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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1600000000.hv', 'ambient_counter': 204, 'ambient_order': 1600000000, 'ambient_tex': 'C_5^4.D_5^4.(C_2^5.D_4)', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 400000000, 'counter': 12, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1600000000.hv.4._.D', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.D', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': None, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '400000000.gwz', 'subgroup_hash': None, 'subgroup_order': 400000000, 'subgroup_tex': 'C_5^4.D_5^4.C_2^4:C_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1600000000.hv', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [26852355356192795859766858499773, 1046417228260009819150419232626000717656145383, 291988463372730454495225109632590433070010189513, 26584446232900328426534563092393984000, 79855162553820297098160, 17988815623552264314662762246423, 565245613963954746852261267443672674577670077413, 409050, 1596967351909073366541136685158980668639114183, 35367047974958807818174464000000, 9157944981056567706147942604896, 35367049913860063234735480627200, 9209579902859282340432265252945920023, 9157942499262960773190185164800, 81591529505720969592979629251469868458860544000, 19939418763395760, 911748359133191228768138133372690432000, 1180931205316361001554690628137513164800], 'label': '1600000000.hv.4._.D', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '4.D', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4._.D', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.21', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[419747434340383859495717916127989995316211987319, 187863751097997441051151525865474368765121471949, 668861061273988753998449774218178570380686529956], [419747434340383859495717268652479853336403097686, 147054223199886186391768604451289479756237227464, 668861060709564885937021706213857275424016008403], [480445585472692041149970738607738213322038183735, 523376272515002983094726757293119380244185500866, 209901061269372672705554594953505539130596485669], [20425788862767498648570715417958860573274447410, 125101037002005307639950011599753053668463945106, 670430496869235607747422932109667595833481006603], [419747434340383859495715977226734578942962943229, 187863751106162719834750176247458385861314305210, 710193611511700446610055579848577319210452167004], [501338963237746317012956705563946833322995987223, 187863752278928637209569279623553659580514716306, 291988457263197261366591433165958746345831187869], [419747434340383833253541893144251145605307569206, 565245614858870954356172614920780237849980134989, 670430500546345038960648410326342029978290716149], [440681730798812340409945758223646195203954830775, 187863751106162719834750176247201579469732399413, 689259315660849308099417797098596498708547913730], [419747435521315038569093725502104887950880100406, 166928711529675560994151515517357954369315635826, 710689478338182099136190975898369400669240113195], [419747434340383859495715977226734578849119590429, 125610295874545962127109064796694760218053116170, 668861061004805907815279455115924627458525499626], [480445585472692041149969447181992938834753874429, 187863751097997441051151525865345944518447060396, 629097950317579796035632557367377144319642974026], [419747434340383859495717916127989995316211987319, 145994414497876789771854283846458767882278271890, 650032247045191123133032013413280908555368676356], [419747435521315055654131997724857500838690150423, 106272223649967848276255753028961945146238029970, 650032246776008267791918550995649176060192321989], [419747434340383859495715977226734578942963987223, 187863751097997468491103731287583472558724425036, 668861060700618437481276203795313022098790158730], [419747434340383859495715434334383062172162841719, 125101037002005307639951849341327112223099385586, 650032247045191131060128325944513023956605095690], [419747434340383859495717916127989995222367590423, 187863751097997468491102540919337671129424627116, 668861060700618437481277394060992796227242299589], [480445585472692041149971386083248355115555609719, 187863751097997441051153360285368901313955135949, 668861061273988753998449774218050146133933585130], [419747434340383850337775416865029222380894546429, 187863751097997450209094025128050999184249549970, 650032247349362138631785057101062192297642950436], [419747434340383859495717916127989995129920281719, 187863751097997441051151525865345944424523071949, 668861061273988753998449774218178570218265070730], [459510546208541159506681935569202767163324967709, 166928711529675560994151515517485290213705644690, 650032246776008267791919195048552773031028289930], [440681731371385050702535479318091700480146790423, 125101037314341638226198253595055422198086140156, 670430500242955176049000975718444853762542023035], [419747434609566714803758349464122990219451161623, 106272223068185576972992740621090746941690175903, 711240028149231700988178800549189956387699026119], [419747434912956569788305155428828567542197503319, 523926822942823234616352656558487089268766929370, 229776299400387548232469167256538342981985206903], [461079986054267743302531446046701564525812079366, 166928711529675560994150325251678180054491955826, 670430500546345039875716762598141103379335662770], [1596973748263264876848116659486870957898111989, 754707491220506741166648944511990639141099547650, 650032246471820814542955945787327249804636984810]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 25600000000, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 50, 2, 1], [2, 625, 2, 1], [2, 1250, 1, 2], [2, 2500, 1, 2], [2, 2500, 2, 1], [2, 5000, 4, 1], [2, 10000, 2, 1], [2, 31250, 2, 1], [2, 390625, 1, 1], [4, 2500, 1, 2], [4, 2500, 2, 1], [4, 12500, 4, 2], [4, 62500, 1, 4], [4, 62500, 2, 6], [4, 62500, 4, 2], [4, 312500, 4, 2], [4, 390625, 1, 2], [4, 390625, 2, 1], [4, 500000, 16, 1], [4, 781250, 1, 2], [4, 781250, 2, 2], [4, 1562500, 1, 4], [4, 1562500, 2, 2], [4, 3125000, 4, 1], [4, 6250000, 2, 1], [5, 16, 2, 1], [5, 32, 2, 2], [5, 64, 1, 1], [5, 64, 2, 2], [5, 64, 4, 1], [5, 64, 8, 1], [5, 128, 2, 1], [5, 128, 4, 2], [5, 128, 8, 1], [5, 256, 2, 4], [5, 256, 4, 1], [5, 256, 16, 1], [5, 256, 32, 1], [5, 512, 2, 1], [5, 512, 4, 3], [5, 512, 8, 7], [5, 512, 16, 3], [5, 1024, 4, 2], [5, 1024, 8, 7], [5, 1024, 16, 7], [5, 1024, 32, 2], [5, 1024, 64, 1], [8, 6250000, 4, 2], [10, 400, 2, 3], [10, 800, 2, 3], [10, 800, 4, 1], [10, 1600, 2, 1], [10, 1600, 4, 2], [10, 1600, 8, 1], [10, 3200, 4, 3], [10, 3200, 8, 3], [10, 6400, 2, 4], [10, 6400, 4, 1], [10, 6400, 32, 1], [10, 10000, 2, 3], [10, 10000, 4, 4], [10, 12800, 4, 2], [10, 12800, 8, 4], [10, 12800, 16, 3], [10, 20000, 2, 4], [10, 20000, 4, 2], [10, 40000, 1, 1], [10, 40000, 2, 8], [10, 40000, 4, 3], [10, 40000, 8, 3], [10, 40000, 16, 1], [10, 80000, 2, 1], [10, 80000, 4, 2], [10, 80000, 8, 7], [10, 160000, 2, 4], [10, 160000, 4, 2], [10, 160000, 16, 1], [10, 160000, 32, 2], [10, 250000, 2, 1], [10, 320000, 2, 1], [10, 320000, 4, 1], [10, 320000, 8, 3], [10, 500000, 2, 1], [20, 10000, 4, 4], [20, 20000, 4, 2], [20, 20000, 8, 1], [20, 40000, 2, 6], [20, 40000, 4, 1], [20, 50000, 4, 6], [20, 80000, 4, 4], [20, 80000, 8, 4], [20, 100000, 8, 2], [20, 160000, 4, 2], [20, 200000, 4, 4], [20, 250000, 2, 8], [20, 250000, 4, 8], [20, 400000, 8, 2], [20, 500000, 2, 2], [20, 500000, 4, 1], [20, 500000, 8, 1], [20, 1000000, 1, 2], [20, 1000000, 2, 5], [20, 1000000, 4, 1], [20, 1250000, 4, 2], [20, 2000000, 32, 1], [20, 4000000, 32, 1]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 50, 2], [2, 625, 2], [2, 1250, 2], [2, 2500, 4], [2, 5000, 4], [2, 10000, 2], [2, 31250, 2], [2, 390625, 1], [4, 2500, 4], [4, 12500, 8], [4, 62500, 24], [4, 312500, 8], [4, 390625, 4], [4, 500000, 16], [4, 781250, 6], [4, 1562500, 8], [4, 3125000, 4], [4, 6250000, 2], [5, 16, 2], [5, 32, 4], [5, 64, 17], [5, 128, 18], [5, 256, 60], [5, 512, 118], [5, 1024, 304], [8, 6250000, 8], [10, 400, 6], [10, 800, 10], [10, 1600, 18], [10, 3200, 36], [10, 6400, 44], [10, 10000, 22], [10, 12800, 88], [10, 20000, 16], [10, 40000, 69], [10, 80000, 66], [10, 160000, 96], [10, 250000, 2], [10, 320000, 30], [10, 500000, 2], [20, 10000, 16], [20, 20000, 16], [20, 40000, 16], [20, 50000, 24], [20, 80000, 48], [20, 100000, 16], [20, 160000, 8], [20, 200000, 16], [20, 250000, 48], [20, 400000, 16], [20, 500000, 16], [20, 1000000, 16], [20, 1250000, 8], [20, 2000000, 32], [20, 4000000, 32]], 'center_label': '1.1', 'center_order': None, 'central_product': None, 'central_quotient': '400000000.gwz', 'commutator_count': None, 'commutator_label': None, 'complements_known': False, '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', '2.1', '5.1', '5.1', '5.1', '5.1', '5.1', '5.1', '5.1', '5.1'], 'composition_length': 18, 'conjugacy_classes_known': False, 'counter': 4654, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': None, 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 50, 1, 2], [2, 625, 1, 2], [2, 1250, 1, 2], [2, 2500, 1, 4], [2, 5000, 1, 4], [2, 10000, 1, 2], [2, 31250, 1, 2], [2, 390625, 1, 1], [4, 2500, 1, 2], [4, 2500, 2, 1], [4, 12500, 2, 4], [4, 62500, 1, 4], [4, 62500, 2, 10], [4, 312500, 2, 4], [4, 390625, 2, 2], [4, 500000, 2, 8], [4, 781250, 1, 2], [4, 781250, 2, 2], [4, 1562500, 1, 6], [4, 1562500, 2, 1], [4, 3125000, 1, 4], [4, 6250000, 1, 2], [5, 16, 1, 2], [5, 32, 1, 4], [5, 64, 1, 17], [5, 128, 1, 18], [5, 256, 1, 60], [5, 512, 1, 118], [5, 1024, 1, 304], [8, 6250000, 2, 4], [10, 400, 1, 6], [10, 800, 1, 10], [10, 1600, 1, 18], [10, 3200, 1, 36], [10, 6400, 1, 44], [10, 10000, 1, 22], [10, 12800, 1, 88], [10, 20000, 1, 16], [10, 40000, 1, 69], [10, 80000, 1, 66], [10, 160000, 1, 96], [10, 250000, 1, 2], [10, 320000, 1, 30], [10, 500000, 1, 2], [20, 10000, 2, 8], [20, 20000, 2, 8], [20, 40000, 1, 4], [20, 40000, 2, 6], [20, 50000, 2, 12], [20, 80000, 1, 8], [20, 80000, 2, 20], [20, 100000, 2, 8], [20, 160000, 2, 4], [20, 200000, 2, 8], [20, 250000, 2, 24], [20, 400000, 2, 8], [20, 500000, 2, 8], [20, 1000000, 1, 4], [20, 1000000, 2, 6], [20, 1250000, 2, 4], [20, 2000000, 2, 16], [20, 4000000, 2, 16]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 40, 'exponents_of_order': [10, 8], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': None, 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '400000000.gwz', 'hash': 6085162300289467760, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': None, 'inner_split': None, 'inner_tex': 'C_5^4.D_5^4.C_2^4:C_4', 'inner_used': None, 'irrC_degree': None, 'irrQ_degree': None, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': None, 'label': '400000000.gwz', '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': None, 'name': 'C5^4.D5^4.C2^4:C4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 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, 1, 0, 0, 0, 0, 3, 0, 0, 0, 11, 0, 0, 0, 19, 0, 0, 0, 35, 0, 0, 51, 0, 0, 51, 0, 35, 0, 19, 7, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 258, 'number_characteristic_subgroups': 92, 'number_conjugacy_classes': 1468, 'number_divisions': 1276, 'number_normal_subgroups': 240, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 400000000, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 506975], [4, 55860000], [5, 390624], [8, 50000000], [10, 36602400], [20, 256640000]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': None, 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 64, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': None, 'perfect': False, 'permutation_degree': 40, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': None, 'representations': {'PC': {'code': '2921310229936893518673515939291875246067782618780276221564717165284572749416431378105757496853282884653666512487517628561646601690508930427053136767480429441849959950106922453563487264775756504725631312546767102446389069676769440711832016403104605567084076344445470161671770023253302614880796494190588909541859351992408169628325468035242771422526019584634934385667609986459067865713176246040327532807725863275571709671597795006834216357297121297685431883645616223089137541946786769810202929689707075587115189483265800381976478969544864022638943688498043037863470899730471357426227949551148571550066203421591656042553501600431092726168223285385128522637588111103903662815313869378369409948521657424889713745905657317988902920482662048284303165884716604159170974943438680360748185926048661615650961807270310610715864746250043973461538631036245965731294041360923760693734223239326274311355055047476402784876104160135440086661974650921188557330993154096594839811467594745255194057454516124818244483053883828359614327102974676695263589939541015435073279701258528211592902560812414763226826348560532809053188800161859063838732034910244863', 'gens': [1, 3, 5, 7, 10, 12, 14, 15, 16, 17, 18], 'pres': [18, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 5, 2, 5, 5, 5, 5, 5, 5, 36, 1983393811, 8063332922, 2805539024, 146, 14043721539, 14234407221, 1013763801, 32489884804, 13901455462, 6024398800, 3133566958, 256, 14327262725, 19568053463, 2806342025, 1097391083, 293786015, 2207586534, 10480163928, 11012454474, 6110549484, 573621126, 1308900444, 366, 1902592519, 12105096217, 7648575019, 2006929213, 2419300303, 391719841, 421, 1439078408, 20352425498, 3478484780, 15614, 5264, 5282, 27117740169, 31717915227, 3803070285, 7254230463, 4125614481, 979207299, 653353317, 146608335, 60391053, 531, 67288320010, 6107904028, 3592512046, 4308670144, 3033391762, 2055255940, 269295958, 134655976, 21456949259, 36397558685, 14296706543, 6188659265, 4007404883, 2252534501, 745934519, 481874537, 72478955, 31534013, 7345271, 641, 19080681996, 29323008030, 5601024048, 1497666, 4118484, 1123302, 2059320, 1123338, 18894, 43868805133, 41852160031, 15442560049, 8064067, 4032085, 4032103, 2016121, 100975, 41900544014, 45097413152, 22521058610, 8523360068, 73440086, 47520104, 382860122, 29970140, 70578176, 54212, 22649978895, 35544342561, 11748593715, 8699904069, 27648087, 71424105, 383040123, 26784141, 72201777, 293973, 75439134736, 26697692194, 14100532, 2252160070, 1226448088, 154224106, 230724124, 154224142, 18543778, 1542454, 104337745937, 1542924323, 13279417397, 9958464071, 3465504089, 3352752107, 460728125, 224532143, 83818979, 8103455]}, 'Perm': {'d': 40, 'gens': [668861061273988753998449774218178570380686529956, 187863751097997441051151525865474368765121471949, 419747434340383859495717916127989995316211987319]}}, 'schur_multiplier': [2, 2, 2, 2, 4, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 25, 'supersolvable': False, 'sylow_subgroups_known': False, 'tex_name': 'C_5^4.D_5^4.C_2^4:C_4', 'transitive_degree': None, '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': '16.10', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[565245613963954746852261267443672674577670077413, 629648495288210494088726564716433284034919127079, 145994415097025670713006457763069174606370181743], [606014222622703549463165931735165924149458846799, 628602078050740897071577681374566612946344356635, 146544593227347934330130213568487431103300261103], [523376277972192564353752598818973059858529616613, 690304984036249960140048509933481928697772806423, 166928711547034000539073297954159964886112238869], [565245613963954747178927271100966114409214730229, 629648495860783187329344787554483889082727127079, 126655969402965021464005580377856924992248474226], [544311318417274651280328187487754503171334077444, 649495455530035136476658019199800057273238433280, 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669370686978672581232698235437974579710959518823, 145994415097025670713008295504259090994927632623], [565245613963954756009397726676962614796107709509, 629648495557393332344797442120750741084338903079, 145994415097025670713007105135910851935049502703], [544311317505526265938032130499408936600790250283, 629648495288210477003687105548783714001513354246, 145994415097025670713008295503772510755713112943], [565245613963954746852261267443801098662002621450, 629648495288210467878814506098435465410050775079, 145994415097025670713007648028519174842643845743], [564695435860216920320583240874421719726055322133, 629648495288210494088726564716433284034919127079, 146544593227347934613641628946884260128745017480], [604967805385234014022977159474922977078534077413, 628602078050740905292674116440160729040320798243, 147054224059271517479360671124868238992391554613], [355344925026829740057583330097675026815937977600, 628051529117232618394153528876374939231630720023, 774596114141561516765800065069186158827393471406], [585629733195389162426273069127086696411323192964, 648986568650566346219804334936155625055180213703, 126655969993693723844376539287593364705071670210]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 25600000000, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 100, 1, 1], [2, 1250, 1, 3], [2, 2500, 1, 2], [2, 5000, 1, 1], [2, 20000, 2, 1], [2, 40000, 1, 1], [2, 62500, 1, 1], [2, 390625, 1, 1], [4, 2500, 1, 2], [4, 5000, 1, 4], [4, 25000, 2, 4], [4, 62500, 1, 4], [4, 125000, 1, 12], [4, 125000, 2, 8], [4, 390625, 1, 2], [4, 625000, 2, 4], [4, 781250, 1, 3], [4, 1000000, 4, 1], [4, 1562500, 1, 6], [4, 2000000, 8, 1], [4, 3125000, 1, 5], [4, 12500000, 2, 1], [4, 25000000, 1, 1], [5, 32, 1, 1], [5, 64, 1, 3], [5, 128, 1, 2], [5, 128, 2, 2], [5, 256, 1, 2], [5, 256, 2, 1], [5, 512, 1, 3], [5, 512, 2, 3], [5, 1024, 1, 1], [5, 1024, 4, 4], [5, 1024, 8, 1], [5, 2048, 1, 3], [5, 2048, 2, 8], [5, 2048, 4, 3], [5, 4096, 2, 8], [5, 4096, 4, 4], [5, 4096, 8, 5], [8, 5000000, 2, 4], [8, 12500000, 4, 1], [8, 25000000, 1, 4], [8, 25000000, 2, 1], [10, 800, 1, 3], [10, 1600, 1, 3], [10, 3200, 1, 2], [10, 3200, 2, 2], [10, 6400, 2, 6], [10, 12800, 1, 6], [10, 12800, 2, 2], [10, 20000, 1, 3], [10, 20000, 2, 4], [10, 25600, 4, 4], [10, 40000, 1, 5], [10, 40000, 2, 2], [10, 51200, 1, 2], [10, 51200, 2, 8], [10, 80000, 1, 8], [10, 80000, 2, 4], [10, 160000, 1, 2], [10, 160000, 2, 5], [10, 160000, 4, 2], [10, 320000, 1, 3], [10, 320000, 2, 7], [10, 320000, 4, 4], [10, 500000, 1, 1], [10, 640000, 1, 1], [10, 640000, 8, 5], [10, 1000000, 1, 1], [10, 1280000, 1, 1], [10, 1280000, 4, 3], [16, 50000000, 8, 1], [20, 20000, 2, 10], [20, 40000, 2, 2], [20, 80000, 1, 12], [20, 80000, 2, 8], [20, 100000, 2, 12], [20, 320000, 1, 4], [20, 320000, 2, 12], [20, 320000, 4, 1], [20, 400000, 2, 12], [20, 500000, 1, 8], [20, 500000, 2, 26], [20, 640000, 2, 2], [20, 1000000, 1, 4], [20, 1600000, 2, 4], [20, 2000000, 1, 12], [20, 2000000, 2, 8], [20, 2500000, 2, 4], [20, 4000000, 4, 2], [20, 8000000, 8, 2], [20, 16000000, 4, 1], [20, 16000000, 16, 1], [40, 20000000, 2, 4]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 100, 1], [2, 1250, 3], [2, 2500, 2], [2, 5000, 1], [2, 20000, 2], [2, 40000, 1], [2, 62500, 1], [2, 390625, 1], [4, 2500, 2], [4, 5000, 4], [4, 25000, 8], [4, 62500, 4], [4, 125000, 28], [4, 390625, 2], [4, 625000, 8], [4, 781250, 3], [4, 1000000, 4], [4, 1562500, 6], [4, 2000000, 8], [4, 3125000, 5], [4, 12500000, 2], [4, 25000000, 1], [5, 32, 1], [5, 64, 3], [5, 128, 6], [5, 256, 4], [5, 512, 9], [5, 1024, 25], [5, 2048, 31], [5, 4096, 72], [8, 5000000, 8], [8, 12500000, 4], [8, 25000000, 6], [10, 800, 3], [10, 1600, 3], [10, 3200, 6], [10, 6400, 12], [10, 12800, 10], [10, 20000, 11], [10, 25600, 16], [10, 40000, 9], [10, 51200, 18], [10, 80000, 16], [10, 160000, 20], [10, 320000, 33], [10, 500000, 1], [10, 640000, 41], [10, 1000000, 1], [10, 1280000, 13], [16, 50000000, 8], [20, 20000, 20], [20, 40000, 4], [20, 80000, 28], [20, 100000, 24], [20, 320000, 32], [20, 400000, 24], [20, 500000, 60], [20, 640000, 4], [20, 1000000, 4], [20, 1600000, 8], [20, 2000000, 28], [20, 2500000, 8], [20, 4000000, 8], [20, 8000000, 16], [20, 16000000, 20], [40, 20000000, 8]], 'center_label': '1.1', 'center_order': None, 'central_product': None, 'central_quotient': '1600000000.hv', 'commutator_count': None, 'commutator_label': None, 'complements_known': False, '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', '2.1', '2.1', '2.1', '5.1', '5.1', '5.1', '5.1', '5.1', '5.1', '5.1', '5.1'], 'composition_length': 20, 'conjugacy_classes_known': False, 'counter': 204, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': None, 'direct_product': None, 'div_stats': [[1, 1, 1, 1], [2, 100, 1, 1], [2, 1250, 1, 3], [2, 2500, 1, 2], [2, 5000, 1, 1], [2, 20000, 1, 2], [2, 40000, 1, 1], [2, 62500, 1, 1], [2, 390625, 1, 1], [4, 2500, 1, 2], [4, 5000, 1, 4], [4, 25000, 2, 4], [4, 62500, 2, 2], [4, 125000, 1, 12], [4, 125000, 2, 8], [4, 390625, 2, 1], [4, 625000, 2, 4], [4, 781250, 1, 3], [4, 1000000, 1, 4], [4, 1562500, 1, 4], [4, 1562500, 2, 1], [4, 2000000, 2, 4], [4, 3125000, 1, 5], [4, 12500000, 1, 2], [4, 25000000, 1, 1], [5, 32, 1, 1], [5, 64, 1, 3], [5, 128, 1, 6], [5, 256, 1, 4], [5, 512, 1, 9], [5, 1024, 1, 25], [5, 2048, 1, 31], [5, 4096, 1, 72], [8, 5000000, 2, 4], [8, 12500000, 2, 2], [8, 25000000, 2, 3], [10, 800, 1, 3], [10, 1600, 1, 3], [10, 3200, 1, 6], [10, 6400, 1, 12], [10, 12800, 1, 10], [10, 20000, 1, 11], [10, 25600, 1, 16], [10, 40000, 1, 9], [10, 51200, 1, 18], [10, 80000, 1, 16], [10, 160000, 1, 20], [10, 320000, 1, 33], [10, 500000, 1, 1], [10, 640000, 1, 41], [10, 1000000, 1, 1], [10, 1280000, 1, 13], [16, 50000000, 4, 2], [20, 20000, 2, 10], [20, 40000, 2, 2], [20, 80000, 1, 12], [20, 80000, 2, 8], [20, 100000, 2, 12], [20, 320000, 1, 12], [20, 320000, 2, 10], [20, 400000, 2, 12], [20, 500000, 2, 30], [20, 640000, 1, 2], [20, 640000, 2, 1], [20, 1000000, 2, 2], [20, 1600000, 2, 4], [20, 2000000, 1, 12], [20, 2000000, 2, 8], [20, 2500000, 2, 4], [20, 4000000, 1, 8], [20, 8000000, 2, 8], [20, 16000000, 1, 4], [20, 16000000, 2, 8], [40, 20000000, 2, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 80, 'exponents_of_order': [12, 8], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': None, 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1600000000.hv', 'hash': 4731928237620763500, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': None, 'inner_split': None, 'inner_tex': 'C_5^4.D_5^4.(C_2^5.D_4)', 'inner_used': None, 'irrC_degree': None, 'irrQ_degree': None, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': None, 'label': '1600000000.hv', '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': None, 'name': 'C5^4.D5^4.(C2^5.D4)', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 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, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 7, 0, 0, 0, 0, 11, 0, 0, 0, 19, 0, 0, 0, 19, 0, 0, 0, 23, 0, 0, 25, 0, 0, 23, 0, 19, 0, 11, 7, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 385, 'number_characteristic_subgroups': 124, 'number_conjugacy_classes': 784, 'number_divisions': 622, 'number_normal_subgroups': 168, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 1600000000, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 546975], [4, 107100000], [5, 390624], [8, 240000000], [10, 61562400], [16, 400000000], [20, 630400000], [40, 160000000]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': None, 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 16, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': None, 'perfect': False, 'permutation_degree': 40, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': None, 'representations': {'PC': {'code': '773868944511600089810185512468133285331618287462095210866229539452638733378461676622195875153235880787722130364757299456017798566663666018736577127066355266073148248673092078844771483838027191119261020600521400631975181417173931114515302764830364389619361534543627184180761174181211937620843210196295064800592881837947082865716267389669802080694078901487659573461264936530012301109766552897593978943830037357031902664507085914626243759396007718543797440948214302013786620483246745636018540063475909947597818159366393397100147500657505799296878877701817478976996514684022451953540857072530086855371191027201724303067864078380861359792489987778175189103129969167341359763960477852789478085697561320680085108525653688273522139572702262132223748648500538577117915772971140189548768380868552650254761225899765943096628537935587977233944223336907135353286451044523377369106883057890256143418448310333899118962186203488458450994189138721507794919508959941319166441137024180121583452095083551464859820775465630212735302455378284794990417234919722255145681958871814750932711134592066259472421541900215182099909749985607618662567538937551245202423545541936817563468052357849058914974156672794578248435146274077889211117548879189676748481948906180549918211297150454004539919193139452421763470375842195640936436416365436037182777113575012436530512833888535766753309842510384435039982896680920738251100859892684633570978700621969512697931752863096051325881594766638250249886199232986275730934925519133339850818171402110553374661448276641706382381495295', 'gens': [1, 3, 5, 6, 8, 10, 13, 15, 17, 18, 19, 20], 'pres': [20, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 5, 2, 5, 2, 5, 5, 5, 5, 5, 40, 10100848661, 6974197202, 3620499262, 162, 94360786563, 18313943383, 153270463, 107744417604, 41759228824, 33627018444, 3137920264, 4658914084, 162528674885, 2741353945, 12987341805, 11798289185, 7180518085, 345, 8825600006, 11379200026, 36200686, 17852281506, 11518981126, 184621479687, 84903646747, 22653690287, 29112593987, 370928087, 260469227, 2552873407, 467, 133403120648, 139794606748, 60579233328, 21738539588, 16118985688, 8616964428, 1254423008, 55125168, 156325448009, 16883140829, 34409062449, 14033184069, 16495443289, 759144109, 21316129, 1578678149, 418921169, 589, 253304832010, 73757016990, 43567194930, 6803948870, 18882167130, 1500628910, 33871330, 437751750, 132530370, 650, 2715648011, 15974400031, 78489937971, 307271, 15244815451, 115311, 76931, 38551, 78353880812, 1154816032, 41362880052, 11933376072, 25780144092, 63024112, 11336132, 2494934152, 375674172, 1783792, 335612, 772, 288872120333, 372736033, 46018560053, 5376073, 17884160093, 23744113, 23296133, 801920153, 5824173, 593793, 582613, 90089568014, 54690057634, 32739840054, 22204800074, 19610880094, 1111200114, 10800134, 1635240154, 419700174, 29220194, 26670214, 57254, 31774, 894, 72089600015, 104857600035, 622592055, 61440075, 205824095, 15360115, 10240135, 6912155, 2560175, 256195, 256215, 25855, 345550888976, 1323008036, 55705600056, 326400076, 21705600096, 108800116, 972400156, 2720196, 13896, 366796800017, 235468892197, 16588800057, 1152000077, 1152000097, 748800117, 72000157, 72000177, 56160197, 7200217, 72297, 528836551698, 1004902438, 99712000058, 6566400078, 17145600098, 182400118, 4446000158, 391400178, 12160198, 39140218, 380298, 144392192019, 204124364839, 153600000059, 49920000079, 8384000099, 6176000119, 2016000159, 520000179, 428800199, 52000219, 20960259, 2000299]}, 'Perm': {'d': 40, 'gens': [565245613963954746852261267443672674577670077413, 145994415097025670713006457763069174606370181743, 629648495288210494088726564716433284034919127079]}}, 'schur_multiplier': None, 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 25, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5^4.D_5^4.(C_2^5.D_4)', 'transitive_degree': None, 'wreath_data': None, 'wreath_product': False}
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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, 3]], 'center_label': '4.2', 'center_order': 4, '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', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
