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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1760.303', 'ambient_counter': 303, 'ambient_order': 1760, 'ambient_tex': '(C_4\\times D_{11}):C_{20}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 44, 'counter': 68, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1760.303.40.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '40.b1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '40.9', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 9, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 40, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_{20}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '44.3', 'subgroup_hash': 3, 'subgroup_order': 44, 'subgroup_tex': 'D_{22}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.303', 'aut_centralizer_order': 32, 'aut_label': '40.b1', 'aut_quo_index': 4, 'aut_stab_index': 2, 'aut_weyl_group': '220.7', 'aut_weyl_index': 64, 'centralizer': '220.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.b1.b1', '20.a1.a1', '20.f1.c1', '20.f1.d1'], 'contains': ['80.a1.a1', '80.d1.b1', '440.c1.b1'], 'core': '40.b1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [980, 1215, 6561, 4025, 2158, 3073, 2823, 3155], 'generators': [25, 40, 880], 'label': '1760.303.40.b1.b1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '40.b1.b1', 'normal_contained_in': ['8.b1.b1', '20.a1.a1', '20.f1.d1', '20.f1.c1'], 'normal_contains': ['80.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '40.b1.b1', 'projective_image': '880.115', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '40.b1.b1', 'subgroup_fusion': None, 'weyl_group': '220.7'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 110, 'aut_gen_orders': [10, 22], 'aut_gens': [[1, 2], [1, 26], [19, 2]], 'aut_group': '220.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 220, 'aut_permdeg': 13, 'aut_perms': [87956214, 965432545], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 2, 1], [11, 2, 5, 1], [22, 2, 5, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_{11}', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 11, 2], [11, 2, 5], [22, 2, 5]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '22.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '11.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['22.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 1, 2], [11, 2, 5, 1], [22, 2, 5, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 22, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 11], 'faithful_reps': [[2, 1, 5]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '44.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [2, 11], 'inner_gens': [[1, 42], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 22, 'inner_split': True, 'inner_tex': 'D_{11}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 10, 'irrQ_dim': 10, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 10]], 'label': '44.3', 'linC_count': 5, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 5, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D22', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 14, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 40, 'old_label': None, 'order': 44, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 23], [11, 10], [22, 10]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [10], 'outer_gen_pows': [0], 'outer_gens': [[23, 38]], 'outer_group': '10.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 10, 'outer_permdeg': 7, 'outer_perms': [810], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{10}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [10, 2]], 'representations': {'PC': {'code': 571694009, 'gens': [1, 2], 'pres': [3, -2, -2, -11, 253, 16, 362]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [1343, 13320, 13311]}, 'Perm': {'d': 13, 'gens': [40284846, 1, 559571286]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{22}', 'transitive_degree': 22, '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': '80.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [20, 10, 2, 10, 20, 10, 20], 'aut_gens': [[1, 10, 440], [21, 1350, 1540], [421, 730, 1320], [401, 430, 660], [1181, 1150, 660], [1241, 1370, 1540], [221, 1130, 1540], [381, 1510, 440]], 'aut_group': None, 'aut_hash': 2218350592113462965, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14080, 'aut_permdeg': 52, 'aut_perms': [43265932199336248298747330972333628833332142954364510301728072866930, 73661165040792846083613675953833887107201934707350275419978959689076, 44909952496432606755273336520927727080207483618664438357448348965982, 7757077595269334130550534445483992872566879027931483952829165839389, 16124915466748922389916755607403411978451456360309410472372448840537, 33803448551021124207050771627559821708011076883911906902785039056330, 24493934983798373993174839143374112066575225818582523115152102434206], 'aut_phi_ratio': 22.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 22, 2, 1], [4, 2, 2, 1], [4, 2, 4, 1], [4, 11, 4, 1], [4, 22, 4, 1], [5, 11, 1, 4], [10, 11, 1, 12], [10, 22, 2, 4], [11, 10, 1, 1], [20, 11, 4, 4], [20, 22, 2, 4], [20, 22, 4, 8], [22, 10, 1, 3], [44, 20, 2, 1], [44, 20, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{22}.(C_{10}\\times D_4).C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '220.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 220, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 22, 2], [4, 2, 6], [4, 11, 4], [4, 22, 4], [5, 11, 4], [10, 11, 12], [10, 22, 8], [11, 10, 1], [20, 11, 16], [20, 22, 40], [22, 10, 3], [44, 20, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '440.42', 'commutator_count': 1, 'commutator_label': '22.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 303, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 22, 1, 2], [4, 2, 1, 2], [4, 2, 2, 2], [4, 11, 2, 2], [4, 22, 2, 2], [5, 11, 4, 1], [10, 11, 4, 3], [10, 22, 4, 2], [11, 10, 1, 1], [20, 11, 8, 2], [20, 22, 4, 2], [20, 22, 8, 4], [22, 10, 1, 3], [44, 20, 1, 2], [44, 20, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 124992, 'exponent': 220, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '440.42', 'hash': 303, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 22, 2], 'inner_gens': [[1, 1010, 440], [1201, 10, 1320], [1, 890, 440]], 'inner_hash': 42, 'inner_nilpotent': False, 'inner_order': 440, 'inner_split': True, 'inner_tex': 'C_2^2\\times F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 20], [10, 8], [20, 2]], 'label': '1760.303', 'linC_count': 80, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 32, 'linQ_dim': 16, 'linQ_dim_count': 32, 'linR_count': 800, 'linR_degree': 16, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C4*D11):C20', 'ngens': 7, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 51, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 110, 'number_divisions': 36, 'number_normal_subgroups': 71, 'number_subgroup_autclasses': 100, 'number_subgroup_classes': 152, 'number_subgroups': 954, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 47], [4, 144], [5, 44], [10, 308], [11, 10], [20, 1056], [22, 30], [44, 120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[881, 210, 440], [1101, 430, 440], [1101, 230, 660], [1101, 670, 660]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [16, 23, 11536, 16583], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 10], [8, 4], [10, 4], [16, 2], [20, 4]], 'representations': {'PC': {'code': 385382076765540558970708603825805575095490300900, 'gens': [1, 3, 6], 'pres': [7, -2, -5, -2, -2, -11, -2, -2, 14, 21212, 3894, 58, 7283, 4210, 80, 2804, 2811, 5563, 124]}, 'GLZN': {'d': 2, 'p': 88, 'gens': [15673879, 682177, 6133329, 1193587, 30843451, 685345, 59634695]}, 'Perm': {'d': 23, 'gens': [51212945749292451360, 91623007, 94525337, 107304560072802892800, 7, 139725360, 1277293892726538086400]}}, 'schur_multiplier': [2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_4\\times D_{11}):C_{20}', 'transitive_degree': 176, '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': '40.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4, 4], 'aut_gens': [[1, 2], [21, 2], [1, 14], [21, 19]], 'aut_group': '32.25', 'aut_hash': 25, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [11527, 17, 11647], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 1, 8, 1], [20, 1, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.25', 'autcent_hash': 25, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times D_4', '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, 3], [4, 1, 4], [5, 1, 4], [10, 1, 12], [20, 1, 16]], 'center_label': '40.9', 'center_order': 40, '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', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [5, 1, 4, 1], [10, 1, 4, 3], [20, 1, 8, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 18, 'exponent': 20, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '20.5', 'hash': 9, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 40]], 'label': '40.9', 'linC_count': 288, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C20', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 40, 'number_divisions': 12, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 40, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 4], [5, 4], [10, 12], [20, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[21, 2], [1, 14], [21, 19]], 'outer_group': '32.25', 'outer_hash': 25, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [11527, 17, 11647], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 4], [8, 2]], 'representations': {'PC': {'code': 5899779, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -5, 21, 34]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729074815868, 91655853488337157]}, 'GLFp': {'d': 2, 'p': 41, 'gens': [1585222, 2756880]}, 'Perm': {'d': 11, 'gens': [131040, 3628800, 96, 41040]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 20], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{20}', 'transitive_degree': 40, 'wreath_data': None, 'wreath_product': False}
