-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '512.6536092', 'ambient_counter': 6536092, 'ambient_order': 512, 'ambient_tex': 'C_2^6:C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 64, 'counter': 43, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '512.6536092.8.n1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '8.n1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '8.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '64.261', 'subgroup_hash': 261, 'subgroup_order': 64, 'subgroup_tex': 'D_4\\times C_2^3', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '512.6536092', 'aut_centralizer_order': None, 'aut_label': '8.n1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '32.g1', 'complements': ['64.bu1'], 'conjugacy_class_count': 2, 'contained_in': ['4.f1', '4.h1'], 'contains': ['16.m1', '16.n1', '16.q1', '16.bk1', '16.dc1'], 'core': '8.n1', 'coset_action_label': None, 'count': 2, 'diagramx': None, 'generators': [1, 8, 4, 32, 160], 'label': '512.6536092.8.n1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.n1', 'normal_contained_in': ['4.f1', '4.h1'], 'normal_contains': ['16.m1', '16.n1', '16.q1'], 'normalizer': '1.a1', 'old_label': '8.n1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.n1', 'subgroup_fusion': None, 'weyl_group': None}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 168, 'aut_gen_orders': [14, 4], 'aut_gens': [[2561, 2565, 1543, 1541, 1587], [1571, 2561, 517, 1589, 2581], [1539, 517, 3623, 1573, 3635]], 'aut_group': '688128.be', 'aut_hash': 4970985371203673109, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 688128, 'aut_permdeg': 32, 'aut_perms': [255721047101209868832962344144395246, 45877573149809264889240548981106065], 'aut_phi_ratio': 21504.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [2, 2, 16, 1], [4, 2, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5.C_2^7:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': None, 'autcent_hash': 6173500181054804164, 'autcent_nilpotent': False, 'autcent_order': 344064, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^5.C_2^6.\\PSL(2,7)', '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, 15], [2, 2, 16], [4, 2, 8]], 'center_label': '16.14', 'center_order': 16, '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', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 261, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [2, 2, 1, 16], [4, 2, 1, 8]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 465, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '32.51', 'hash': 261, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 1, 1, 2, 2], 'inner_gens': [[2561, 2565, 1543, 1541, 1587], [2561, 2565, 1543, 1541, 1587], [2561, 2565, 1543, 1541, 1587], [2561, 2565, 1543, 1541, 1555], [2561, 2565, 1543, 1573, 1587]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8]], 'label': '64.261', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, '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': 'D4*C2^3', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 40, 'number_divisions': 40, 'number_normal_subgroups': 425, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 681, 'number_subgroups': 937, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 47], [4, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [8, 4], 'outer_gen_pows': [513, 513], 'outer_gens': [[3619, 3591, 1571, 567, 3607], [3587, 517, 2565, 1541, 533]], 'outer_group': None, 'outer_hash': 7746879706912876876, 'outer_nilpotent': False, 'outer_order': 172032, 'outer_permdeg': 24, 'outer_perms': [293898027136311178599264, 162476987823691988414844], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^9.C_2.\\PSL(2,7)', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 8]], 'representations': {'PC': {'code': 68288528, 'gens': [1, 2, 3, 4, 5], 'pres': [6, -2, 2, 2, 2, 2, -2, 202, 88]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 705686952884, 706461792886, 706460730980, 706461792404]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [4562006, 7292993, 5128257, 20225089, 19896385, 6210625]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [529, 2565, 545, 515, 2611, 1539]}, 'Perm': {'d': 10, 'gens': [127, 368045, 806416, 136, 368056, 16]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4\\times C_2^3', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 8, 4, 4, 4, 12], 'aut_gens': [[1, 2, 8, 16, 64, 128, 256], [321, 214, 40, 368, 416, 128, 224], [25, 418, 40, 20, 428, 448, 204], [157, 386, 172, 340, 252, 160, 412], [161, 454, 8, 48, 407, 484, 215], [321, 193, 360, 336, 327, 324, 7], [185, 229, 44, 180, 295, 164, 71]], 'aut_group': None, 'aut_hash': 7005986967837481181, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1572864, 'aut_permdeg': 128, 'aut_perms': [290577495005461418801578561681232612690383335688490467100243427561680746507851815756474329065921130773065637090092936923920709807151400303708531560628104917333898923025583936129663304613789974258781075661018509855190, 214192655390529831380265542275805876742697208507426762651355436921335606044962941164082874589388242918301285594458112875473755066348724852361786690674083461310483256902471982560902629357521492222032419638835728675153, 45567365078339297672575430564295506152148672938904885193882889903831166531456692693734362407973421051482075984635147862352820351351132639634549597008035734933727024352562782584084279673932031239963270871589740670802, 7505808638515643529411877003462211432240595112787513551881134257486180751651153470364586235543489300956680536417632095472115243541787760734849497825652835900622354091093232888490423004230426744650768613311823181630, 157991081548546616323369167040374124236571658535144742846242513773456320962330260604381313959146575582481480154079041960662656445679266554071242514447666662923136403673419995807932533851842219582737026062091812078046, 354190476846111780475802354681621945651245017445608192449529093594617431685140121757212654009335646695057992672260188870345511792134212396258588944690404533586050973889234933901231237526261383958061860642897536659873], 'aut_phi_ratio': 6144.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 6, 1], [2, 2, 8, 1], [2, 4, 4, 1], [2, 4, 24, 1], [2, 8, 4, 1], [4, 4, 4, 1], [4, 4, 24, 1], [4, 16, 2, 1], [4, 16, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^7.C_2^6.C_6.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '1024.djt', 'autcent_hash': 1957374279625110839, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '1536.408632661', 'autcentquo_hash': 6321312703651895429, 'autcentquo_nilpotent': False, 'autcentquo_order': 1536, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^6:S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 14], [2, 4, 28], [2, 8, 4], [4, 4, 28], [4, 16, 14]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '128.2163', 'commutator_count': 1, 'commutator_label': '16.14', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 6536092, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 14], [2, 4, 1, 28], [2, 8, 1, 4], [4, 4, 1, 28], [4, 16, 1, 14]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6666240, 'exponent': 4, 'exponents_of_order': [9], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.14', 'frattini_quotient': '32.51', 'hash': 6536092, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2, 2, 2, 2, 2], 'inner_gens': [[1, 6, 360, 368, 384, 128, 192], [5, 2, 328, 368, 384, 160, 192], [353, 322, 8, 48, 96, 128, 288], [353, 354, 40, 16, 64, 128, 256], [449, 450, 40, 16, 64, 128, 256], [1, 34, 8, 16, 64, 128, 256], [449, 450, 40, 16, 64, 128, 256]], 'inner_hash': 2163, 'inner_nilpotent': True, 'inner_order': 128, 'inner_split': False, 'inner_tex': 'C_2^4:D_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 56], [8, 4]], 'label': '512.6536092', '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': 'C2^6:C2^3', 'ngens': 5, 'nilpotency_class': 3, 'nilpotent': True, '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': 12, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 92, 'number_divisions': 92, 'number_normal_subgroups': 946, 'number_subgroup_autclasses': 384, 'number_subgroup_classes': 17912, 'number_subgroups': 43474, 'old_label': None, 'order': 512, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 175], [4, 336]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [6, 4, 12, 4, 12, 6], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[129, 314, 328, 144, 459, 4, 139], [1, 498, 456, 372, 300, 480, 108], [61, 274, 456, 372, 183, 4, 471], [153, 354, 456, 464, 235, 484, 395], [189, 434, 136, 20, 151, 36, 503], [33, 94, 460, 372, 471, 36, 151]], 'outer_group': None, 'outer_hash': 9005961918598495056, 'outer_nilpotent': False, 'outer_order': 12288, 'outer_permdeg': 28, 'outer_perms': [95482230429449251061783237543, 144042279291798415223154773160, 84308168210768426066881471329, 258830344143471157503623983574, 19857954166231118565543124458, 16631595706659758099465999760], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_4^2:A_4.C_2^5.C_2', 'pc_rank': None, 'perfect': False, 'permutation_degree': None, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 56], [8, 4]], 'representations': {'PC': {'code': '3144692845907993302700666034774412832627977717037491727634', 'gens': [1, 2, 4, 5, 7, 8, 9], 'pres': [9, 2, 2, 2, 2, 2, 2, 2, 2, 2, 109, 46, 12963, 5916, 16564, 8293, 301, 130, 24198, 12111, 1536, 789, 5776, 15560, 7793, 5858, 2951]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 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_2^6:C_2^3', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 84, 'aut_gen_orders': [3, 3], 'aut_gens': [[1, 2, 4], [4, 5, 3], [2, 4, 1]], 'aut_group': '168.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 7, 'aut_perms': [4361, 244], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 168, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\PSL(2,7)', '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, 7]], 'center_label': '8.5', 'center_order': 8, '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', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 8]], 'label': '8.5', 'linC_count': 28, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 28, 'linQ_dim': 3, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3', 'ngens': 3, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 8, 'number_divisions': 8, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 5, 3], [2, 4, 1]], 'outer_group': '168.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 168, 'outer_permdeg': 7, 'outer_perms': [4361, 244], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PSL(2,7)', '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]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -2, 2, 2]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16482, 16322, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 13286, 13933]}, 'Perm': {'d': 6, 'gens': [120, 6, 1]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
