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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '4608.x', 'ambient_counter': 24, 'ambient_order': 4608, 'ambient_tex': 'D_{192}:C_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 2304, 'characteristic': True, 'core_order': 1152, 'counter': 16, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '4608.x.4.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.d1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, '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': False, 'standard_generators': False, 'stem': False, 'subgroup': '1152.2487', 'subgroup_hash': 1994011958017864, 'subgroup_order': 1152, 'subgroup_tex': 'C_6\\times C_{192}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4608.x', 'aut_centralizer_order': 384, 'aut_label': '4.d1', 'aut_quo_index': 6, 'aut_stab_index': 1, 'aut_weyl_group': '256.55608', 'aut_weyl_index': 384, 'centralizer': '2.d1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1', '2.d1.a1', '2.e1.a1'], 'contains': ['8.b1.a1', '8.g1.a1', '8.f1.a1', '12.c1.a1', '12.g1.a1', '12.s1.a1'], 'core': '4.d1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7945, 7273, 9582, 7902, 868, 7811, 9120, 7487], 'generators': [611069954, 603880896, 833930617, 352263923, 330696643, 179726564, 603880789, 611069846, 1337164657], 'label': '4608.x.4.d1.a1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.d1.a1', 'normal_contained_in': ['2.c1.a1', '2.d1.a1', '2.e1.a1'], 'normal_contains': ['8.b1.a1', '8.g1.a1', '8.f1.a1', '12.c1.a1', '12.g1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.d1.a1', 'projective_image': '384.1943', 'quotient_action_image': '2.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '4.d1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '1152.2487', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 48, 'aut_gen_orders': [12, 16, 8, 16, 8, 8], 'aut_gens': [[1, 6], [193, 9], [195, 896], [771, 631], [581, 406], [385, 584], [773, 859]], 'aut_group': None, 'aut_hash': 6822997134833335799, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6144, 'aut_permdeg': 72, 'aut_perms': [20817057574854664151720247626264111410927375236769114812043011385522213953457670472999077956064812002105, 16983839237301713850689045154202258717170429319687681854401619388219071583978646309100900585439459094621, 16660217546733708744463765962809037718285434783040765313634454829921308306504582267758683265957141334500, 2263978436756074651513570755237033472163012926351330960855484246451572976171998939568330370212423143220, 52315478203190619149474102162840274703616987276627554911472152110733102995064909242140244942052946598544, 24889720351368934544696146583127437163847478294371978612594759901091991541609666084636792810948001637899], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 8, 1], [4, 1, 2, 2], [6, 1, 8, 1], [6, 1, 16, 1], [8, 1, 4, 2], [12, 1, 16, 2], [16, 1, 8, 2], [24, 1, 32, 2], [32, 1, 16, 2], [48, 1, 64, 2], [64, 1, 64, 1], [96, 1, 128, 2], [192, 1, 512, 1]], 'aut_supersolvable': False, 'aut_tex': 'Q_8.(C_8\\times S_3).C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 48, 'autcent_group': None, 'autcent_hash': 6822997134833335799, 'autcent_nilpotent': False, 'autcent_order': 6144, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'Q_8.(C_8\\times S_3).C_2^4', '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], [3, 1, 8], [4, 1, 4], [6, 1, 24], [8, 1, 8], [12, 1, 32], [16, 1, 16], [24, 1, 64], [32, 1, 32], [48, 1, 128], [64, 1, 64], [96, 1, 256], [192, 1, 512]], 'center_label': '1152.2487', 'center_order': 1152, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 2487, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 2], ['64.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 4], [4, 1, 2, 2], [6, 1, 2, 12], [8, 1, 4, 2], [12, 1, 4, 8], [16, 1, 8, 2], [24, 1, 8, 8], [32, 1, 16, 2], [48, 1, 16, 8], [64, 1, 32, 2], [96, 1, 32, 8], [192, 1, 64, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 192, 'exponents_of_order': [7, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '32.1', 'frattini_quotient': '36.14', 'hash': 2487, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 6], [1, 6]], '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': None, 'irrep_stats': [[1, 1152]], 'label': '1152.2487', 'linC_count': None, 'linC_degree': 2, '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': True, 'monomial': True, 'name': 'C6*C192', 'ngens': 9, 'nilpotency_class': 1, 'nilpotent': True, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 1152, 'number_divisions': 70, 'number_normal_subgroups': 120, 'number_subgroup_autclasses': 54, 'number_subgroup_classes': 120, 'number_subgroups': 120, 'old_label': None, 'order': 1152, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [3, 8], [4, 4], [6, 24], [8, 8], [12, 32], [16, 16], [24, 64], [32, 32], [48, 128], [64, 64], [96, 256], [192, 512]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 48, 'outer_gen_orders': [16, 8, 2, 8, 6, 8], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[771, 992], [771, 11], [387, 958], [197, 967], [771, 391], [581, 246]], 'outer_group': None, 'outer_hash': 6822997134833335799, 'outer_nilpotent': False, 'outer_order': 6144, 'outer_permdeg': 72, 'outer_perms': [16600337509530112352605498561060338790508899177700567715705063707885383076294867567615241824624562475281, 17007699457252362201288294675421038156280852210354369867175035617595033718822186025148586681496441552825, 51523171173590983227729609542760335249265045078005003058435313015290122115323196502735343966065113575807, 25512745768756749664819262842503612279791798581726423255061838407533046892322936415515333202597708621769, 17007699457252362201288294675421041140633944920620758840814236356540209168936721277176435205272404901945, 1796845228191424223938264651289708893563697401144422442509571428307381513071386955493175919563603113265], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'Q_8.(C_8\\times S_3).C_2^4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 72, 'pgroup': 0, 'primary_abelian_invariants': [2, 64, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 10], [8, 10], [16, 10], [32, 10], [64, 8]], 'representations': {'PC': {'code': '21107466645086460155280070344309377', 'gens': [1, 3], 'pres': [9, -2, -3, -2, -2, -2, -2, -2, -2, -3, 18, 74, 102, 130, 158, 186, 214]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [150970234, 783607298]}, 'Perm': {'d': 72, 'gens': [10936698103042336594564247058340931269153989331327735474351138529059130591750442445021642752000000000, 850478588567862317521167644239926010288584608120796235886430763388588680378079017697280000000000000000, 5380233725346735740173297861243879235056180696140434424014659655756092419555920294365064205434880000, 2602001536498935312977823262830338642891006871324546356325377297179336415052750820456684567894425600, 1212885442075042831269847865593910103426536423573289051191849910519922050834904472071948260745497600, 518327453366024775318526185783187460018203549388420119811611554629313423845005581357479711432618880, 171158931788693208174569230918329200531692585670287224724129043040955365371372288938583379392681520, 240, 4]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 192], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times C_{192}', 'transitive_degree': 1152, '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': '48.44', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 192, 'aut_gen_orders': [48, 64, 64, 16, 32, 96, 32, 4], 'aut_gens': [[833930617, 37442, 352263794], [1049602400, 7133087, 1380299074], [122214128, 2933600, 805174493], [647015308, 2768006, 783607294], [1279652236, 791107, 934577411], [1099925803, 5998247, 934577602], [1114303901, 5182436, 582313725], [107835958, 1084081, 7189106], [1351542793, 4458300, 7189106]], 'aut_group': None, 'aut_hash': 1169717094561038857, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 98304, 'aut_permdeg': 198, 'aut_perms': [10612460235486022502783232621374079469799730512814642477677700315964396111621674611079497600781722663160135589495177444915229940769105209085358042861010443629912989931891788970781435338219303092327457476453378439273373979366567480860052500790980123737144031346855417604209901295810755244642651246874369541694219402719287219689504282024638217150868232494411877708969979464, 3696997192540310171574241090439629820957448384800341199438956047635966202013987799992037174883558903206806697264680174491736325713893851132920118251128934225374845404732195051484967048965187911586263802536117010028581534869342703552342899106155396136626939615638072776352009778510013741907854519621031330067600285244118002445111051856531979167368693631637448970001845510, 11521120379147265230903020992362792288544439002516239721652267187430560555609407761660475531911521355822629738015104435803486081119811222142739041556327939746699600086352741958393962704196155351029564868987426028613576887603043167767106031279671210697106109397501058345933701517743824603725708239150365687751342705693079766084407660129810980560871445891465031278846027944, 2501639853317065693812733819469466302669076443488847400571887363065975496009912346750107466841732151166204954220818038576100723218478212668077444146308845336917816707022737676693971201902838428623978229350353200913749353455497568599904775240442704285995888512151040067626502041056168624089300217154531016833716354141056776512022297760692077299056847703545603005282740888, 5515238027241039765462318404035738054784741480794028870815004121444749602629024201836262514003791833103393073166192194540812205325484286636248569252585360058027595967501644847268998806200869646061633041537506995492624234205398514379478313305741464766353237105887993169614358446003924477066124321356472817724355179825215098781176503722876703280183498182233132465072746595, 1322685663562470387881729293531421655085895045506275968097906634492475473081584771791509948176802880557355581747656391431959895414982549512979473209333116569614570653919674979366434475178279328361762308155570540575142788361937855557985721868846546669744428549974210106233877867006222687331760007567993435424033864259386448719923792525482335459216910419256724498768216575, 7202678656039741921128300481205647755245907400297882865795988971299961987649724816573271278872239391807673414639964156179728069744750741676270058023061522860720223783737096099982089270579998268566373568105344038978899816613611778794674131591439264215411500435316990050563911688181717338701015178904130279227805564768835266416143241592651442536968161500917319212847482382, 12306963688834076324754122861906795889381033964534865608927399411877753785951787096080597479935973236255877805769562475538401814301060562641541611094521573588533990070038598602650485180843428903256256862820029902283569037655876102932245299184041955849986802831089219160160872401274261287814951041474652893960036190632806673117739112870836811356498908972550461830091967451], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 96, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 4, 1], [4, 96, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 3], [6, 2, 4, 1], [6, 96, 4, 1], [8, 1, 4, 1], [8, 2, 2, 3], [8, 96, 4, 1], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 2], [12, 2, 8, 2], [12, 2, 16, 1], [12, 96, 4, 1], [16, 2, 4, 2], [16, 2, 8, 1], [24, 1, 8, 1], [24, 2, 4, 7], [24, 2, 8, 4], [24, 96, 8, 1], [32, 2, 8, 2], [32, 2, 16, 1], [48, 2, 8, 4], [48, 2, 16, 4], [48, 2, 32, 1], [64, 2, 16, 2], [64, 2, 32, 1], [96, 2, 16, 4], [96, 2, 32, 4], [96, 2, 64, 1], [192, 2, 32, 4], [192, 2, 64, 4], [192, 2, 128, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_{32}.C_8.C_2^6.C_2)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 96, 'autcentquo_group': '3072.bkp', 'autcentquo_hash': 837218907410379413, 'autcentquo_nilpotent': False, 'autcentquo_order': 3072, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times D_{32}:C_8', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 96, 2], [3, 1, 2], [3, 2, 3], [4, 1, 2], [4, 2, 5], [4, 96, 2], [6, 1, 2], [6, 2, 11], [6, 96, 4], [8, 1, 4], [8, 2, 6], [8, 96, 4], [12, 1, 4], [12, 2, 46], [12, 96, 4], [16, 2, 16], [24, 1, 8], [24, 2, 60], [24, 96, 8], [32, 2, 32], [48, 2, 128], [64, 2, 64], [96, 2, 256], [192, 2, 512]], 'center_label': '24.2', 'center_order': 24, 'central_product': True, 'central_quotient': '192.7', 'commutator_count': 1, 'commutator_label': '96.2', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 24, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1536.10843761', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 96, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 2, 2], [4, 96, 1, 2], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 5], [6, 96, 2, 2], [8, 1, 4, 1], [8, 2, 2, 3], [8, 96, 2, 2], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 10], [12, 96, 2, 2], [16, 2, 4, 2], [16, 2, 8, 1], [24, 1, 8, 1], [24, 2, 4, 7], [24, 2, 8, 4], [24, 96, 4, 2], [32, 2, 8, 2], [32, 2, 16, 1], [48, 2, 8, 4], [48, 2, 16, 6], [64, 2, 16, 2], [64, 2, 32, 1], [96, 2, 16, 4], [96, 2, 32, 6], [192, 2, 32, 4], [192, 2, 64, 6]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 279552, 'exponent': 192, 'exponents_of_order': [9, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 256]], 'familial': False, 'frattini_label': '64.50', 'frattini_quotient': '72.48', 'hash': 1025750976874433368, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 96, 'inner_gen_orders': [96, 2, 12], 'inner_gens': [[833930617, 958052, 352263794], [35945401, 37442, 7189106], [833930617, 4851827, 352263794]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 192, 'inner_split': False, 'inner_tex': 'D_{96}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 128, 'irrQ_dim': 128, 'irrR_degree': None, 'irrep_stats': [[1, 48], [2, 1140]], 'label': '4608.x', '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': 'D192:C12', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 82, 'number_characteristic_subgroups': 126, 'number_conjugacy_classes': 1188, 'number_divisions': 98, 'number_normal_subgroups': 154, 'number_subgroup_autclasses': 339, 'number_subgroup_classes': 495, 'number_subgroups': 4282, 'old_label': None, 'order': 4608, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 195], [3, 8], [4, 204], [6, 408], [8, 400], [12, 480], [16, 32], [24, 896], [32, 64], [48, 256], [64, 128], [96, 512], [192, 1024]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [16, 2, 16, 8, 8, 2], 'outer_gen_pows': [7189058, 7189058, 7189058, 1380298945, 783607214, 603880789], 'outer_gens': [[1279652236, 3187202, 452910783], [35945401, 1251798, 452910783], [970522878, 3149760, 934577411], [158159333, 6562386, 1380299088], [797985367, 5714923, 582313702], [1143060080, 2043870, 7189120]], 'outer_group': '512.10493039', 'outer_hash': 3046548263559564634, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 512, 'outer_perms': 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'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\times C_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 134, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [4, 17], [8, 15], [16, 13], [32, 11], [64, 8], [128, 8]], 'representations': {'PC': {'code': '14393957789053804268956540308875743268145250761846466575781081300132358858346636226689', 'gens': [1, 2, 5], 'pres': [11, -2, -2, -2, -3, -2, -2, -2, -2, -2, -2, -3, 8493, 56, 25478, 90, 67939, 125404, 158, 300965, 192, 347430, 226, 388615, 260, 418184, 294, 422409, 328, 371722]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [833930617, 37442, 352263794]}, 'Perm': {'d': 134, 'gens': [15323440425147759346105929691235740094797998041214723154793837873806460408845517749286203907891819082654102269512039347198437564788995594926652544294339236871035378175633539249182058164394068977775024037955935485063498402528826, 30304590484582350738273793010033522003569044222252547546483750918560584928223178892561208997596487526913410136935021658190949837278156990838551319176606870162108279696739468222256683073451162374647970954646961167443375353910335, 45294153624597919675427713709151706669764738488280611073726455578099423141683547800808556340349071612789519298826386087665254458886250123290333509674937792239707029044572009951202341393331601343077987148535074166063887131183036]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{192}:C_{12}', 'transitive_degree': 384, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
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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}
