel_order': 3, 'quotient_fusion': None, 'short_label': '6.b1', 'subgroup_fusion': None, 'weyl_group': '1944.3575'}
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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': 5, 'aut_exponent': 24, 'aut_gen_orders': [6, 24, 6, 12], 'aut_gens': [[174356939664, 93405312000, 174436416690, 3, 87218208000, 43545600, 144, 45360], [93405633987, 174356582400, 80191317, 45604, 180667065600, 93448857600, 80883, 80788], [80190723, 43545600, 174356628454, 80880, 6266937600, 93485145600, 80644, 244], [43729890, 79833600, 174356628210, 80788, 6310483200, 93448857600, 80883, 45604], [93449179444, 174436416000, 174356801280, 144, 87261753600, 174400128000, 45360, 3]], 'aut_group': None, 'aut_hash': 7791745375834213671, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 139968, 'aut_permdeg': 36, 'aut_perms': [42618432066393486161397247462072779959086, 262190067912347966149362535574214586975649, 263405786544940354835383993713069662440116, 44607373718449944370126309614168249973060], 'aut_phi_ratio': 144.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 81, 1, 1], [3, 1, 2, 1], [3, 2, 4, 1], [3, 2, 8, 1], [3, 3, 6, 1], [3, 6, 1, 1], [3, 6, 24, 1], [3, 12, 4, 1], [3, 18, 1, 1], [3, 36, 4, 1], [6, 9, 2, 2], [6, 9, 6, 1], [6, 18, 4, 1], [6, 18, 8, 1], [6, 18, 24, 1], [6, 27, 6, 1], [6, 54, 1, 1], [6, 81, 2, 1], [6, 81, 6, 1], [6, 162, 1, 1], [9, 18, 2, 1], [9, 36, 8, 1], [18, 162, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3\\times (C_3\\times \\PSU(3,2)).S_3^3', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 3, 'autcent_group': '3.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 3, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': None, 'autcentquo_hash': 7640073310130559848, 'autcentquo_nilpotent': False, 'autcentquo_order': 46656, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3:S_3.C_6^2.S_3^2.C_2', 'cc_stats': [[1, 1, 1], [2, 9, 2], [2, 81, 1], [3, 1, 2], [3, 2, 12], [3, 3, 6], [3, 6, 25], [3, 12, 4], [3, 18, 1], [3, 36, 4], [6, 9, 10], [6, 18, 36], [6, 27, 6], [6, 54, 1], [6, 81, 8], [6, 162, 1], [9, 18, 2], [9, 36, 8], [18, 162, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '972.788', 'commutator_count': 1, 'commutator_label': '243.62', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 306, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['162.10', 1], ['18.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 81, 1, 1], [3, 1, 2, 1], [3, 2, 1, 4], [3, 2, 2, 4], [3, 3, 2, 3], [3, 6, 1, 1], [3, 6, 2, 12], [3, 12, 1, 4], [3, 18, 1, 1], [3, 36, 1, 4], [6, 9, 2, 5], [6, 18, 1, 4], [6, 18, 2, 16], [6, 27, 2, 3], [6, 54, 1, 1], [6, 81, 2, 4], [6, 162, 1, 1], [9, 18, 2, 1], [9, 36, 2, 4], [18, 162, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 58968, 'exponent': 18, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '324.166', 'hash': 6948860401795433662, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 6, 3, 2, 3, 3, 3], 'inner_gens': [[174356939664, 93405312000, 174436599744, 144, 180623520000, 43545600, 45360, 3], [174356939664, 93405312000, 174436416690, 3, 6266937600, 43545600, 144, 45360], [174356801974, 93405312000, 174436416690, 144, 180667065600, 43545600, 3, 45360], [174356859028, 93405312000, 174436416454, 3, 87218208000, 43545600, 144, 45360], [93405669264, 174356582400, 93448858290, 3, 87218208000, 79833600, 144, 45360], [174356939664, 93405312000, 174436416690, 3, 87181920000, 43545600, 144, 45360], [174356939523, 93405312000, 174436416597, 3, 87218208000, 43545600, 144, 45360], [174356904480, 93405312000, 174436416690, 3, 87218208000, 43545600, 144, 45360]], 'inner_hash': 788, 'inner_nilpotent': False, 'inner_order': 972, 'inner_split': False, 'inner_tex': 'C_3^3:S_3^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 12], [2, 30], [3, 24], [4, 12], [6, 50], [12, 4]], 'label': '2916.lt', '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': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^5:D6', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 27, 'number_characteristic_subgroups': 26, 'number_conjugacy_classes': 132, 'number_divisions': 78, 'number_normal_subgroups': 58, 'number_subgroup_autclasses': 291, 'number_subgroup_classes': 1240, 'number_subgroups': 16160, 'old_label': None, 'order': 2916, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 99], [3, 404], [6, 1764], [9, 324], [18, 324]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [6, 6, 6, 6, 3, 3], 'outer_gen_pows': [87218208000, 0, 0, 87218208000, 87218208000, 0], 'outer_gens': [[174436773264, 93448857600, 93405393238, 45508, 87218208000, 93485145600, 45603, 80787], [93449214864, 174436416000, 93405357813, 80880, 87218208000, 43545600, 80644, 244], [93405634080, 174356582400, 93448903653, 80883, 87218208000, 79833600, 80788, 45604], [93485467680, 174400128000, 43626694, 45504, 87218208000, 93448857600, 45363, 147], [80190864, 43545600, 93448903413, 80880, 87218208000, 174400128000, 80644, 244], [174356939664, 93405312000, 174436461813, 80880, 87218208000, 43545600, 80644, 244]], 'outer_group': '144.188', 'outer_hash': 188, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 9, 'outer_perms': [41070, 768, 31, 131809, 121728, 48], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times S_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 14], [4, 14], [6, 14], [8, 4], [12, 28]], 'representations': {'PC': {'code': '391394161083480753899192144549215320228817080695', 'gens': [1, 3, 4, 5, 7, 8], 'pres': [8, 2, 3, 3, 3, 2, 3, 3, 3, 16, 290, 90484, 21492, 620, 116, 10661, 3613, 1029, 710, 2343]}, 'Perm': {'d': 15, 'gens': [174356939664, 93405312000, 174436416690, 3, 87218208000, 43545600, 144, 45360]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^5:D_6', 'transitive_degree': 54, '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [6, 6, 6, 6, 4, 6], 'aut_gens': [[358477216929361, 732484456705585, 753501170919147], [1576184985542403, 732484456917363, 91544474247986], [1576097328206162, 732311058246482, 1133935737678243], [358652048884561, 2973839804128081, 1134195835631907], [795246639386429, 70876473343925, 1805917026912748], [1176919868055843, 2973839804103147, 2952823169772722], [133845102036025, 2868010627509481, 1428162035022243]], 'aut_group': '13872.e', 'aut_hash': 6831934589354346940, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 139968, 'aut_permdeg': 173, 'aut_perms': [23662795993880940803523727486323957916473953032524383125822033588936285632313026959290989274780569541042739773573286409846228683190773268622961638048152775011398140719626682875551914331917254650255392467846576659541999688024104450801242294669748270636100517558555230291756035758456685433758187657306882306348020396, 7669231622983187717684856467244639441023237560403020100501229960544254704093512496805085803423558562403931234874747729977069289014263809892848453774068235322350767057479017647024520123636748936447632839866956496443103014227876804300456807439735349920446276414194196128296839917606942606029537301022987424162153635, 23695017472248542912863806389500349961752231853198032450729485235397753953634296135919233681262717858294335135693054127836435381173865658700980927256178712093510284586984120214482545485269685435070318129601361621584034386455171370351780548686145647195143236522185871605237603441410535216761083078726894347993982645, 15640311343860873874700195583702172586333638840314718701798726849574815427276203558320919350384846131308523151914863869309573277451649411106391499074306243091910922709747863213529801977601006390716073856082188267319171841132432973171967076798711804528338945838381311509289247812280247047401207563215593743907479087, 23016997127733408089560081300813584128096632439710522583173943812760890716146372104258775886597719677462062348027642622756049447517078502436924929945766852267061988528981929277716078707916264855905916666156967715083246098808479663812523322379073632044559180872193049264614136466457346209982633893385558274267453980, 17906380238018095266273828037961247532247734319075507350257832653201224811174666832833918911785999712182128412731400538858631131094852138287551444382477940950175312450042527355005586231093927347512524661678249163820168065898829151537712512555231920521926125072317963460084443741484564803516895417008578048958732084], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 81, 1, 1], [2, 81, 2, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 2], [3, 2, 4, 1], [3, 4, 1, 1], [3, 4, 2, 3], [3, 4, 4, 2], [3, 6, 1, 1], [3, 6, 2, 1], [3, 6, 3, 1], [3, 6, 6, 1], [3, 12, 2, 1], [3, 12, 4, 1], [3, 12, 6, 2], [3, 12, 12, 2], [3, 18, 1, 1], [3, 18, 2, 1], [3, 24, 1, 1], [3, 24, 2, 1], [3, 36, 2, 1], [3, 36, 4, 1], [3, 72, 1, 1], [3, 72, 2, 1], [6, 9, 2, 2], [6, 18, 1, 2], [6, 18, 2, 3], [6, 18, 3, 1], [6, 18, 4, 1], [6, 18, 6, 1], [6, 36, 1, 1], [6, 36, 2, 3], [6, 36, 4, 2], [6, 36, 6, 2], [6, 36, 12, 2], [6, 54, 1, 1], [6, 54, 2, 1], [6, 54, 3, 1], [6, 54, 6, 1], [6, 81, 2, 1], [6, 81, 4, 2], [6, 162, 1, 2], [6, 162, 2, 6], [6, 162, 3, 1], [6, 162, 4, 4], [6, 162, 6, 1], [6, 324, 2, 2], [6, 324, 4, 2], [9, 36, 1, 1], [9, 36, 2, 1], [9, 72, 2, 2], [9, 72, 4, 2], [18, 324, 1, 1], [18, 324, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{17}^2:(C_2\\times C_{24})', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 5851352362639871032, 'autcentquo_nilpotent': False, 'autcentquo_order': 69984, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^5.C_6.C_6.C_2^3', 'cc_stats': [[1, 1, 1], [2, 9, 2], [2, 81, 5], [3, 1, 2], [3, 2, 9], [3, 4, 15], [3, 6, 12], [3, 12, 42], [3, 18, 3], [3, 24, 3], [3, 36, 6], [3, 72, 3], [6, 9, 4], [6, 18, 21], [6, 36, 51], [6, 54, 12], [6, 81, 10], [6, 162, 39], [6, 324, 12], [9, 36, 3], [9, 72, 12], [18, 324, 3]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '5832.ir', 'commutator_count': 1, 'commutator_label': '729.479', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 215, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['5832.ir', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 81, 1, 5], [3, 1, 2, 1], [3, 2, 1, 3], [3, 2, 2, 3], [3, 4, 1, 3], [3, 4, 2, 6], [3, 6, 1, 4], [3, 6, 2, 4], [3, 12, 1, 8], [3, 12, 2, 17], [3, 18, 1, 1], [3, 18, 2, 1], [3, 24, 1, 1], [3, 24, 2, 1], [3, 36, 1, 2], [3, 36, 2, 2], [3, 72, 1, 1], [3, 72, 2, 1], [6, 9, 2, 2], [6, 18, 1, 7], [6, 18, 2, 7], [6, 36, 1, 9], [6, 36, 2, 21], [6, 54, 1, 4], [6, 54, 2, 4], [6, 81, 2, 5], [6, 162, 1, 13], [6, 162, 2, 13], [6, 324, 1, 4], [6, 324, 2, 4], [9, 36, 1, 1], [9, 36, 2, 1], [9, 72, 1, 2], [9, 72, 2, 5], [18, 324, 1, 1], [18, 324, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 7547904, 'exponent': 18, 'exponents_of_order': [7, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 24]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '1944.3934', 'hash': 1991687623141296861, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 6, 6], 'inner_gens': [[358477216929361, 70876473294385, 1868848772495164], [2177564748359761, 732484456705585, 795334296738364], [2513457471693629, 732311058175925, 753501170919147]], 'inner_hash': 4679729500980651093, 'inner_nilpotent': False, 'inner_order': 5832, 'inner_split': True, 'inner_tex': 'C_3^3:S_3^3', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 24], [2, 48], [4, 48], [6, 48], [8, 15], [12, 84], [24, 3]], 'label': '17496.ig', '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': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^4:S3^3', '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 88, 'number_characteristic_subgroups': 60, 'number_conjugacy_classes': 270, 'number_divisions': 171, 'number_normal_subgroups': 128, 'number_subgroup_autclasses': 3389, 'number_subgroup_classes': 10609, 'number_subgroups': 312824, 'old_label': None, 'order': 17496, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 423], [3, 1214], [6, 13914], [9, 972], [18, 972]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[358477216999225, 732484456755121, 753501171003963], [358477216999225, 732484456706281, 753501170919843], [358477217089468, 71051348975164, 753501170873525]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [24, 40320, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 24], [4, 28], [6, 16], [8, 21], [12, 32], [16, 6], [24, 35], [48, 1]], 'representations': {'PC': {'code': '20485277682706432621590233005481730769303710344323141258322473236487343726739941644945846657117262427784869132920754925324320625356684330947935', 'gens': [1, 3, 5, 7, 8, 9], 'pres': [10, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 20, 333542, 258222, 82, 963, 487804, 422114, 1524, 144, 522725, 1465, 211686, 5066, 155527, 492497, 57627, 30767, 17337, 5827, 1517, 1108088, 349938, 207388, 10848, 13018, 4388, 1158, 358, 1166409, 194429]}, 'Perm': {'d': 18, 'gens': [358477216929361, 732484456705585, 753501170919147]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:S_3^3', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [5]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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], [3, 1, 2], [6, 1, 2]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 6]], 'label': '6.2', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6', 'ngens': 2, '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': 6, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5]], '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': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2]], 'representations': {'PC': {'code': 21, 'gens': [1], 'pres': [2, -2, -3, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [73]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 56]}, 'Perm': {'d': 5, 'gens': [24, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6', 'transitive_degree': 6, 'wreath_data': None, 'wreath_product': False}
