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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '115200.d', 'ambient_counter': 4, 'ambient_order': 115200, 'ambient_tex': 'S_5^2:D_4', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 4, 'counter': 198, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '115200.d.120.C', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '120.c1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 120, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '960.11377', 'subgroup_hash': 11377, 'subgroup_order': 960, 'subgroup_tex': 'C_2^2\\times D_6\\times F_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '115200.d', 'aut_centralizer_order': None, 'aut_label': '120.C', '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': ['12.a1', '20.C'], 'contains': ['240.A', '240.B', '240.D', '240.E', '240.F', '240.J', '240.K', '240.W', '240.X', '240.Y', '240.Z', '240.CA', '240.CB', '360.A', '600.M'], 'core': '28800.A', 'coset_action_label': None, 'count': 120, 'diagramx': None, 'generators': [23, 408240, 806400, 7, 39916816, 31658422320, 62098687, 3396965776], 'label': '115200.d.120.C', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.A', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '120.C', 'old_label': '120.c1', 'projective_image': '57600.l', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '120.C', 'subgroup_fusion': None, 'weyl_group': '120.36'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '64.260', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [12, 70, 12], 'aut_gens': [[297011, 27451, 27311, 27013, 783149], [297011, 513469, 783319, 783737, 27511], [297461, 783479, 513479, 513637, 513589], [783029, 513469, 27761, 513817, 783539]], 'aut_group': None, 'aut_hash': 2599079116499701836, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1290240, 'aut_permdeg': 64, 'aut_perms': [40417275602318616534353664770512953089635897522803820529930392908177784695757393220538397, 7739040962335637502667124071744488660917718133358536406598724472024535905165323540840532, 123879998036930808577088311703415295591188216362487670861013833190712899656484487251388260], 'aut_phi_ratio': 5040.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [2, 3, 8, 1], [2, 5, 1, 1], [2, 5, 7, 1], [2, 15, 8, 1], [3, 2, 1, 1], [4, 5, 8, 2], [4, 15, 8, 2], [5, 4, 1, 1], [6, 2, 7, 1], [6, 10, 1, 1], [6, 10, 7, 1], [10, 4, 7, 1], [10, 12, 8, 1], [12, 10, 8, 2], [15, 8, 1, 1], [30, 8, 7, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times C_2^6.\\PSL(2,7)\\times F_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '10752.bb', 'autcent_hash': 6417384533392091029, 'autcent_nilpotent': False, 'autcent_order': 10752, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^6:\\GL(3,2)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '120.36', 'autcentquo_hash': 36, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 3, 8], [2, 5, 8], [2, 15, 8], [3, 2, 1], [4, 5, 16], [4, 15, 16], [5, 4, 1], [6, 2, 7], [6, 10, 8], [10, 4, 7], [10, 12, 8], [12, 10, 16], [15, 8, 1], [30, 8, 7]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '120.36', 'commutator_count': 1, 'commutator_label': '15.1', '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', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 11377, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['20.3', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 3, 1, 8], [2, 5, 1, 8], [2, 15, 1, 8], [3, 2, 1, 1], [4, 5, 2, 8], [4, 15, 2, 8], [5, 4, 1, 1], [6, 2, 1, 7], [6, 10, 1, 8], [10, 4, 1, 7], [10, 12, 1, 8], [12, 10, 2, 8], [15, 8, 1, 1], [30, 8, 1, 7]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 185702400, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '960.11377', 'hash': 11377, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [1, 1, 2, 4, 15], 'inner_gens': [[297011, 27451, 27311, 27013, 783149], [297011, 27451, 27311, 27013, 783149], [297011, 27451, 27311, 27013, 783449], [297011, 27451, 27311, 27013, 783869], [297011, 27451, 27011, 27373, 783149]], 'inner_hash': 36, 'inner_nilpotent': False, 'inner_order': 120, 'inner_split': True, 'inner_tex': 'S_3\\times F_5', 'inner_used': [3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 64], [2, 32], [4, 16], [8, 8]], 'label': '960.11377', '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': 'C2^2*D6*F5', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 21, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 120, 'number_divisions': 96, 'number_normal_subgroups': 882, 'number_subgroup_autclasses': 152, 'number_subgroup_classes': 2724, 'number_subgroups': 13984, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 191], [3, 2], [4, 320], [5, 4], [6, 94], [10, 124], [12, 160], [15, 8], [30, 56]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 84, 'outer_gen_orders': [7, 7], 'outer_gen_pows': [27001, 27001], 'outer_gens': [[513469, 513019, 783169, 513007, 297491], [783479, 783029, 783619, 513007, 513889]], 'outer_group': '10752.bb', 'outer_hash': 6417384533392091029, 'outer_nilpotent': False, 'outer_order': 10752, 'outer_permdeg': 16, 'outer_perms': [7073899272447, 5666632857298], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:\\GL(3,2)', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 4], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 32], [4, 24], [8, 8]], 'representations': {'PC': {'code': 4492774340465766361790246732158368577990688775, 'gens': [1, 2, 3, 4, 6], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -5, 91, 4245, 1373, 1861, 141, 9878, 3166, 934, 222, 3103, 3111]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [27301, 27007, 27019, 297011, 27451, 783619, 27181, 513019]}, 'Perm': {'d': 14, 'gens': [522587527, 136, 40327, 7, 40336, 1037836800, 840, 6785856000]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times D_6\\times F_5', 'transitive_degree': 120, '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': 2, 'aut_exponent': 120, 'aut_gen_orders': [6, 12, 6, 8, 6], 'aut_gens': [[14386070169, 7672776605, 20773152734], [6314288405, 12545301009, 21088812974], [59118635525, 8797829769, 51931812245], [51991683129, 58613522414, 7678677005], [59575270454, 57975568569, 52909039454], [13905684014, 58456500609, 52865131085]], 'aut_group': '460800.bo', 'aut_hash': 1673116337738338913, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 460800, 'aut_permdeg': 320, 'aut_perms': [9126974453583707549525523729537312145254597616333907906670064285036387897771296850570178512518395872091024247774536714863370674655851850720319141038814994605437945633970799237340994982407870471641534642617200568455413571764310432960090731171588497957550605838659929178313708764107006707659306399317253313634874944599077702303826035556008433748062527347832886649794563010668081373278702015248976638265207820080823090565464984216835709974072722436090952959982614582146629041483484815485452146397237855475995996500663356895473416189806907003200791699962893355960401471048800426916408211132062804998112619470879596892803884480853778889059915693892423930843123383126852, 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4555383485297111708774865307189393950948460186710216424158828236017349785098832805341225503031832346266283606557422690524773810193813364318424469845411026109640660492048248945537302786530006376331685052383799730432717455767853185505680872116137109959788874277867503214673999490987239574326436558085404959056614918241004895398392775595582801038348296691090366241935636056763529870377715618852826543086685108154544562004338882156525971074618713005156733171722598432800410999123856964222467161902565844445121286020211502108059170976841437115763997948481935505679711374546135724520300499181014208614295077627664855859137989262506428553976415173158203471664685316337049, 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20300733636529429203902232606038876829369391396196122912796887295014979709166123619548763778540889732709786474923069968372054118664442327341989969720300749275582618699828561303537626773565565955536550808256726328171114158741841221542267705285896794092998660346636992423965375932925770395040909298159846826965793502542679431179913244668058345645369673341647142022305621696767992790277977614269292296822117336718950331365072216356675611105912322856050285943929444445679339776640640951857442132330493746034373152944656702334426363348535084093606548408342049625134500396819696867314866742423496348213882553668912717860496437570998489546145954310532280454477749206533356], 'aut_phi_ratio': 15.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 20, 4, 1], [2, 30, 1, 2], [2, 30, 2, 1], [2, 100, 2, 1], [2, 200, 1, 1], [2, 225, 1, 2], [2, 240, 1, 1], [2, 300, 4, 1], [2, 450, 1, 1], [3, 40, 1, 1], [3, 400, 1, 1], [4, 60, 4, 1], [4, 240, 1, 1], [4, 600, 2, 2], [4, 900, 2, 1], [4, 900, 4, 1], [4, 1800, 1, 1], [4, 2400, 2, 1], [4, 3600, 1, 2], [5, 48, 1, 1], [5, 576, 1, 1], [6, 40, 1, 1], [6, 40, 2, 1], [6, 40, 4, 1], [6, 400, 1, 1], [6, 400, 2, 3], [6, 400, 4, 1], [6, 600, 1, 2], [6, 600, 2, 1], [6, 600, 4, 1], [6, 800, 1, 2], [6, 800, 4, 1], [6, 4800, 1, 1], [8, 7200, 2, 1], [10, 48, 1, 1], [10, 48, 2, 1], [10, 480, 4, 1], [10, 576, 1, 1], [10, 720, 1, 2], [10, 720, 2, 1], [10, 1152, 1, 1], [10, 5760, 1, 1], [12, 1200, 2, 2], [12, 1200, 4, 1], [12, 4800, 1, 1], [12, 4800, 2, 1], [15, 960, 1, 1], [20, 1440, 4, 1], [20, 5760, 1, 1], [30, 960, 1, 1], [30, 960, 2, 1], [30, 960, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times A_5^2).D_4^2', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '57600.l', 'autcentquo_hash': 3328040073188890159, 'autcentquo_nilpotent': False, 'autcentquo_order': 57600, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_5^2:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 20, 4], [2, 30, 4], [2, 100, 2], [2, 200, 1], [2, 225, 2], [2, 240, 1], [2, 300, 4], [2, 450, 1], [3, 40, 1], [3, 400, 1], [4, 60, 4], [4, 240, 1], [4, 600, 4], [4, 900, 6], [4, 1800, 1], [4, 2400, 2], [4, 3600, 2], [5, 48, 1], [5, 576, 1], [6, 40, 7], [6, 400, 11], [6, 600, 8], [6, 800, 6], [6, 4800, 1], [8, 7200, 2], [10, 48, 3], [10, 480, 4], [10, 576, 1], [10, 720, 4], [10, 1152, 1], [10, 5760, 1], [12, 1200, 8], [12, 4800, 3], [15, 960, 1], [20, 1440, 4], [20, 5760, 1], [30, 960, 7]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '57600.l', 'commutator_count': 1, 'commutator_label': '14400.ca', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '60.5', '60.5'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 20, 1, 4], [2, 30, 1, 4], [2, 100, 1, 2], [2, 200, 1, 1], [2, 225, 1, 2], [2, 240, 1, 1], [2, 300, 1, 4], [2, 450, 1, 1], [3, 40, 1, 1], [3, 400, 1, 1], [4, 60, 1, 4], [4, 240, 1, 1], [4, 600, 1, 4], [4, 900, 1, 6], [4, 1800, 1, 1], [4, 2400, 1, 2], [4, 3600, 1, 2], [5, 48, 1, 1], [5, 576, 1, 1], [6, 40, 1, 7], [6, 400, 1, 11], [6, 600, 1, 8], [6, 800, 1, 6], [6, 4800, 1, 1], [8, 7200, 1, 2], [10, 48, 1, 3], [10, 480, 1, 4], [10, 576, 1, 1], [10, 720, 1, 4], [10, 1152, 1, 1], [10, 5760, 1, 1], [12, 1200, 1, 8], [12, 4800, 1, 3], [15, 960, 1, 1], [20, 1440, 1, 4], [20, 5760, 1, 1], [30, 960, 1, 7]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 120, 'exponents_of_order': [9, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 1, 8], [10, 1, 8], [12, 1, 4], [32, 1, 4], [40, 1, 8], [48, 1, 4], [50, 1, 4], [60, 1, 4], [72, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '57600.l', 'hash': 8438217488598235417, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 120, 'inner_gen_orders': [4, 8, 4], 'inner_gens': [[14386070169, 15023731814, 58199112845], [6398100138, 7672776605, 14490337694], [58217248218, 52948355885, 20773152734]], 'inner_hash': 3328040073188890159, 'inner_nilpotent': False, 'inner_order': 57600, 'inner_split': True, 'inner_tex': 'S_5^2:C_2^2', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 6], [8, 16], [10, 16], [12, 8], [16, 8], [25, 8], [32, 6], [36, 4], [40, 16], [48, 8], [50, 6], [60, 8], [72, 1]], 'label': '115200.d', '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': False, 'name': 'S5^2:D4', '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, 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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 65, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 119, 'number_divisions': 119, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 4913, 'number_subgroup_classes': 14962, 'number_subgroups': 5371298, 'old_label': None, 'order': 115200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 2943], [3, 440], [4, 22080], [5, 624], [6, 19080], [8, 14400], [10, 12432], [12, 24000], [15, 960], [20, 11520], [30, 6720]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[14386070169, 7672776614, 20773152734], [14386070174, 7672776609, 20773152734]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [8, 16], [10, 16], [12, 8], [16, 8], [25, 8], [32, 6], [36, 4], [40, 16], [48, 8], [50, 6], [60, 8], [72, 1]], 'representations': {'Perm': {'d': 14, 'gens': [14386070169, 7672776605, 20773152734]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 720, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_5^2:D_4', 'transitive_degree': 20, 'wreath_data': ['C_2\times S_5', 'C_2', '2T1'], 'wreath_product': True}
