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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '448.729', 'ambient_counter': 729, 'ambient_order': 448, 'ambient_tex': 'C_{28}.D_8', 'central': False, 'central_factor': False, 'centralizer_order': 112, 'characteristic': True, 'core_order': 28, 'counter': 41, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '448.729.16.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '16.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '16.11', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 11, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times D_4', 'simple': False, 'solvable': True, 'special_labels': ['L2', 'C4'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '28.2', 'subgroup_hash': 2, 'subgroup_order': 28, 'subgroup_tex': 'C_{28}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '448.729', 'aut_centralizer_order': 896, 'aut_label': '16.c1', 'aut_quo_index': 8, 'aut_stab_index': 1, 'aut_weyl_group': '12.5', 'aut_weyl_index': 896, 'centralizer': '4.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '8.b1.a1', '8.c1.a1', '8.g1.a1', '8.i1.a1', '8.j1.a1', '8.j1.b1'], 'contains': ['32.a1.a1', '112.c1.a1'], 'core': '16.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5087, 5886, 4628, 3132, 4494, 3651, 4560, 4220], 'generators': [8, 224, 64], 'label': '448.729.16.c1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '16.c1.a1', 'normal_contained_in': ['8.a1.a1', '8.b1.a1', '8.c1.a1'], 'normal_contains': ['32.a1.a1', '112.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '16.c1.a1', 'projective_image': '224.126', 'quotient_action_image': '4.2', 'quotient_action_kernel': '4.2', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '16.c1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '28.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1], [13], [11]], 'aut_group': '12.5', 'aut_hash': 5, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 12, 'aut_permdeg': 7, 'aut_perms': [24, 723], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [7, 1, 6, 1], [14, 1, 6, 1], [28, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_6', '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], [4, 1, 2], [7, 1, 6], [14, 1, 6], [28, 1, 12]], 'center_label': '28.2', 'center_order': 28, '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', '7.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [7, 1, 6, 1], [14, 1, 6, 1], [28, 1, 12, 1]], 'element_repr_type': 'PC', 'elementary': 14, 'eulerian_function': 1, 'exponent': 28, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7], 'faithful_reps': [[1, 0, 12]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '14.2', 'hash': 2, 'hyperelementary': 14, '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': 12, 'irrQ_dim': 12, 'irrR_degree': 2, 'irrep_stats': [[1, 28]], 'label': '28.2', 'linC_count': 12, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 6, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C28', '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': 28, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 28, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [7, 6], [14, 6], [28, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[13], [11]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [24, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [4, 7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 2], [12, 1]], 'representations': {'PC': {'code': 242573, 'gens': [1], 'pres': [3, -2, -2, -7, 6, 16]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [14163]}, 'Perm': {'d': 11, 'gens': [11612160, 4320, 3669120]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [28], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{28}', 'transitive_degree': 28, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [24, 12, 28, 8, 4, 4, 6], 'aut_gens': [[1, 2, 16], [5, 186, 80], [233, 178, 400], [13, 446, 16], [237, 306, 208], [225, 202, 208], [9, 298, 432], [13, 126, 176]], 'aut_group': None, 'aut_hash': 915345584156982159, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10752, 'aut_permdeg': 120, 'aut_perms': [5189160623743292394326997292390136096172771787222596555299575805495276298806435006629859534846438301425688656125335779820278304015782180153375095933115482816871506675160215071586209757910398227567944, 294367106395905943312349474378039513863758637554866560732640726308096106937433413612903095109183230568120130859394446612251706115161463741211120634368715008547934513901031287401973474643205708897944, 2799315922701923931072529279815023944088318053824845878714732169525599491794445987820852757349105192466152045302352758073949382185845819034055629912199712990836687485794721564762496331603709529047140, 323042497395235681142563740666195532196592994988673173508998658284807477170552086468956996048499438828555156723552892804804605386020067782271630337478668038274979771376905429747074267760759910067490, 5584410506346153617660820773264011014991486897280304653677928279853619356931783792186551424912220411925411032527780744949142208484213417452979909127246438229249108465339430572983651107127640721142024, 190378110914109617395331092110319273467281586077326040521553103712256668974238815434750986951642041434226261287301660599703801553533216571895609530700259115853551836769785847977862130902851189230664, 3736653368024896189831437568919833238836502048322939493699624658742626793345391625583713168742302460896818454998758341224124076244484340335653835777782795924731026914014810691585272889343660818427156], 'aut_phi_ratio': 56.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 8, 1, 1], [4, 2, 1, 2], [4, 8, 1, 1], [4, 56, 2, 1], [7, 2, 3, 1], [8, 2, 2, 1], [8, 4, 1, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 8, 6, 1], [16, 28, 4, 1], [28, 2, 6, 1], [28, 4, 3, 1], [28, 8, 6, 1], [56, 4, 6, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_{28}.(C_2^3\\times C_6).C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '1344.11434', 'autcentquo_hash': 11434, 'autcentquo_nilpotent': False, 'autcentquo_order': 1344, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times D_4\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 8, 1], [4, 2, 2], [4, 8, 1], [4, 56, 2], [7, 2, 3], [8, 2, 2], [8, 4, 1], [14, 2, 3], [14, 4, 3], [14, 8, 6], [16, 28, 4], [28, 2, 6], [28, 4, 3], [28, 8, 6], [56, 4, 12]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '224.126', 'commutator_count': 1, 'commutator_label': '56.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 729, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 8, 1, 1], [4, 2, 1, 2], [4, 8, 1, 1], [4, 56, 1, 2], [7, 2, 3, 1], [8, 2, 2, 1], [8, 4, 1, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 8, 6, 1], [16, 28, 2, 2], [28, 2, 6, 1], [28, 4, 3, 1], [28, 8, 6, 1], [56, 4, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2688, 'exponent': 112, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, -1, 12]], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '56.12', 'hash': 729, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 56, 'inner_gen_orders': [2, 8, 14], 'inner_gens': [[1, 14, 16], [229, 2, 432], [1, 34, 16]], 'inner_hash': 126, 'inner_nilpotent': False, 'inner_order': 224, 'inner_split': False, 'inner_tex': 'C_{14}:D_8', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 48, 'irrQ_dim': 96, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 30], [4, 20]], 'label': '448.729', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 4, 'linQ_dim': 22, 'linQ_dim_count': 4, 'linR_count': 12, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C28.D8', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 20, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 58, 'number_divisions': 22, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 70, 'number_subgroup_classes': 82, 'number_subgroups': 372, 'old_label': None, 'order': 448, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 11], [4, 124], [7, 6], [8, 8], [14, 66], [16, 112], [28, 72], [56, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 6], 'outer_gen_pows': [0, 112, 236, 112], 'outer_gens': [[1, 2, 240], [1, 10, 16], [13, 2, 240], [1, 114, 400]], 'outer_group': '48.52', 'outer_hash': 52, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [744, 3628800, 24, 40323], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 39, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 2], [6, 4], [8, 1], [12, 4], [48, 1]], 'representations': {'PC': {'code': 61345983602586317507811766492269195195653, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -7, 197, 36, 4958, 58, 6499, 808, 7571, 102, 8748, 124, 9421]}, 'Perm': {'d': 39, 'gens': [581171734436213155343639334193351065273886207, 3358581232284905593655692454848031230444800, 1114992791509442462723297543934924207624729600, 1655508588254668913257004991005151597264399360, 2192674621859360853748592118133297044515748480, 2728729891619589842712253818784929760092725760, 973]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{28}.D_8', 'transitive_degree': 224, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2], 'aut_gens': [[126, 55, 289, 288], [127, 55, 289, 288], [126, 265, 1, 288], [54, 127, 289, 288], [414, 55, 289, 288], [127, 54, 289, 288], [414, 265, 289, 288]], 'aut_group': '64.138', 'aut_hash': 138, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 8, 'aut_perms': [2309, 526, 5329, 3043, 12316, 18498], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [4, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [4, 2, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [4, 2, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 21, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 1], 'inner_gens': [[126, 265, 289, 288], [414, 55, 289, 288], [126, 55, 289, 288], [126, 55, 289, 288]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.11', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 8, 'linQ_dim': 3, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 10, 'number_divisions': 10, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 27, 'number_subgroups': 35, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 11], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 151, 0], 'outer_gens': [[54, 127, 289, 288], [127, 55, 289, 288], [415, 54, 1, 288], [127, 54, 289, 288]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2]], 'representations': {'PC': {'code': 8772, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 78, 34]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16322, 16432, 3198]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8912, 8156, 13286, 14044]}, 'Perm': {'d': 6, 'gens': [126, 55, 289, 288]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times D_4', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
