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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '486.260', 'ambient_counter': 260, 'ambient_order': 486, 'ambient_tex': 'C_3^4:S_3', 'central': False, 'central_factor': False, 'centralizer_order': 243, 'characteristic': False, 'core_order': 3, 'counter': 10, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '486.260.162.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '162.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '162.54', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 54, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 162, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3^3:S_3', 'simple': True, 'solvable': True, 'special_labels': ['C5'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '486.260', 'aut_centralizer_order': 477531624960, 'aut_label': '162.a1', 'aut_quo_index': 1, 'aut_stab_index': 121, 'aut_weyl_group': '2.1', 'aut_weyl_index': 57781326620160, 'centralizer': '2.a1', 'complements': ['3.a1'], 'conjugacy_class_count': 121, 'contained_in': ['54.a1', '81.a1'], 'contains': ['486.a1'], 'core': '162.a1', 'coset_action_label': None, 'count': 121, 'diagramx': [3818, 5000, 3578, 5000], 'generators': [2351581309], 'label': '486.260.162.a1', 'mobius_quo': -1, 'mobius_sub': -59049, 'normal_closure': '162.a1', 'normal_contained_in': ['54.a1'], 'normal_contains': ['486.a1'], 'normalizer': '1.a1', 'old_label': '162.a1', 'projective_image': '486.260', 'quotient_action_image': '2.1', 'quotient_action_kernel': '81.15', 'quotient_action_kernel_order': 81, 'quotient_fusion': None, 'short_label': '162.a1', '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': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, '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': 'C3', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[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': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 1132560, 'aut_gen_orders': [121, 2, 3], 'aut_gens': [[1435351114, 2892333419, 2351581309, 25154876, 576898846, 368382181], [2121265556, 2355113313, 1224828790, 25154876, 2218581694, 1383477188], [3310511604, 2892333419, 2351581309, 25154876, 576898846, 2756441483], [1435351114, 2892333419, 2351581309, 2272325038, 576898846, 368382181]], 'aut_group': '115562653240320.a', 'aut_hash': 632204379810423377, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 115562653240320, 'aut_permdeg': 243, 'aut_perms': [20728704531154982206662521097902511754177402768278494075677729284289892425366327596855843177220767607829751256028955690014933080739997550555566714727070152262148167099502170605702110174550404931037061638709515654148761127191201767962708075219195032701006975414419032584848862082250533628235959860238328144223946610438927759372494982426093294467079476033319886380830037975767370306989762608820974761368578456089633874261664687778750611442365830795037966920543751010899719653747, 56586848241311755480324177830747876934255663941799753464590974081145213407850762118057190435535264441369029249766429774619841135240589475958832831045882423500670322784263026519756637746749607888555756867666002071084784537672496008513874582728101222316300366224062834026321980413193847919586444892407783229044351264012883486874074494890259944919317766769461479631049703709179750061411899495019150216457840364876782165190881325062352727415227794158003467845752520668972725774376, 56587165617778856460414566316333672859058005447954158112655509383146035903729620511213614842108640159147312816052480628440187614781941632724127326172788160561720496978431319738904637174728322656982029594530453798326307433128263684744238169932829851420279076028784728148685808298205422527803037001925727493448093962522607142022901980170481070371416323966694399612090578298504115447506792900173096573588570032622671631969408228318715987240442596988451237255386464022984681852401], 'aut_phi_ratio': 713349711360.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 121, 1]], 'aut_supersolvable': False, 'aut_tex': '\\AGL(5,3)', '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': 1132560, 'autcentquo_group': '115562653240320.a', 'autcentquo_hash': 632204379810423377, 'autcentquo_nilpotent': False, 'autcentquo_order': 115562653240320, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\AGL(5,3)', 'cc_stats': [[1, 1, 1], [2, 243, 1], [3, 2, 121]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '486.260', 'commutator_count': 1, 'commutator_label': '243.67', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 260, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 1, 121]], 'element_repr_type': 'GLFq', 'elementary': 1, 'eulerian_function': 63, 'exponent': 6, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 5, 11, 13], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '486.260', 'hash': 260, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 3, 2, 3, 3], 'inner_gens': [[1435351114, 2892333419, 2351581309, 1445625641, 576898846, 368382181], [1435351114, 2892333419, 2351581309, 2888790477, 576898846, 368382181], [1435351114, 2892333419, 2351581309, 2345912606, 576898846, 368382181], [2468342098, 2321716242, 1185357808, 25154876, 1137795295, 191484025], [1435351114, 2892333419, 2351581309, 586110491, 576898846, 368382181], [1435351114, 2892333419, 2351581309, 364839242, 576898846, 368382181]], 'inner_hash': 260, 'inner_nilpotent': False, 'inner_order': 486, 'inner_split': True, 'inner_tex': 'C_3^4:S_3', 'inner_used': [1, 2, 3, 4, 5, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 121]], 'label': '486.260', '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': 'C3^4:S3', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 123, 'number_divisions': 123, 'number_normal_subgroups': 2665, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 5328, 'number_subgroups': 56632, 'old_label': None, 'order': 486, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 243], [3, 242]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 1132560, 'outer_gen_orders': [3, 5], 'outer_gen_pows': [25154876, 14348908], 'outer_gens': [[3271394916, 1525020304, 1185357808, 25154876, 350488148, 980207619], [3069386852, 2848079468, 211510383, 25154876, 980207619, 104628271]], 'outer_group': '237783237120.a', 'outer_hash': 3786363380318986680, 'outer_nilpotent': False, 'outer_order': 237783237120, 'outer_permdeg': 121, 'outer_perms': [642918142256503458059786895071656860016602763450765560460182761926889417092061927780677813852000684007776826010790345747415962834908298009276145043588052997583163728419278935428749310397948537655519395, 778851398001434174457157092584816524824863900696212592598860293238764876631262858196023194421496120800146484570038422576952145838237524793804747553128815814661010408453797895640713692477494802296460116], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\SL(5,3)', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 6, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 121]], 'representations': {'PC': {'code': 3767160318631933983, 'gens': [1, 2, 3, 4, 5, 6], 'pres': [6, -2, -3, -3, -3, -3, -3, 49, 218, 867, 3244, 11669]}, 'GLFq': {'d': 2, 'q': 243, 'gens': [1435351114, 2892333419, 2351581309, 25154876, 576898846, 368382181]}, 'Perm': {'d': 15, 'gens': [6230654665, 93448903104, 174400128000, 147, 148, 43545744]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:S_3', 'transitive_degree': 243, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 9360, 'aut_gen_orders': [2, 13, 3], 'aut_gens': [[24577758, 24507118, 2939606, 36731395, 14159168], [24577758, 24950012, 3035065, 37169915, 14159168], [14481270, 11495158, 20254942, 11661233, 14159168], [24577758, 24507118, 2939606, 36731395, 13835790]], 'aut_group': '1965150720.a', 'aut_hash': 6119322437363060898, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1965150720, 'aut_permdeg': 81, 'aut_perms': [657771367610839812722531985099075193581105650644659615214452318713464489102759582557145784570236080374102250639668958972, 808612898731331102202427324917560800475004880744121933655982886623123936826672862299267279681933963840112953340251939252, 707884959424340928995589841961242110118180352872775572462057266195870131286433642548161864689177486446723818580494968017], 'aut_phi_ratio': 36391680.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 40, 1]], 'aut_supersolvable': False, 'aut_tex': '\\AGL(4,3)', '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': 9360, 'autcentquo_group': '1965150720.a', 'autcentquo_hash': 6119322437363060898, 'autcentquo_nilpotent': False, 'autcentquo_order': 1965150720, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\AGL(4,3)', 'cc_stats': [[1, 1, 1], [2, 81, 1], [3, 2, 40]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '162.54', 'commutator_count': 1, 'commutator_label': '81.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 54, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 2, 1, 40]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 31, 'exponent': 6, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 5, 13], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '162.54', 'hash': 54, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 3, 3, 2], 'inner_gens': [[24577758, 24507118, 2939606, 36731395, 8188666], [24577758, 24507118, 2939606, 36731395, 8314856], [24577758, 24507118, 2939606, 36731395, 39300942], [24577758, 24507118, 2939606, 36731395, 17350487], [20201582, 20254942, 32864349, 11225629, 14159168]], 'inner_hash': 54, 'inner_nilpotent': False, 'inner_order': 162, 'inner_split': False, 'inner_tex': 'C_3^3:S_3', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 40]], 'label': '162.54', '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': 'C3^3:S3', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 42, 'number_divisions': 42, 'number_normal_subgroups': 213, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 424, 'number_subgroups': 2664, 'old_label': None, 'order': 162, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 81], [3, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4680, 'outer_gen_orders': [2, 13], 'outer_gen_pows': [14408200, 14408200], 'outer_gens': [[7411244, 32705081, 20646633, 11444201, 14159168], [3035065, 11225629, 14686477, 32653905, 14159168]], 'outer_group': '12130560.c', 'outer_hash': 8461112410546189005, 'outer_nilpotent': False, 'outer_order': 12130560, 'outer_permdeg': 40, 'outer_perms': [19409857794192766541401284787497792047355199198, 743042578780903555112641255592744448909112275761], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PGL(4,3)', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 40]], 'representations': {'PC': {'code': 2271081930255, 'gens': [1, 2, 3, 4, 5], 'pres': [5, -2, -3, -3, -3, -3, 41, 182, 723, 2704]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 24507118, 2939606, 36731395, 14159168]}, 'Perm': {'d': 12, 'gens': [3633865, 43545747, 79833843, 43591200, 79879200]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3', 'transitive_degree': 81, 'wreath_data': None, 'wreath_product': False}
