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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '486.213', 'ambient_counter': 213, 'ambient_order': 486, 'ambient_tex': 'C_3\\times C_6.\\He_3', 'central': False, 'central_factor': False, 'centralizer_order': 18, 'characteristic': False, 'core_order': 27, 'counter': 18, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '486.213.18.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '18.c1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '18.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 18, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3\\times C_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '27.4', 'subgroup_hash': 4, 'subgroup_order': 27, 'subgroup_tex': 'C_9:C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '486.213', 'aut_centralizer_order': 54, 'aut_label': '18.c1', 'aut_quo_index': 4, 'aut_stab_index': 9, 'aut_weyl_group': '54.8', 'aut_weyl_index': 486, 'centralizer': '27.a1', 'complements': [], 'conjugacy_class_count': 9, 'contained_in': ['6.b1', '6.c1', '9.c1'], 'contains': ['54.b1', '54.f1'], 'core': '18.c1', 'coset_action_label': None, 'count': 9, 'diagramx': [9518, 3860, 3948, 2627], 'generators': [376801, 373987], 'label': '486.213.18.c1', 'mobius_quo': 0, 'mobius_sub': -3, 'normal_closure': '18.c1', 'normal_contained_in': ['6.b1', '6.c1', '9.c1'], 'normal_contains': ['54.b1'], 'normalizer': '1.a1', 'old_label': '18.c1', 'projective_image': '162.48', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.c1', 'subgroup_fusion': None, 'weyl_group': '27.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [3, 3, 3, 2], 'aut_gens': [[1, 3], [1, 5], [19, 21], [1, 12], [19, 24]], 'aut_group': '54.8', 'aut_hash': 8, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 54, 'aut_permdeg': 9, 'aut_perms': [232440, 53286, 123858, 3838], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 1, 2], [9, 3, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2:S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '9.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 9, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 3, 2], [9, 3, 6]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [9, 3, 2, 3]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 8, 'exponent': 9, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [[3, 0, 2]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '9.2', 'hash': 4, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3], 'inner_gens': [[1, 21], [10, 3]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': True, 'inner_tex': 'C_3^2', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 9], [3, 2]], 'label': '27.4', 'linC_count': 2, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C9:C3', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 11, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 8], [9, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [2, 0], 'outer_gens': [[1, 6], [1, 5]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 1]], 'representations': {'PC': {'code': 1766, 'gens': [1, 2], 'pres': [3, -3, 3, -3, 127, 22]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [52802554844085621, 108238938226410615]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [36709477, 10475581, 23068812]}, 'Perm': {'d': 9, 'gens': [357120, 243, 80884]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9:C_3', 'transitive_degree': 9, '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': '54.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [18, 6, 3, 3, 3, 3, 3, 3], 'aut_gens': [[373996, 376801, 374473, 164375], [373996, 497143, 373987, 171008], [197326, 333019, 373987, 348164], [373996, 199177, 374230, 518408], [373996, 376801, 374473, 164618], [373996, 199159, 374473, 341504], [373996, 22507, 374473, 164618], [373996, 376567, 374473, 341504], [373996, 376792, 374473, 518660]], 'aut_group': '26244.nn', 'aut_hash': 4111858139665495509, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 26244, 'aut_permdeg': 164, 'aut_perms': [288645379492613763798016448661585266967800031834704951218902371043617473540105955903168340926200669932807551677291279063946032098958632708831443610791295817678125442558661592886789482072097058897708099444150279884738297009822675453529274539587417949119410338050985082556740401181409434954807879, 170974799226641484416506336080741019295702793739718869088575951640988282247687033131958509686544327352863065519351601749997341796217711004617751797715684335567490252628216054891446211263090339722798409176010071082879334156493737167162836200097051269616602927743814516397331823355845210200404263, 44614166820706534634304472808697702934078813986584989516320002705230229151888377374088320809843126696833060239336551884382005068253010413308849696179015187083428814172430905280406126938766758372405650267006652903323392178233561864805663474096476965663185474196152774481719329334913146001036234, 44118089607025723558107143042086181753749394977388893940453973198986693702608893106256299997753562459359348504182558145226381873948837824882566092258576113387770447587847932439487147418283129308635799663384813901308458073125122435449904266058661486852700959618625715425385900072927896635847918, 322794886124971235012581304166397749817379237897845983003383843621360166075828327834150776062510668754122857637810366356711708170266583125810305252269486188276590815703109853941585730421137640326064169328797177252162519768443099976386112287191109322123776823421875612771262935787974503107710279, 175323465798509822056898340933208587416555154106083393834015042943506971865221176250531434254330865179559115604246898902468901249695445235112392037122740215803439826386637146040054469383505622969628241423204738387734189958564206380203514377049807493456496583353284366010673089877279819232473346, 305175944664006076493283409958334929087805627626926922593844790239977606049567509604917722441062508230351518651872095537614579613251778332492038571794498607266402933186759021634454101761899186873427369650139446037485982292341972663105314499262199588778469107638269711858502424524340008313416930, 174956658348109472542914849688742482459474245851478014594175949159865921241161368453729808239062812424472972180339600004254497322937363759643257858763299748129232792025538609783172407013862176361260140796912440019863108610716690589990433582113223482620351270862423162820560068598377100846590975], 'aut_phi_ratio': 162.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [3, 3, 1, 2], [3, 3, 2, 2], [6, 1, 2, 1], [6, 1, 6, 1], [6, 3, 1, 2], [6, 3, 2, 2], [9, 3, 18, 1], [9, 9, 18, 1], [18, 3, 18, 1], [18, 9, 18, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^5.(C_3\\times S_3^2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '486.183', 'autcent_hash': 183, 'autcent_nilpotent': False, 'autcent_order': 486, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^4:S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '54.5', 'autcentquo_hash': 5, 'autcentquo_nilpotent': False, 'autcentquo_order': 54, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2:C_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 8], [3, 3, 6], [6, 1, 8], [6, 3, 6], [9, 3, 18], [9, 9, 18], [18, 3, 18], [18, 9, 18]], 'center_label': '18.5', 'center_order': 18, 'central_product': True, 'central_quotient': '27.3', 'commutator_count': 1, 'commutator_label': '9.2', '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': 213, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['81.10', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [3, 3, 2, 3], [6, 1, 2, 4], [6, 3, 2, 3], [9, 3, 6, 3], [9, 9, 2, 9], [18, 3, 6, 3], [18, 9, 2, 9]], 'element_repr_type': 'GLZq', 'elementary': 3, 'eulerian_function': 2184, 'exponent': 18, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '54.15', 'hash': 213, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [1, 3, 3, 3], 'inner_gens': [[373996, 376801, 374473, 164375], [373996, 376801, 373987, 341513], [373996, 376558, 374473, 164375], [373996, 22255, 374473, 164375]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 27, 'inner_split': False, 'inner_tex': '\\He_3', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 54], [3, 48]], 'label': '486.213', '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*C6.He3', 'ngens': 6, 'nilpotency_class': 3, 'nilpotent': True, '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': 18, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 102, 'number_divisions': 40, 'number_normal_subgroups': 72, 'number_subgroup_autclasses': 44, 'number_subgroup_classes': 132, 'number_subgroups': 252, 'old_label': None, 'order': 486, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 1], [3, 26], [6, 26], [9, 216], [18, 216]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 3, 3, 6, 2, 6, 6], 'outer_gen_pows': [196849, 196849, 373987, 19684, 196849, 19684, 19684], 'outer_gens': [[373996, 510166, 373987, 348164], [373996, 28915, 374473, 164375], [373996, 35314, 374473, 164375], [196840, 509680, 373987, 348407], [196840, 510166, 373987, 348164], [197083, 148996, 373987, 525536], [197083, 510166, 373987, 347921]], 'outer_group': '972.646', 'outer_hash': 646, 'outer_nilpotent': False, 'outer_order': 972, 'outer_permdeg': 20, 'outer_perms': [66952927654650, 243332058851570820, 128069711284859888, 243332058855110400, 3628800, 1013731400541677084, 898488680544403200], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^4:D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 32, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 26], [6, 6], [18, 6]], 'representations': {'PC': {'code': 69213602426858443835067, 'gens': [1, 2, 3, 4], 'pres': [6, 3, 3, 3, 2, 3, 3, 655, 1034, 1593, 69, 3970, 118]}, 'GLZN': {'d': 2, 'p': 54, 'gens': [157483, 2137243, 158437, 211285, 158923, 2991835]}, 'GLZq': {'d': 2, 'q': 27, 'gens': [511784, 196840, 376558, 387505, 19693, 19927]}, 'Perm': {'d': 32, 'gens': [8222838654177922817725562880000000, 9823410235015926371682349167360, 18982253052262400815558566534507, 28140211547958303057602869753108, 1840764313927677088715674302720, 28140211547958303057602869753104]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_6.\\He_3', 'transitive_degree': 162, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 3], [1, 17], [7, 3]], 'aut_group': '48.29', 'aut_hash': 29, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 8, 'aut_perms': [475, 23888], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 8, 1], [6, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', '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, 8], [6, 1, 8]], 'center_label': '18.5', 'center_order': 18, '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', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [6, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '18.5', 'hash': 5, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 18]], 'label': '18.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 18, 'linQ_dim': 4, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C6', '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 18, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 12, 'number_subgroups': 12, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 8], [6, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 17], [7, 3]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [475, 23888], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8]], 'representations': {'PC': {'code': 277, 'gens': [1, 2], 'pres': [3, -3, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931151, 16858245]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 1374]}, 'Perm': {'d': 8, 'gens': [5040, 240, 4]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_6', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
