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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '52800.f', 'ambient_counter': 6, 'ambient_order': 52800, 'ambient_tex': '\\GL(2,11):C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 40, 'characteristic': False, 'core_order': 5, 'counter': 551, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '52800.f.1056.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '1056.e1.a1', 'outer_equivalence': False, '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': 1056, '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': '50.3', 'subgroup_hash': 3, 'subgroup_order': 50, 'subgroup_tex': 'C_5\\times D_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '52800.f', 'aut_centralizer_order': None, 'aut_label': '1056.e1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1320.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['528.d1.a1', '528.m1.a1', '528.m1.b1'], 'contains': ['2112.a1.a1', '5280.i1.a1', '5280.l1.a1'], 'core': '10560.a1.a1', 'coset_action_label': None, 'count': 132, 'diagramx': [1120, -1, 1553, -1, 1802, -1, 3411, -1], 'generators': [19458836602050490, 1714344618444302, 37595466963011736], 'label': '52800.f.1056.e1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '132.g1.a1', 'old_label': '1056.e1.a1', 'projective_image': '10560.e', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1056.e1.a1', 'subgroup_fusion': None, 'weyl_group': '10.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 20], 'aut_gens': [[1, 10], [1, 30], [27, 10]], 'aut_group': '80.30', 'aut_hash': 30, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 80, 'aut_permdeg': 9, 'aut_perms': [6496, 131169], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [10, 5, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [5, 1, 4], [5, 2, 10], [10, 5, 4]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [10, 5, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 0, 8]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '50.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 40], [21, 10]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 10], [2, 10]], 'label': '50.3', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*D5', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 20, 'number_divisions': 7, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 10, 'number_subgroups': 20, 'old_label': None, 'order': 50, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 5], [5, 24], [10, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [1, 0], 'outer_gens': [[9, 30], [7, 10]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 3], [8, 2]], 'representations': {'PC': {'code': 318031, 'gens': [1, 3], 'pres': [3, -2, -5, -5, 6, 362]}, 'Lie': [{'d': 1, 'q': 25, 'gens': [1818624673360232724641098, 4935872799604142007576, 800006261860070844758196], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 11, 'gens': [132, 3997, 5325]}, 'Perm': {'d': 10, 'gens': [2045400, 408993, 1451553]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_5', 'transitive_degree': 10, 'wreath_data': ['C_5', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [20, 22, 12, 4, 20, 20], 'aut_gens': [[12531822321003912, 15132536926350214, 11046409626844756, 40324864276851944, 37884680616577116, 27136409472509464, 33418192856010432], [37595466963011736, 6271083270386120, 41217938635736154, 45225922333974920, 32670378695680573, 23373744708349568, 8354548214002608], [16709096428005216, 40649872330802616, 13260044168661714, 15089486603864456, 27972317258104669, 23373744708349568, 29240918749009128], [16709096428005216, 6606824134882883, 10118503963757260, 15089486603864456, 5916706299688826, 27136409472509464, 29240918749009128], [20886370535006520, 28124374035010643, 1352354993236855, 12085751058942530, 23383275105687753, 19990544660987432, 25063644642007824], [20886370535006520, 7269364482440532, 32967917561469070, 12085751058942530, 16507475848864984, 30519609519871600, 25063644642007824], [37595466963011736, 32028058385184734, 32828686343133432, 30899074022311810, 45453762846123775, 30519609519871600, 8354548214002608]], 'aut_group': None, 'aut_hash': 2045418814995964927, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 84480, 'aut_permdeg': 140, 'aut_perms': [12332465421102242427446350401132538122721201430557817106058271912949157346854159218116365058419685782202666573192995375871304970868499064031429188492260467214975383385753454249036595024427498130524941798384412552326976285100702585438542071527, 7218200806279447298672610202419958362031370214943472130619209114620962788528119459031731909806813482914647318994928281242191795502815490177009737598899641113182958486381665490245005410841688158151334702417435607976612585805552217170199397103, 1145498047135498711374607264710668661947645103264503701541171100378524999924637330988642406744636734786043321480096875915716308343273751997554376757066323812031607274580137404785324274389982754807753019872274516940725967660849095628074485511, 6772886901681524229275191197747391083868768276756352800288888736921328250200912975024430081126167137802870570834667856453371594084150386898951771232053224384146760330360762086318259021157272670138368933145766611249105586020214700125962662193, 1565727394649537754562775799007078414694731516154805065278401135577931463963836101376159952742117992478118690029735103171364728168406391571864742783839911295847629085457762651886477087802301515355818825458877358393061381238408507878525187871, 13186282387754704336677155155838035258868156220412280684968262663103794973759746122834425876878934367424012302767146290952200891787766364835480082363967671709096199556008761193097455858007629880120553974218292131878976044014606361070359166534], 'aut_phi_ratio': 6.6, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [2, 110, 1, 1], [2, 132, 1, 1], [2, 132, 2, 1], [3, 110, 1, 1], [4, 2, 1, 1], [4, 110, 1, 1], [4, 110, 2, 1], [4, 132, 1, 1], [5, 1, 4, 1], [5, 132, 1, 2], [5, 132, 4, 2], [6, 110, 1, 1], [6, 220, 1, 1], [6, 220, 2, 1], [8, 110, 2, 1], [8, 220, 1, 1], [8, 220, 2, 1], [10, 1, 4, 1], [10, 2, 8, 1], [10, 110, 4, 1], [10, 132, 1, 2], [10, 132, 2, 2], [10, 132, 4, 3], [10, 132, 8, 3], [10, 264, 2, 4], [10, 264, 8, 4], [11, 120, 1, 1], [12, 110, 1, 2], [12, 220, 1, 1], [12, 220, 2, 1], [15, 110, 4, 1], [20, 2, 4, 1], [20, 110, 4, 1], [20, 110, 8, 1], [20, 132, 4, 1], [20, 264, 1, 4], [20, 264, 4, 4], [22, 120, 1, 1], [22, 240, 2, 1], [24, 110, 2, 2], [24, 220, 1, 2], [24, 220, 2, 2], [30, 110, 4, 1], [30, 220, 4, 1], [30, 220, 8, 1], [40, 110, 8, 1], [40, 220, 4, 1], [40, 220, 8, 1], [44, 240, 1, 1], [55, 120, 4, 1], [60, 110, 4, 2], [60, 220, 4, 1], [60, 220, 8, 1], [110, 120, 4, 1], [110, 240, 8, 1], [120, 110, 8, 2], [120, 220, 4, 2], [120, 220, 8, 2], [220, 240, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_4\\times \\PSL(2,11).C_2\\times D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.45', 'autcent_hash': 45, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 660, 'autcentquo_group': '2640.a', 'autcentquo_hash': 9195845394065659656, 'autcentquo_nilpotent': False, 'autcentquo_order': 2640, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times \\PGL(2,11)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [2, 110, 1], [2, 132, 3], [3, 110, 1], [4, 2, 1], [4, 110, 3], [4, 132, 1], [5, 1, 4], [5, 132, 10], [6, 110, 1], [6, 220, 3], [8, 110, 2], [8, 220, 3], [10, 1, 4], [10, 2, 8], [10, 110, 4], [10, 132, 42], [10, 264, 40], [11, 120, 1], [12, 110, 2], [12, 220, 3], [15, 110, 4], [20, 2, 4], [20, 110, 12], [20, 132, 4], [20, 264, 20], [22, 120, 1], [22, 240, 2], [24, 110, 4], [24, 220, 6], [30, 110, 4], [30, 220, 12], [40, 110, 8], [40, 220, 12], [44, 240, 1], [55, 120, 4], [60, 110, 8], [60, 220, 12], [110, 120, 4], [110, 240, 8], [120, 110, 16], [120, 220, 24], [220, 240, 4]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': None, 'commutator_count': 1, 'commutator_label': '1320.13', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '660.13'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['10560.e', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [2, 110, 1, 1], [2, 132, 1, 3], [3, 110, 1, 1], [4, 2, 1, 1], [4, 110, 1, 3], [4, 132, 1, 1], [5, 1, 4, 1], [5, 132, 2, 1], [5, 132, 4, 2], [6, 110, 1, 1], [6, 220, 1, 3], [8, 110, 2, 1], [8, 220, 1, 3], [10, 1, 4, 1], [10, 2, 4, 2], [10, 110, 4, 1], [10, 132, 2, 1], [10, 132, 4, 10], [10, 264, 2, 4], [10, 264, 4, 8], [11, 120, 1, 1], [12, 110, 2, 1], [12, 220, 1, 3], [15, 110, 4, 1], [20, 2, 4, 1], [20, 110, 4, 3], [20, 132, 4, 1], [20, 264, 2, 2], [20, 264, 4, 4], [22, 120, 1, 1], [22, 240, 1, 2], [24, 110, 4, 1], [24, 220, 2, 3], [30, 110, 4, 1], [30, 220, 4, 3], [40, 110, 8, 1], [40, 220, 4, 3], [44, 240, 1, 1], [55, 120, 4, 1], [60, 110, 8, 1], [60, 220, 4, 3], [110, 120, 4, 1], [110, 240, 4, 2], [120, 110, 16, 1], [120, 220, 8, 3], [220, 240, 4, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 562776480, 'exponent': 1320, 'exponents_of_order': [6, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[20, 0, 24], [24, 0, 20]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': None, 'hash': 7693100823582725605, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 660, 'inner_gen_orders': [1, 2, 10, 2, 6, 2, 1], 'inner_gens': [[12531822321003912, 15132536926350214, 11046409626844756, 40324864276851944, 37884680616577116, 27136409472509464, 33418192856010432], [12531822321003912, 15132536926350214, 7039805157630056, 40324864276851944, 13915786098996659, 27136409472509464, 33418192856010432], [12531822321003912, 18992209968962666, 11046409626844756, 10185289904007088, 11682981661140105, 23373744708349568, 33418192856010432], [12531822321003912, 15132536926350214, 39118541029782127, 40324864276851944, 37884680616577116, 23373744708349568, 33418192856010432], [12531822321003912, 44933021264876801, 27311851406251630, 40324864276851944, 37884680616577116, 23373744708349568, 33418192856010432], [12531822321003912, 15132536926350214, 39118541029782127, 10185289904007088, 8482773849563087, 27136409472509464, 33418192856010432], [12531822321003912, 15132536926350214, 11046409626844756, 40324864276851944, 37884680616577116, 27136409472509464, 33418192856010432]], 'inner_hash': 1422479616026575418, 'inner_nilpotent': False, 'inner_order': 5280, 'inner_split': True, 'inner_tex': 'C_2^2.\\PSL(2,11).C_2', 'inner_used': [2, 3, 4, 6], 'irrC_degree': 20, 'irrQ_degree': 96, 'irrQ_dim': 96, 'irrR_degree': 40, 'irrep_stats': [[1, 40], [10, 100], [11, 40], [12, 80], [20, 30], [24, 25]], 'label': '52800.f', 'linC_count': 24, 'linC_degree': 20, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 28, 'linQ_degree_count': 8, 'linQ_dim': 28, 'linQ_dim_count': 8, 'linR_count': 16, 'linR_degree': 26, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'GL(2,11):C2^2', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 90, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 315, 'number_divisions': 99, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 608, 'number_subgroup_classes': 986, 'number_subgroups': 82612, 'old_label': None, 'order': 52800, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 511], [3, 110], [4, 464], [5, 1324], [6, 770], [8, 880], [10, 16564], [11, 120], [12, 880], [15, 440], [20, 7136], [22, 600], [24, 1760], [30, 3080], [40, 3520], [44, 240], [55, 480], [60, 3520], [110, 2400], [120, 7040], [220, 960]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [4177274107001304, 4177274107001304, 4177274107001304], 'outer_gens': [[12531822321003912, 15132536926350214, 11046409626844756, 34661738572114156, 20906487832601043, 19990544660987432, 33418192856010432], [12531822321003912, 15132536926350214, 39118541029782127, 40324864276851944, 37884680616577116, 27136409472509464, 33418192856010432], [37595466963011736, 40457382395368985, 11046409626844756, 5284231846884112, 20960243757025835, 27136409472509464, 8354548214002608]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [120, 5040, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 53, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 8], [10, 12], [11, 8], [20, 4], [24, 9], [40, 13], [44, 8], [48, 16], [80, 5], [96, 6], [160, 1], [320, 1]], 'representations': {'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 15132536926350214, 11046409626844756, 40324864276851944, 37884680616577116, 27136409472509464, 33418192856010432]}, 'Perm': {'d': 53, 'gens': [46, 80813352015732470688313784482467135809845499609386893118061471188383, 170688313841736247916850850062744224944679651693256863382842379999360, 253075796313307954193651261800427805934891582153648474020907709324206, 335322130990318569682065861289587220304565520268526364800111245959037, 411358025327350499740048471215077183319541396219269828561873888014748, 492047283688462284232185771730623398173536976273510549966257077573806]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\GL(2,11):C_2^2', 'transitive_degree': 240, 'wreath_data': None, 'wreath_product': False}
