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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '700.35', 'ambient_counter': 35, 'ambient_order': 700, 'ambient_tex': 'C_5:D_{70}', 'central': False, 'central_factor': False, 'centralizer_order': 350, 'characteristic': False, 'core_order': 10, 'counter': 55, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '700.35.70.a1.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '70.a1.e1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '70.3', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 70, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{35}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '10.2', 'subgroup_hash': 2, 'subgroup_order': 10, 'subgroup_tex': 'C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '700.35', 'aut_centralizer_order': 42000, 'aut_label': '70.a1', 'aut_quo_index': 1, 'aut_stab_index': 6, 'aut_weyl_group': '4.1', 'aut_weyl_index': 252000, 'centralizer': '2.a1.a1', 'complements': ['10.b1.a1', '10.b1.g1', '10.b1.b1', '10.b1.h1', '10.b1.f1', '10.b1.l1', '10.b1.c1', '10.b1.i1', '10.b1.d1', '10.b1.j1'], 'conjugacy_class_count': 1, 'contained_in': ['10.a1.e1', '14.a1.a1', '35.a1.e1'], 'contains': ['140.a1.e1', '350.a1.a1'], 'core': '70.a1.e1', 'coset_action_label': None, 'count': 1, 'diagramx': [3327, 4987, 5378, 9736, 6460, 6898, 8927, 6796], 'generators': [350, 142], 'label': '700.35.70.a1.e1', 'mobius_quo': 1, 'mobius_sub': -35, 'normal_closure': '70.a1.e1', 'normal_contained_in': ['10.a1.e1', '14.a1.a1'], 'normal_contains': ['140.a1.e1', '350.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '70.a1.e1', 'projective_image': '350.9', 'quotient_action_image': '2.1', 'quotient_action_kernel': '35.1', 'quotient_action_kernel_order': 35, 'quotient_fusion': None, 'short_label': '70.a1.e1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [7]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', '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': 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, 1], [5, 1, 4], [10, 1, 4]], 'center_label': '10.2', 'center_order': 10, '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', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 10, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '10.2', 'hash': 2, 'hyperelementary': 10, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 10]], 'label': '10.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C10', 'ngens': 2, 'nilpotency_class': 1, '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': 10, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 10, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [5, 4], [10, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[7]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2]], 'representations': {'PC': {'code': 83, 'gens': [1], 'pres': [2, -2, -5, 4]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16717348]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504]}, 'Perm': {'d': 7, 'gens': [720, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}', 'transitive_degree': 10, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, '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': 840, 'aut_gen_orders': [20, 12, 12, 60], 'aut_gens': [[1, 2, 10], [117, 288, 558], [371, 6, 192], [369, 560, 32], [145, 2, 178]], 'aut_group': None, 'aut_hash': 4468114598930034496, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1008000, 'aut_permdeg': 350, 'aut_perms': [104080732471909570825580642783248479484184400390198667687036619034599541325724880497045688653332286896822819189286356093280987617636854687481705635007707754498421834289713179488403277148443352975265013182063002392829469244600946713608996213296249650354282138467647514665070649979000320275098643903206098082059919410426143584973484268566262132305669514364943207655075872168000458760283842927089401713685013924391118667903656850507149795198134012991229296674909991661958074191814779373755033594458440066255455888223110855440086753365373173004812323744485404189974074685756815047596027720748335673927791450164529754428519967060571655433835389427404381006859875556151199791348027595034448590528294766837423982128853916330642203847777535848338925, 37147017644171539086213347816407575892421140685694492242661185889847481020095601188922696322603426235202964856019636134721218281057584374302775184142271864619456096030927512118547547331174542669008354298122533053541519417437133318325056951992496391147970461634233920926517950101416799938810000145257290544639524330194420136875532388977334017258491398525006031869272048356606531029247837517659586985316774089356624687480853604606884903467720859512958826833096751299681298515853176081538016512662296097245111736182512160021188706578987749070173766231922944624291203387705708543043085416958784310358617909503519334809272914773692173215069987661042715369503200461760879481331586820656779419649286126041257793871719310409098739104762735937341834, 516487198518758462075166593164103403193752498441937680973757768969626967444368770225089919556837134959633367884313046519165660705571839420913635443764209287784897908248545722139329438870649455871905915800429483818357724574001996584345641677444543249979295928212957642533552429575453319580739824700532376146859939313677129186730584722496520538147315726823833823058948135378929280444300453006062256721052016787487224038242876299512794628663318037782098942772149001474521626013969042905660561995322384742246355991704494416403381328312073132312124442605525334197124724624997987697532364246786404543005674501754362433245568502052311881565290749388547596847666243246751715201436994966805311375019281255484839689414564505885843821873449499579690, 119346322791076150473230902030132095135488274616570028558666290657115082818511698764683489886221108515852096537844260776704204900845926074311408707243280717913158961248677095520511286830289208081242964622183693874753223796266460154836734602343807775153237761098572623179796563040816787148435008966839892575085612008829115041581715873981656663480211106602573247805102933399738350113006118162742079603761267686057066512666800683376009665584389781544170431394630221623356378169054970645075101505742764600124035515377478328781034833741409898223606307076462863590058300320213464468058005170023274955630981232922798849909340618564931103154959439889499121357088212300701217898171193588611662773443648168059668298454693669692852246367289050921048456], 'aut_phi_ratio': 4200.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 175, 2, 1], [5, 2, 12, 1], [7, 2, 3, 1], [10, 2, 12, 1], [14, 2, 3, 1], [35, 2, 72, 1], [70, 2, 72, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_5^2:C_4.S_5\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 840, 'autcentquo_group': None, 'autcentquo_hash': 482492227677916899, 'autcentquo_nilpotent': False, 'autcentquo_order': 504000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^2:C_4.S_5\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 175, 2], [5, 2, 12], [7, 2, 3], [10, 2, 12], [14, 2, 3], [35, 2, 72], [70, 2, 72]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '350.9', 'commutator_count': 1, 'commutator_label': '175.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '5.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 35, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['350.9', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 175, 1, 2], [5, 2, 2, 6], [7, 2, 3, 1], [10, 2, 2, 6], [14, 2, 3, 1], [35, 2, 12, 6], [70, 2, 12, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 168, 'exponent': 70, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '700.35', 'hash': 35, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [2, 5, 35], 'inner_gens': [[1, 8, 690], [5, 2, 10], [21, 2, 10]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 350, 'inner_split': True, 'inner_tex': 'C_5:D_{35}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 174]], 'label': '700.35', 'linC_count': 8640, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 105, 'linQ_dim': 14, 'linQ_dim_count': 105, 'linR_count': 8640, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5:D70', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 178, 'number_divisions': 30, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 80, 'number_subgroups': 1376, 'old_label': None, 'order': 700, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 351], [5, 24], [7, 6], [10, 24], [14, 6], [35, 144], [70, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [3, 4, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 284, 572], [1, 144, 290], [351, 2, 310]], 'outer_group': '2880.g', 'outer_hash': 6844346583817555215, 'outer_nilpotent': False, 'outer_order': 2880, 'outer_permdeg': 29, 'outer_perms': [2545696549197715182288323235120, 1453094803747342386324563477040, 49], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times \\GL(2,5)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 12], [6, 2], [24, 12]], 'representations': {'PC': {'code': 107889664967768436762911, 'gens': [1, 2, 3], 'pres': [5, -2, -5, -2, -5, -7, 81, 10352, 42, 13603, 118, 15004]}, 'Perm': {'d': 19, 'gens': [356996140305966, 1, 7115063980147200, 7115069691878400, 50646]}}, 'schur_multiplier': [10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5:D_{70}', 'transitive_degree': 350, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [2, 12, 35], 'aut_gens': [[1, 2], [1, 12], [1, 46], [65, 2]], 'aut_group': '840.139', 'aut_hash': 139, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 840, 'aut_permdeg': 12, 'aut_perms': [15729840, 8427613, 258013504], 'aut_phi_ratio': 35.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 35, 1, 1], [5, 2, 2, 1], [7, 2, 3, 1], [35, 2, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_5\\times F_7', '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': 420, 'autcentquo_group': '840.139', 'autcentquo_hash': 139, 'autcentquo_nilpotent': False, 'autcentquo_order': 840, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5\\times F_7', 'cc_stats': [[1, 1, 1], [2, 35, 1], [5, 2, 2], [7, 2, 3], [35, 2, 12]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '70.3', 'commutator_count': 1, 'commutator_label': '35.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '7.1'], '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, 35, 1, 1], [5, 2, 2, 1], [7, 2, 3, 1], [35, 2, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 70, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': [[2, 1, 12]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '70.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [2, 35], 'inner_gens': [[1, 68], [5, 2]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 70, 'inner_split': True, 'inner_tex': 'D_{35}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 17]], 'label': '70.3', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 12, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D35', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 19, 'number_divisions': 5, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 52, 'old_label': None, 'order': 70, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 35], [5, 4], [7, 6], [35, 24]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [12], 'outer_gen_pows': [0], 'outer_gens': [[1, 34]], 'outer_group': '12.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [867], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 1], [6, 1], [24, 1]], 'representations': {'PC': {'code': 182417639, 'gens': [1, 2], 'pres': [3, -2, -5, -7, 409, 34, 542]}, 'GLFp': {'d': 2, 'p': 71, 'gens': [10379468, 5112]}, 'Perm': {'d': 12, 'gens': [3669847, 37, 47255760]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{35}', 'transitive_degree': 35, 'wreath_data': None, 'wreath_product': False}
