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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '15552.fd', 'ambient_counter': 134, 'ambient_order': 15552, 'ambient_tex': 'C_6^4:D_6', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 1944, 'counter': 63, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '15552.fd.8.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '8.3', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '1944.2519', 'subgroup_hash': 2519, 'subgroup_order': 1944, 'subgroup_tex': 'C_3^4:S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '15552.fd', 'aut_centralizer_order': None, 'aut_label': '8.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '3888.A', 'complements': ['1944.B', '1944.C', '1944.D', '1944.E', '1944.F', '1944.G', '1944.H', '1944.I', '1944.J', '1944.K', '1944.L', '1944.M', '1944.N'], 'conjugacy_class_count': 2, 'contained_in': ['4.d1', '4.i1', '4.j1'], 'contains': ['16.a1', '24.dn1', '24.do1', '24.dp1', '24.dq1', '24.dr1', '24.ds1', '24.dt1', '32.e1'], 'core': '8.b1', 'coset_action_label': None, 'count': 2, 'diagramx': None, 'generators': [378098439554953288, 1395415288877, 144, 1440696333, 1520467200, 482630400, 336782, 25590], 'label': '15552.fd.8.b1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.b1', 'normal_contained_in': ['4.d1'], 'normal_contains': ['16.a1'], 'normalizer': '1.a1', 'old_label': '8.b1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.b1', '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [12, 18, 6, 18], 'aut_gens': [[274479569545, 1589335292307, 2976642990724, 355546235520, 3063734150400, 4359553591680, 744, 1680], [274479570842, 9367003337160, 2976642990723, 4010201395203, 2708917706880, 4359553591680, 2424, 1680], [9262026468865, 7180241969640, 2976642990723, 2708877830403, 355586112000, 4359553591680, 1680, 2424], [1334102958746, 9367003336560, 4359553591683, 4359433478404, 355586112000, 2976642990720, 1680, 744], [5619168537989, 1944794437464, 4359553591683, 4272382195204, 355586112000, 4359553591680, 1680, 2424]], 'aut_group': None, 'aut_hash': 6385099260404231401, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 629856, 'aut_permdeg': 162, 'aut_perms': [7558159489891272845758702856513416232733340027495862561586510629316934769086892777013078511931868432142293697391367878581615398065249081585669779559055548626277927912988051510705944910675656352115584397274340102807826052161074431909868112293489794845434008093785482248341079643101823998757, 1008895045953047039039681336932077831405629140682026533428973552411926223029196097813535494250654615194502683007112533076219856585523674703194767924358067513564632113261859540751433403673003267944600479994797381133021970752301865177356892080761573004599005503874814134100996368658796621356, 7595733664113734465658933474306453519052183847565037851957126020817854088727100048584840232174293696045931961393769500228325214497977500940800486013285024497793365459046128279516980609507387495021872617518179029454636121975467485584102404927413887491193550818474565792854235036013635655697, 2437450208537865373330179650162047240849488778322804380660455193377470397086579348201993879855285029794982467349325384840094193554660643385430792377393252807491255527784171347700492681430768756129472304050782870602385501912174396612336885596808129488484020524870280519437888236692408077711], 'aut_phi_ratio': 972.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 162, 1, 1], [3, 2, 1, 1], [3, 2, 3, 1], [3, 3, 2, 1], [3, 6, 2, 1], [3, 6, 9, 1], [3, 72, 3, 1], [4, 162, 1, 1], [6, 3, 2, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 6, 3, 1], [6, 6, 6, 1], [6, 6, 9, 1], [6, 6, 18, 1], [6, 162, 2, 1], [9, 72, 6, 1], [12, 162, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_6^2.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': 36, 'autcentquo_group': None, 'autcentquo_hash': 6385099260404231401, 'autcentquo_nilpotent': False, 'autcentquo_order': 629856, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_6^2.S_3^3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 162, 1], [3, 2, 4], [3, 3, 2], [3, 6, 11], [3, 72, 3], [4, 162, 1], [6, 3, 2], [6, 6, 39], [6, 162, 2], [9, 72, 6], [12, 162, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1944.2519', 'commutator_count': 2, 'commutator_label': '972.590', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2519, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 162, 1, 1], [3, 2, 1, 4], [3, 3, 2, 1], [3, 6, 1, 9], [3, 6, 2, 1], [3, 72, 1, 3], [4, 162, 1, 1], [6, 3, 2, 1], [6, 6, 1, 13], [6, 6, 2, 13], [6, 162, 2, 1], [9, 72, 1, 6], [12, 162, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 11941020, 'exponent': 36, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '216.165', 'hash': 2519, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [2, 3, 3, 3, 3, 3, 2, 2], 'inner_gens': [[274479569545, 5264227584244, 4359553591683, 442757508480, 4010325177600, 2976642990720, 2424, 1680], [5357644548605, 1589335292307, 2976642990724, 2708877830400, 355586112000, 4359553591680, 1680, 2424], [4545856224266, 1589335292307, 2976642990724, 355546235520, 3063734150400, 4359553591680, 744, 1680], [623312121865, 6912559906707, 2976642990724, 355546235520, 3063734150400, 4359553591680, 744, 1680], [4545856224265, 7093107262227, 2976642990724, 355546235520, 3063734150400, 4359553591680, 744, 1680], [3138981673225, 1589335292307, 2976642990724, 355546235520, 3063734150400, 4359553591680, 744, 1680], [274479570841, 1589335294443, 2976642990724, 355546235520, 3063734150400, 4359553591680, 744, 1680], [274479569545, 1589335293147, 2976642990724, 355546235520, 3063734150400, 4359553591680, 744, 1680]], 'inner_hash': 2519, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': False, 'inner_tex': 'C_3^4:S_4', 'inner_used': [1, 2, 3, 4, 7], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 13], [3, 10], [6, 50]], 'label': '1944.2519', '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:S4', 'ngens': 8, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 75, 'number_divisions': 57, 'number_normal_subgroups': 47, 'number_subgroup_autclasses': 184, 'number_subgroup_classes': 660, 'number_subgroups': 9890, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 165], [3, 296], [4, 162], [6, 564], [9, 432], [12, 324]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [3, 6, 3, 3, 3, 6], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[274479569545, 1589335292307, 4, 4010201395203, 3063734150400, 4359553591680, 744, 1680], [274479569545, 1589335292307, 4359553591683, 3063854263680, 3063734150400, 4359553591680, 744, 1680], [274479569545, 8464525090708, 4, 3063854263683, 3063734150400, 4359553591680, 744, 1680], [274479569545, 1589335292307, 4359553591684, 355546235524, 3063734150400, 4359553591680, 744, 1680], [274479569545, 1589335292307, 4, 3063854263680, 3063734150400, 4359553591680, 744, 1680], [274479569545, 1589335292307, 4359553591683, 4272342318720, 4272258816000, 2976642990720, 744, 1680]], 'outer_group': '324.122', 'outer_hash': 122, 'outer_nilpotent': False, 'outer_order': 324, 'outer_permdeg': 12, 'outer_perms': [93074664, 121939441, 179360164, 306069840, 120486960, 121204081], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:S_3^2', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 13], [3, 2], [6, 26], [12, 14]], 'representations': {'PC': {'code': 244922911743685123802970949054274845088504857693, 'gens': [1, 2, 3, 5, 6, 7], 'pres': [8, 2, 3, 2, 3, 3, 3, 2, 3, 65, 24050, 4834, 66, 771, 11147, 2884, 10373, 37814, 166, 23055]}, 'Perm': {'d': 16, 'gens': [274479569545, 1589335292307, 2976642990724, 355546235520, 3063734150400, 4359553591680, 744, 1680]}}, 'schur_multiplier': [3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:S_4', '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': '8.5', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [12, 6, 18, 6, 6, 2, 6], 'aut_gens': [[122003490882966572, 122002183212146089, 6403768638273453], [256095035451158335, 256097564539603765, 371695978271726428], [256096255947275454, 122002184206635619, 371694758171133736], [256097650836245135, 122003404227318216, 371695979149783058], [256095036409220015, 256097564619531619, 371695978228244495], [256097651757749181, 122003403708556959, 6404990495160819], [122003490926580042, 122003403744745288, 6404989098170814], [256097651757799052, 122002183212151849, 6403769999114258]], 'aut_group': None, 'aut_hash': 4206657513537088607, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 186624, 'aut_permdeg': 116, 'aut_perms': [31293814257700036787374800120896077677581725017421359421127778414839858362977490278432716016601675104888313392421777995889639790478433676624674168521856539771544907716483047424822449655387622, 580582938151057214957526275536283152974267718568751385270436501863654210890597727522245561814911303477449951435278722152975514548070839798003247183737421679342098790833114152000083132927983, 294698932557323877649086163712804071328249810015500515729299643446018505562955521710854973630784156576822721184750969889552660608888401369132670654273986616472338853992167672943961921317441, 31294240847289021212101935828557554962163783236421068200934488759812919627834505409924456586609396307239677869959173944189173663220628425865808918108144389145527682985371912475514751640216178, 293234532366098969682010846330750571680430954090578002312523801579365602543675102583292104071904724481609465284705981890547392849587804642903563961088200807018522068517882578688903376636783, 3778150099720208099401475930210739688665767342860285932383552164675745734459611160126890023108597589300709147239852223273135629987287417382401800906826672065915305331977344702987104915521799, 31037490105008431130089946672773096291368778529541539018778599905647638827047903822307290134944259714524228577258644620240839646973406535867664048891581796310443206434849837383316885316578385], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 6, 1, 2], [2, 18, 1, 1], [2, 108, 1, 1], [2, 162, 2, 1], [2, 324, 1, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 2, 1], [3, 6, 3, 1], [3, 12, 3, 1], [3, 72, 1, 1], [3, 144, 1, 1], [4, 6, 1, 1], [4, 18, 1, 1], [4, 108, 1, 3], [4, 162, 2, 1], [4, 324, 1, 1], [6, 2, 1, 2], [6, 3, 2, 3], [6, 4, 1, 3], [6, 4, 2, 1], [6, 6, 1, 4], [6, 6, 2, 7], [6, 6, 3, 3], [6, 6, 6, 2], [6, 12, 1, 5], [6, 12, 2, 6], [6, 12, 3, 5], [6, 12, 4, 1], [6, 12, 6, 5], [6, 12, 12, 1], [6, 18, 2, 2], [6, 36, 1, 1], [6, 36, 2, 1], [6, 36, 3, 2], [6, 36, 6, 1], [6, 72, 1, 1], [6, 108, 2, 1], [6, 144, 1, 2], [6, 144, 2, 1], [6, 162, 4, 1], [6, 216, 1, 1], [6, 216, 2, 1], [6, 324, 2, 1], [6, 432, 1, 1], [9, 72, 2, 1], [9, 144, 2, 1], [12, 12, 1, 1], [12, 18, 2, 2], [12, 36, 1, 1], [12, 36, 2, 1], [12, 36, 3, 2], [12, 36, 6, 1], [12, 108, 2, 3], [12, 162, 4, 1], [12, 216, 1, 3], [12, 216, 2, 3], [12, 324, 2, 1], [12, 432, 1, 1], [18, 72, 2, 1], [18, 144, 2, 2], [18, 144, 4, 1], [18, 432, 2, 1], [36, 432, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_6^2.C_3^4.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 2074683354935437783, 'autcentquo_nilpotent': False, 'autcentquo_order': 23328, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.C_3^4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 3, 2], [2, 6, 2], [2, 18, 1], [2, 108, 1], [2, 162, 2], [2, 324, 1], [3, 2, 2], [3, 3, 2], [3, 4, 1], [3, 6, 5], [3, 12, 3], [3, 72, 1], [3, 144, 1], [4, 6, 1], [4, 18, 1], [4, 108, 3], [4, 162, 2], [4, 324, 1], [6, 2, 2], [6, 3, 6], [6, 4, 5], [6, 6, 39], [6, 12, 78], [6, 18, 4], [6, 36, 15], [6, 72, 1], [6, 108, 2], [6, 144, 4], [6, 162, 4], [6, 216, 3], [6, 324, 2], [6, 432, 1], [9, 72, 2], [9, 144, 2], [12, 12, 1], [12, 18, 4], [12, 36, 15], [12, 108, 6], [12, 162, 4], [12, 216, 9], [12, 324, 2], [12, 432, 1], [18, 72, 2], [18, 144, 8], [18, 432, 2], [36, 432, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '7776.ga', 'commutator_count': 2, 'commutator_label': '1944.2845', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 134, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 6, 1, 2], [2, 18, 1, 1], [2, 108, 1, 1], [2, 162, 1, 2], [2, 324, 1, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 1, 3], [3, 6, 2, 1], [3, 12, 1, 3], [3, 72, 1, 1], [3, 144, 1, 1], [4, 6, 1, 1], [4, 18, 1, 1], [4, 108, 1, 3], [4, 162, 1, 2], [4, 324, 1, 1], [6, 2, 1, 2], [6, 3, 2, 3], [6, 4, 1, 5], [6, 6, 1, 13], [6, 6, 2, 13], [6, 12, 1, 34], [6, 12, 2, 22], [6, 18, 2, 2], [6, 36, 1, 7], [6, 36, 2, 4], [6, 72, 1, 1], [6, 108, 2, 1], [6, 144, 1, 4], [6, 162, 2, 2], [6, 216, 1, 1], [6, 216, 2, 1], [6, 324, 2, 1], [6, 432, 1, 1], [9, 72, 1, 2], [9, 144, 1, 2], [12, 12, 1, 1], [12, 18, 2, 2], [12, 36, 1, 7], [12, 36, 2, 4], [12, 108, 2, 3], [12, 162, 2, 2], [12, 216, 1, 3], [12, 216, 2, 3], [12, 324, 2, 1], [12, 432, 1, 1], [18, 72, 1, 2], [18, 144, 1, 8], [18, 432, 1, 2], [36, 432, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 36, 'exponents_of_order': [6, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 12], [12, 1, 6]], 'familial': False, 'frattini_label': '18.5', 'frattini_quotient': '864.4692', 'hash': 2581741099378992623, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [18, 12, 6], 'inner_gens': [[122003490882966572, 122003404702730755, 371695978710702092], [122002095555033257, 122002183212146089, 371694758690052136], [256096257344184333, 256096344598827235, 6403768638273453]], 'inner_hash': 1987921750283067703, 'inner_nilpotent': False, 'inner_order': 7776, 'inner_split': True, 'inner_tex': 'C_6^3:S_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 22], [3, 40], [4, 21], [6, 90], [12, 80]], 'label': '15552.fd', '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': 'C6^4:D6', '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, 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': 120, 'number_characteristic_subgroups': 104, 'number_conjugacy_classes': 261, 'number_divisions': 195, 'number_normal_subgroups': 124, 'number_subgroup_autclasses': 4400, 'number_subgroup_classes': 8582, 'number_subgroups': 239372, 'old_label': None, 'order': 15552, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 795], [3, 296], [4, 996], [6, 5064], [9, 432], [12, 4944], [18, 2160], [36, 864]], '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': [[122002096030307372, 122003403708222889, 6404989134350253], [122003491322109676, 256096344123374179, 6403768602058156], [122003490882936861, 256096343604676479, 6403768638273453]], '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': None, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 22], [3, 8], [4, 21], [6, 50], [12, 64], [24, 22]], 'representations': {'PC': {'code': '64525695607265450788082105100841322233256513463459378752526363681753504710646325910509318180164637519306856929342697356954097222900152561930449060372841920402926170081438950380310875501042836256859091564161890652', 'gens': [1, 2, 4, 6, 8, 10, 11], 'pres': [11, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 2, 129888, 88925, 56, 494474, 32100, 545427, 115382, 142717, 124, 555724, 286455, 142916, 776957, 245932, 172287, 48746, 20047, 192, 310470, 471257, 252280, 94287, 28694, 765, 608263, 551250, 294653, 103528, 33843, 3758, 1129, 348, 128323, 1283049, 71322, 53513, 17884, 6015, 2066, 470458, 705693, 117655, 19656, 6599, 9877, 3354]}, 'Perm': {'d': 20, 'gens': [122003490882966572, 122002183212146089, 6403768638273453]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 64, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^4:D_6', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, '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': 'D4', 'ngens': 2, 'nilpotency_class': 2, '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': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], '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': 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], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
