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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '540.49', 'ambient_counter': 49, 'ambient_order': 540, 'ambient_tex': '\\He_3:D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 270, 'characteristic': True, 'core_order': 5, 'counter': 100, 'cyclic': True, 'direct': False, 'hall': 5, 'label': '540.49.108.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '108.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '108.28', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 28, 'quotient_metabelian': False, 'quotient_nilpotent': False, 'quotient_order': 108, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3^2:D_6', 'simple': True, 'solvable': True, 'special_labels': ['C5'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '5.1', 'subgroup_hash': 1, 'subgroup_order': 5, 'subgroup_tex': 'C_5', 'supersolvable': True, 'sylow': 5}
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gps_subgroup_data • Show schema
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{'ambient': '540.49', 'aut_centralizer_order': 2160, 'aut_label': '108.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 2160, 'centralizer': '2.b1.a1', 'complements': ['5.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['36.a1.a1', '36.b1.a1', '36.b1.b1', '36.b1.c1', '36.b1.d1', '54.a1.a1', '54.b1.a1', '54.c1.a1'], 'contains': ['540.a1.a1'], 'core': '108.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3222, 2935, 8328, 5719, 7006, 7471, 5667, 5543], 'generators': [108], 'label': '540.49.108.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '108.a1.a1', 'normal_contained_in': ['36.a1.a1', '54.a1.a1'], 'normal_contains': ['540.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '108.a1.a1', 'projective_image': '540.49', 'quotient_action_image': '2.1', 'quotient_action_kernel': '54.8', 'quotient_action_kernel_order': 54, 'quotient_fusion': None, 'short_label': '108.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '5.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [2]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': '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], [5, 1, 4]], 'center_label': '5.1', 'center_order': 5, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['5.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '5.1', 'hash': 1, 'hyperelementary': 5, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 5]], 'label': '5.1', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 5, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 5, 'order_factorization_type': 1, 'order_stats': [[1, 1], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 5, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -5]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26329372]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131]}, 'Perm': {'d': 5, 'gens': [96]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [5], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5', 'transitive_degree': 5, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [12, 8], 'aut_gens': [[1, 2, 12, 36], [209, 62, 24, 312], [181, 186, 24, 172]], 'aut_group': '8640.be', 'aut_hash': 9203803121985669655, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 8640, 'aut_permdeg': 14, 'aut_perms': [76788890400, 47313692802], 'aut_phi_ratio': 60.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [2, 9, 1, 1], [2, 45, 1, 1], [3, 1, 2, 1], [3, 6, 4, 1], [5, 2, 2, 1], [6, 5, 2, 1], [6, 9, 2, 1], [6, 30, 4, 1], [6, 45, 2, 1], [10, 18, 2, 1], [15, 2, 4, 1], [15, 12, 8, 1], [30, 18, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\times C_3^2:\\GL(2,3)', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '8640.be', 'autcentquo_hash': 9203803121985669655, 'autcentquo_nilpotent': False, 'autcentquo_order': 8640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 5, 1], [2, 9, 1], [2, 45, 1], [3, 1, 2], [3, 6, 4], [5, 2, 2], [6, 5, 2], [6, 9, 2], [6, 30, 4], [6, 45, 2], [10, 18, 2], [15, 2, 4], [15, 12, 8], [30, 18, 4]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '180.27', 'commutator_count': 1, 'commutator_label': '135.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 49, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['54.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [2, 9, 1, 1], [2, 45, 1, 1], [3, 1, 2, 1], [3, 6, 1, 4], [5, 2, 2, 1], [6, 5, 2, 1], [6, 9, 2, 1], [6, 30, 1, 4], [6, 45, 2, 1], [10, 18, 2, 1], [15, 2, 4, 1], [15, 12, 2, 4], [30, 18, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6804, 'exponent': 30, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 8]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '180.27', 'hash': 49, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 6, 1, 15], 'inner_gens': [[1, 10, 12, 396], [5, 2, 12, 156], [1, 2, 12, 36], [181, 458, 12, 36]], 'inner_hash': 27, 'inner_nilpotent': False, 'inner_order': 180, 'inner_split': False, 'inner_tex': 'C_{15}:D_6', 'inner_used': [1, 2, 4], 'irrC_degree': 6, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 12], [3, 8], [4, 8], [6, 8]], 'label': '540.49', 'linC_count': 32, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 8, 'linQ_dim': 10, 'linQ_dim_count': 8, 'linR_count': 16, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'He3:D10', 'ngens': 6, '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': 15, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 40, 'number_divisions': 24, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 50, 'number_subgroup_classes': 110, 'number_subgroups': 754, 'old_label': None, 'order': 540, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 59], [3, 26], [5, 4], [6, 238], [10, 36], [15, 104], [30, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 6, 0, 1, 1], 'outer_gens': [[1, 366, 24, 224], [1, 2, 12, 252], [1, 366, 12, 424], [1, 382, 12, 68], [1, 214, 12, 52]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [143, 127, 576, 126, 121], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 2], [6, 4], [8, 4], [24, 2]], 'representations': {'PC': {'code': 1725514634910927040576579117143, 'gens': [1, 2, 4, 5], 'pres': [6, -2, -2, -3, -3, 3, -5, 121, 31, 146, 11884, 2350, 466, 118, 7787]}, 'Perm': {'d': 14, 'gens': [7, 999059040, 1489350960, 7266072240, 37, 14026769280]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\He_3:D_{10}', 'transitive_degree': 45, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 4, 2, 4, 3, 2, 6, 6], 'aut_gens': [[7426, 8101, 13286, 13259, 15473], [7426, 8101, 13286, 6751, 15473], [10315, 8101, 13286, 11098, 16985], [7399, 8101, 13286, 13259, 18389], [11044, 8101, 13286, 8911, 16229], [6697, 8101, 13286, 19091, 15473], [8155, 7372, 13286, 13259, 18443], [8155, 8101, 13286, 11881, 16202], [8128, 8101, 13286, 10423, 17687]], 'aut_group': '864.4661', 'aut_hash': 4661, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 11, 'aut_perms': [1, 29083801, 8719926, 27297007, 11007408, 1514929, 28008751, 13317145], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [3, 1, 2, 1], [3, 6, 4, 1], [6, 1, 2, 1], [6, 6, 4, 1], [6, 9, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_3^2:\\GL(2,3)', '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': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 1, 2], [3, 6, 4], [6, 1, 2], [6, 6, 4], [6, 9, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '27.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 28, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['54.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 1, 2, 1], [3, 6, 1, 4], [6, 1, 2, 1], [6, 6, 1, 4], [6, 9, 2, 2]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 567, 'exponent': 6, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 0, 4]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '36.13', 'hash': 28, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 1, 1, 2, 3], 'inner_gens': [[7426, 8101, 13286, 13205, 16931], [7426, 8101, 13286, 13259, 15473], [7426, 8101, 13286, 13259, 15473], [7399, 8101, 13286, 13259, 18389], [6697, 8101, 13286, 19091, 15473]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 4, 5], 'irrC_degree': 3, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 4], [2, 8], [3, 8]], 'label': '108.28', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 2, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:D6', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, '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': 8, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 20, 'number_divisions': 16, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 55, 'number_subgroups': 167, 'old_label': None, 'order': 108, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 19], [3, 26], [6, 62]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [6643, 6643, 6643, 13934, 13934], 'outer_gens': [[10288, 7372, 13286, 17579, 13340], [7426, 8101, 13286, 6751, 15473], [8857, 8101, 13286, 18335, 13313], [11071, 8101, 13286, 16175, 19145], [12502, 8101, 13286, 16877, 16229]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [143, 127, 576, 126, 121], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [6, 4]], 'representations': {'PC': {'code': 47077684069063437, 'gens': [1, 2, 3, 4], 'pres': [5, 2, 3, 3, 2, 3, 41, 1803, 308, 58, 1804, 534]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [78750878879449598, 41624316883886018, 74635755094754554]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8101, 13286, 13259, 15473]}, 'Perm': {'d': 11, 'gens': [769375, 1, 4478784, 1150590, 8891526]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:D_6', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
