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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '6948.a', 'ambient_counter': 1, 'ambient_order': 6948, 'ambient_tex': 'C_{579}:C_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 579, 'counter': 14, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '6948.a.12.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '12.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '12.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 12, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{12}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '579.1', 'subgroup_hash': 1, 'subgroup_order': 579, 'subgroup_tex': 'C_{193}:C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6948.a', 'aut_centralizer_order': None, 'aut_label': '12.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2316.a1.a1', 'complements': ['579.a1.a1', '579.b1.b1', '579.b1.c1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '6.b1.a1'], 'contains': ['36.a1.a1', '2316.b1.a1'], 'core': '12.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1831, 3372, 2557, 9364, 2600, 5072, 3365, 3210], 'generators': [28, 36], 'label': '6948.a.12.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '12.b1.a1', 'normal_contained_in': ['4.a1.a1', '6.b1.a1'], 'normal_contains': ['36.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '12.b1.a1', 'projective_image': '6948.a', 'quotient_action_image': '4.1', 'quotient_action_kernel': '3.1', 'quotient_action_kernel_order': 3, 'quotient_fusion': None, 'short_label': '12.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2316.b'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 37056, 'aut_gen_orders': [192, 193], 'aut_gens': [[1, 3], [1, 114], [445, 3]], 'aut_group': '37056.a', 'aut_hash': 1432757702296672986, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 37056, 'aut_permdeg': 193, 'aut_perms': [39053339670721327572543569039395005031096653245802774324204298263788953285647090763476541643345676324473056783378543774660072842471514711015868302482279618194714261105949788702112347134548908091797187306612050281053946057960581089182560863568602379974419627409015496038525465690691519350455551181816446171767151758025418672254963521648341121096265656543605694, 27537046770698028029509447452483255633857520585283838421883440607221768231104384860166214394878916709368227289860720561987161029954363059081082583776521393368876604263321802053274125013685434525755757109968295753899028688290909515762484435944252483952769187280784953622110633939104871157686235761278347542756654691358393510201744955724775767189520816064076473], 'aut_phi_ratio': 96.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 193, 1, 2], [193, 3, 64, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_{193}', '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': 37056, 'autcentquo_group': '37056.a', 'autcentquo_hash': 1432757702296672986, 'autcentquo_nilpotent': False, 'autcentquo_order': 37056, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{193}', 'cc_stats': [[1, 1, 1], [3, 193, 2], [193, 3, 64]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '579.1', 'commutator_count': 1, 'commutator_label': '193.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '193.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 193, 2, 1], [193, 3, 64, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 8, 'exponent': 579, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3, 193], 'factors_of_order': [3, 193], 'faithful_reps': [[3, 0, 64]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '579.1', 'hash': 1, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 579, 'inner_gen_orders': [3, 193], 'inner_gens': [[1, 252], [331, 3]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 579, 'inner_split': True, 'inner_tex': 'C_{193}:C_3', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 192, 'irrQ_dim': 192, 'irrR_degree': 6, 'irrep_stats': [[1, 3], [3, 64]], 'label': '579.1', 'linC_count': 64, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 192, 'linQ_degree_count': 1, 'linQ_dim': 192, 'linQ_dim_count': 1, 'linR_count': 32, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C193:C3', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, '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': 3, 'number_conjugacy_classes': 67, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 196, 'old_label': None, 'order': 579, 'order_factorization_type': 11, 'order_stats': [[1, 1], [3, 386], [193, 192]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 64, 'outer_gen_orders': [64], 'outer_gen_pows': [2], 'outer_gens': [[1, 447]], 'outer_group': '64.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 64, 'outer_perms': [2014594330940748694500315964401030837224914941382363373987443624058107002128920420940313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{64}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 193, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [192, 1]], 'representations': {'PC': {'code': 12312959, 'gens': [1, 2], 'pres': [2, -3, -193, 1009]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [7189251, 603880789]}, 'Perm': {'d': 193, 'gens': [155072081769025479153068005764655310398056985534484553249510570014180068596765767602220702796160851423904603713380353967374496827693048038873577863314427425322295450355127868085281304176469528282592426059218603719048013636231661630498986145587733965119292349387964516295495591000760117599793495606069314966880735163520645227705730930758607614125986326901066, 68159384284216415434244229769722897785298203470789342442007462716964519312047994015473537393503254737883254692840162175520283792238796194464802516244938515792342454228783658226402112128483243230028631870556954631033853483990335822197126471859553676149202878029250924171717175622127511797106029233701101543903723520000000000000000000000000000000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{193}:C_3', 'transitive_degree': 193, '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': '36.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 37056, 'aut_gen_orders': [96, 192, 3], 'aut_gens': [[1, 12], [6541, 5280], [2941, 264], [457, 3612]], 'aut_group': None, 'aut_hash': 9034218873832238734, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 222336, 'aut_permdeg': 772, 'aut_perms': 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'aut_phi_ratio': 96.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 193, 1, 1], [3, 1, 2, 1], [3, 193, 3, 2], [4, 193, 1, 2], [6, 193, 2, 1], [6, 193, 3, 2], [12, 193, 2, 2], [12, 193, 3, 4], [193, 12, 16, 1], [579, 12, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{579}.C_{192}.C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 37056, 'autcentquo_group': '37056.a', 'autcentquo_hash': 1432757702296672986, 'autcentquo_nilpotent': False, 'autcentquo_order': 37056, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{193}', 'cc_stats': [[1, 1, 1], [2, 193, 1], [3, 1, 2], [3, 193, 6], [4, 193, 2], [6, 193, 8], [12, 193, 16], [193, 12, 16], [579, 12, 32]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '2316.b', 'commutator_count': 1, 'commutator_label': '193.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '193.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2316.b', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 193, 1, 1], [3, 1, 2, 1], [3, 193, 2, 3], [4, 193, 2, 1], [6, 193, 2, 4], [12, 193, 4, 4], [193, 12, 16, 1], [579, 12, 32, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 96, 'exponent': 2316, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 193], 'factors_of_order': [2, 3, 193], 'faithful_reps': [[12, 0, 32]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '6948.a', 'hash': 163829408175284635, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 2316, 'inner_gen_orders': [12, 193], 'inner_gens': [[1, 1560], [5401, 12]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 2316, 'inner_split': False, 'inner_tex': 'C_{193}:C_{12}', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 384, 'irrQ_dim': 384, 'irrR_degree': 24, 'irrep_stats': [[1, 36], [12, 48]], 'label': '6948.a', 'linC_count': 32, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 194, 'linQ_degree_count': 6, 'linQ_dim': 194, 'linQ_dim_count': 6, 'linR_count': 192, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C579:C12', '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': 18, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 84, 'number_divisions': 17, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 36, 'number_subgroups': 3108, 'old_label': None, 'order': 6948, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 193], [3, 1160], [4, 386], [6, 1544], [12, 3088], [193, 192], [579, 384]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 48, 'outer_gen_orders': [2, 48], 'outer_gen_pows': [0, 3], 'outer_gens': [[2317, 6936], [4633, 5124]], 'outer_group': '96.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 19, 'outer_perms': [6423384156578665, 13560771052696588], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times C_{16}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 196, 'pgroup': 0, 'primary_abelian_invariants': [4, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 4], [192, 1], [384, 1]], 'representations': {'PC': {'code': '2484544393403494715474679500204522112733094646143', 'gens': [1, 4], 'pres': [5, -2, -2, -3, -3, -193, 10, 26, 31203, 13088, 18073, 78, 117004, 49059, 24314]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [7189251, 7189106, 776418240]}, 'Perm': {'d': 196, 'gens': [26997883370683755077834348632583544075689123646545924239333415195270970618446762803151534820839084168332574930202330878136717844416248344506188842014859582665659028857979420994463281937424786636465603603264193802991238759711988872255230924689688520127256054785069161550511582761091078406738638113786583221698027154119610457882727594361829212191882754524688969640, 2631777980035845350135382426734565011034447922052599891420990730113738056753597827295822581371823538584720697912300444475715628525068356947731137665342306565309372655949933937527381197282002066466381013963617266693057375177357599189542857037592345127941010427549992337437299501736403248100214772049765657481237711923522648211336620799430801152351097935929224721400, 53649271295476239427485909099775454234166999615723762995659120830603826843282000977475945190144260550089319006283869137285687970646473984268535334212926921323592118929967245128994273681836852701890011310405127668484185849915818135096279080876220528005067781820885549539553055118957409493114150463364122295060375962689736182697797098731623576456621354100625274163]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 12], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{579}:C_{12}', 'transitive_degree': 579, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '12.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2], 'aut_gens': [[1], [5], [7]], 'aut_group': '4.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [1, 6], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2', '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': 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], [3, 1, 2], [4, 1, 2], [6, 1, 2], [12, 1, 4]], 'center_label': '12.2', 'center_order': 12, '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', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 12, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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, 12]], 'label': '12.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C12', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, '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': 12, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [12, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5], [7]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [4, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 3865, 'gens': [1], 'pres': [3, -2, -2, -3, 6, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20970031]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [619]}, 'Perm': {'d': 7, 'gens': [2400, 4, 744]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [12], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
