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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1760.333', 'ambient_counter': 333, 'ambient_order': 1760, 'ambient_tex': 'C_{22}.(D_4\\times C_{10})', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 88, 'counter': 57, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1760.333.20.a1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '20.a1.c1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '20.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{20}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '88.6', 'subgroup_hash': 6, 'subgroup_order': 88, 'subgroup_tex': 'C_{22}:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.333', 'aut_centralizer_order': 8, 'aut_label': '20.a1', 'aut_quo_index': 4, 'aut_stab_index': 4, 'aut_weyl_group': '880.118', 'aut_weyl_index': 32, 'centralizer': '220.a1.a1', 'complements': ['88.f1.a1', '88.g1.a1', '88.f1.b1', '88.g1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.c1', '10.b1.a1'], 'contains': ['40.b1.b1', '40.e1.a1', '40.e1.d1', '220.c1.c1'], 'core': '20.a1.c1', 'coset_action_label': None, 'count': 1, 'diagramx': [8130, 7105, 1735, 5676, 9798, 8873, 2003, 5925], 'generators': [882, 40, 160, 20], 'label': '1760.333.20.a1.c1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '20.a1.c1', 'normal_contained_in': ['4.a1.c1', '10.b1.a1'], 'normal_contains': ['40.b1.b1', '40.e1.d1', '40.e1.a1'], 'normalizer': '1.a1.a1', 'old_label': '20.a1.c1', 'projective_image': '440.8', 'quotient_action_image': '10.2', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '20.a1.c1', '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [2, 10, 44], 'aut_gens': [[1, 4], [3, 84], [3, 38], [39, 6]], 'aut_group': '880.118', 'aut_hash': 118, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 880, 'aut_permdeg': 15, 'aut_perms': [6321546127, 45234459503, 177511906097], 'aut_phi_ratio': 22.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 11, 4, 1], [11, 2, 5, 1], [22, 2, 5, 1], [22, 2, 10, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times F_{11}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', '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, 3], [4, 11, 4], [11, 2, 5], [22, 2, 15]], 'center_label': '4.2', 'center_order': 4, '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', '2.1', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['44.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 11, 2, 2], [11, 2, 5, 1], [22, 2, 5, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 44, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '44.3', 'hash': 6, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [2, 11], 'inner_gens': [[1, 84], [9, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 22, 'inner_split': True, 'inner_tex': 'D_{11}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 20]], 'label': '88.6', 'linC_count': 60, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 2, 'linR_count': 10, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C22:C4', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 28, 'number_divisions': 10, 'number_normal_subgroups': 13, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 46, 'old_label': None, 'order': 88, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 44], [11, 10], [22, 30]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [2, 20], 'outer_gen_pows': [0, 0], 'outer_gens': [[45, 84], [45, 38]], 'outer_group': '40.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 40, 'outer_permdeg': 9, 'outer_perms': [41040, 120993], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_5\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [10, 4]], 'representations': {'PC': {'code': 31799286290009, 'gens': [1, 3], 'pres': [4, -2, -2, -2, -11, 8, 1010, 34, 1283]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [826574, 359380, 36037, 48113]}, 'Perm': {'d': 17, 'gens': [1313941673671, 1, 288, 23550673230000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{22}:C_4', 'transitive_degree': 88, '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, 44, 20, 10, 20, 20], 'aut_gens': [[1, 4, 80], [41, 1244, 722], [881, 1324, 82], [43, 1406, 1400], [881, 1244, 1520], [43, 1084, 602], [921, 1046, 402]], 'aut_group': None, 'aut_hash': 6780658973947388987, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 28160, 'aut_permdeg': 96, 'aut_perms': [822392712663829253938126073340244770437551457114732992942432069397375636108204942445857375663687325092056364874538237358224754047710920396241939663428, 174465967064000151481540948799484214647539670346091533086956948306222678755639776694296883343731133326270242210656227018576884106866604373898656574957, 234409047193745673737656279287167890570662891520407819868927791767646616063019781439954570345889267792511483591038733147191713877707211611029923439672, 938958934038211664120320098854569200330068531965691386372159751329675779650239432716904969818179602746682762300663259677375257143636210331132161291875, 585552957916130296342058092839326583777701022337208212226138241950200295442247477046897150179820742461136512148403165180260164437166188118877514721822, 304734337613489040384019090254642373516331697636754189817144780288744443935215082659667948353038342946638115263725958848520112670157495197134859391834], 'aut_phi_ratio': 44.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 4, 1], [4, 2, 4, 1], [4, 22, 4, 2], [5, 11, 1, 4], [10, 11, 1, 12], [10, 11, 4, 4], [11, 10, 1, 1], [20, 22, 4, 12], [22, 10, 1, 3], [22, 10, 4, 1], [44, 10, 8, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2^3\\times C_{22}).C_5.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.8935', 'autcent_hash': 8935, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6:C_2^2', '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, 7], [4, 2, 4], [4, 22, 8], [5, 11, 4], [10, 11, 28], [11, 10, 1], [20, 22, 48], [22, 10, 7], [44, 10, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '220.7', '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': 333, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['880.15', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 2, 2, 2], [4, 22, 1, 4], [4, 22, 2, 2], [5, 11, 4, 1], [10, 11, 4, 7], [11, 10, 1, 1], [20, 22, 4, 4], [20, 22, 8, 4], [22, 10, 1, 7], [44, 10, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 62496, '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': 333, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [2, 10, 11], 'inner_gens': [[1, 44, 80], [41, 4, 1040], [1, 804, 80]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 220, 'inner_split': False, 'inner_tex': 'C_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, 16]], 'label': '1760.333', 'linC_count': 9600, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 40, 'linQ_dim': 14, 'linQ_dim_count': 16, 'linR_count': 80, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C22.(D4*C10)', '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': 46, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 116, 'number_divisions': 42, 'number_normal_subgroups': 95, 'number_subgroup_autclasses': 96, 'number_subgroup_classes': 184, 'number_subgroups': 1026, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [4, 184], [5, 44], [10, 308], [11, 10], [20, 1056], [22, 70], [44, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 46, 1682], [1, 6, 120], [1, 886, 122], [3, 44, 80], [881, 4, 1720]], 'outer_group': '128.1135', 'outer_hash': 1135, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [16686, 40296367, 127382422, 127387567, 216091583], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 8], [8, 8], [10, 4], [20, 6]], 'representations': {'PC': {'code': 1531510643122417946662766973770025208383170000309, 'gens': [1, 3, 6], 'pres': [7, -2, -2, -2, -2, -5, -2, -11, 14, 926, 58, 80, 10939, 6326, 1083, 124, 3940, 3947, 2484]}, 'GLZN': {'d': 2, 'p': 88, 'gens': [1198451, 46003299, 30666285, 36885894, 6133249, 14655541, 682177]}, 'Perm': {'d': 21, 'gens': [122002108923708849, 368040, 811447, 256851058042656000, 367920, 16, 2797904269515801600]}}, 'schur_multiplier': [2, 2, 2], '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_{22}.(D_4\\times C_{10})', 'transitive_degree': 352, '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': '20.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1], [11], [17]], 'aut_group': '8.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 6, 'aut_perms': [120, 9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_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, 1], [4, 1, 2], [5, 1, 4], [10, 1, 4], [20, 1, 8]], 'center_label': '20.2', 'center_order': 20, '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', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 8]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '10.2', 'hash': 2, 'hyperelementary': 10, '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': 8, 'irrQ_dim': 8, 'irrR_degree': 2, 'irrep_stats': [[1, 20]], 'label': '20.2', 'linC_count': 8, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 2, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 4, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C20', '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': 20, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 4], [10, 4], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[11], [17]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [4, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1]], 'representations': {'PC': {'code': 51395, 'gens': [1], 'pres': [3, -2, -2, -5, 6, 16]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58438750843733785]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504, 252]}, 'Perm': {'d': 9, 'gens': [131040, 96, 41040]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [20], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
