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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '9072.b', 'ambient_counter': 2, 'ambient_order': 9072, 'ambient_tex': 'S_3\\times {}^2G(2,3)', 'central': True, 'central_factor': False, 'centralizer_order': 9072, 'characteristic': True, 'core_order': 1, 'counter': 144, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '9072.b.9072.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '9072.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '9072.b', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 6831087576112572302, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 9072, 'quotient_simple': False, 'quotient_solvable': False, 'quotient_supersolvable': False, 'quotient_tex': 'S_3\\times {}^2G(2,3)', 'simple': False, 'solvable': True, 'special_labels': ['Z', 'Phi', 'U0', 'C4'], 'split': True, 'standard_generators': False, 'stem': True, 'subgroup': '1.1', 'subgroup_hash': 1, 'subgroup_order': 1, 'subgroup_tex': 'C_1', 'supersolvable': True, 'sylow': 1}
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gps_subgroup_data • Show schema
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{'ambient': '9072.b', 'aut_centralizer_order': None, 'aut_label': '9072.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1.a1', 'complements': ['1.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1296.a1.a1', '3024.a1.a1', '3024.b1.a1', '3024.c1.a1', '3024.d1.a1', '3024.e1.a1', '4536.a1.a1', '4536.b1.a1', '4536.c1.a1'], 'contains': [], 'core': '9072.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4454, 5648, 5403, 5648, 4055, 5484, 4069, 5484], 'generators': [], 'label': '9072.b.9072.a1.a1', 'mobius_quo': 1, 'mobius_sub': 1512, 'normal_closure': '9072.a1.a1', 'normal_contained_in': ['18.a1.a1', '3024.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '9072.a1.a1', 'projective_image': '9072.b', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '9072.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 126, 'aut_gen_orders': [2, 18], 'aut_gens': [[40764962, 88197241], [165314402, 131091269], [142619285, 249062665]], 'aut_group': '9072.b', 'aut_hash': 6831087576112572302, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9072, 'aut_permdeg': 87, 'aut_perms': [1348919322402289824191205030496493336609836198598674786996276903301977612018404917849810498464920394535114974921729650654595264606058, 1174957425817158728937100615259526548911223014102431515037781059966883898522075254444424481096591777037819477118804480912040497065513], 'aut_phi_ratio': 3.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 63, 1, 1], [2, 189, 1, 1], [3, 2, 1, 1], [3, 56, 1, 1], [3, 84, 1, 2], [3, 112, 1, 1], [3, 168, 1, 2], [6, 126, 1, 1], [6, 168, 1, 1], [6, 252, 1, 4], [6, 504, 1, 2], [6, 756, 1, 2], [7, 216, 1, 1], [9, 168, 1, 3], [9, 336, 1, 3], [14, 648, 1, 1], [18, 504, 1, 3], [21, 432, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times {}^2G(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': 126, 'autcentquo_group': '9072.b', 'autcentquo_hash': 6831087576112572302, 'autcentquo_nilpotent': False, 'autcentquo_order': 9072, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times {}^2G(2,3)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 63, 1], [2, 189, 1], [3, 2, 1], [3, 56, 1], [3, 84, 2], [3, 112, 1], [3, 168, 2], [6, 126, 1], [6, 168, 1], [6, 252, 4], [6, 504, 2], [6, 756, 2], [7, 216, 1], [9, 168, 3], [9, 336, 3], [14, 648, 1], [18, 504, 3], [21, 432, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '9072.b', 'commutator_count': 1, 'commutator_label': '1512.780', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '504.156'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1512.779', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 63, 1, 1], [2, 189, 1, 1], [3, 2, 1, 1], [3, 56, 1, 1], [3, 84, 2, 1], [3, 112, 1, 1], [3, 168, 2, 1], [6, 126, 1, 1], [6, 168, 1, 1], [6, 252, 2, 2], [6, 504, 2, 1], [6, 756, 2, 1], [7, 216, 1, 1], [9, 168, 1, 1], [9, 168, 2, 1], [9, 336, 1, 1], [9, 336, 2, 1], [14, 648, 1, 1], [18, 504, 1, 1], [18, 504, 2, 1], [21, 432, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3408, 'exponent': 126, 'exponents_of_order': [4, 4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[14, 0, 2], [14, 1, 1], [16, 0, 2], [16, 1, 1], [42, 1, 1], [54, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '9072.b', 'hash': 6831087576112572302, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 126, 'inner_gen_orders': [2, 18], 'inner_gens': [[40764962, 255563309], [186240509, 88197241]], 'inner_hash': 6831087576112572302, 'inner_nilpotent': False, 'inner_order': 9072, 'inner_split': True, 'inner_tex': 'S_3\\times {}^2G(2,3)', 'inner_used': [1, 2], 'irrC_degree': 14, 'irrQ_degree': 14, 'irrQ_dim': 14, 'irrR_degree': 14, 'irrep_stats': [[1, 6], [2, 3], [7, 6], [8, 6], [14, 3], [16, 3], [21, 2], [27, 2], [42, 1], [54, 1]], 'label': '9072.b', 'linC_count': 18, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 2, 'linQ_dim': 9, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'S3*2G(2,3)', 'ngens': 2, '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, 0, 0, 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': 33, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 33, 'number_divisions': 24, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 144, 'number_subgroup_classes': 144, 'number_subgroups': 13956, 'old_label': None, 'order': 9072, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 255], [3, 674], [6, 3822], [7, 216], [9, 1512], [14, 648], [18, 1512], [21, 432]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': None, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [7, 2], [8, 2], [14, 3], [16, 3], [21, 2], [27, 2], [28, 1], [32, 1], [42, 1], [54, 1]], 'representations': {'Perm': {'d': 12, 'gens': [40764962, 88197241]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times {}^2G(2,3)', 'transitive_degree': 27, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 126, 'aut_gen_orders': [2, 18], 'aut_gens': [[40764962, 88197241], [165314402, 131091269], [142619285, 249062665]], 'aut_group': '9072.b', 'aut_hash': 6831087576112572302, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9072, 'aut_permdeg': 87, 'aut_perms': [1348919322402289824191205030496493336609836198598674786996276903301977612018404917849810498464920394535114974921729650654595264606058, 1174957425817158728937100615259526548911223014102431515037781059966883898522075254444424481096591777037819477118804480912040497065513], 'aut_phi_ratio': 3.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 63, 1, 1], [2, 189, 1, 1], [3, 2, 1, 1], [3, 56, 1, 1], [3, 84, 1, 2], [3, 112, 1, 1], [3, 168, 1, 2], [6, 126, 1, 1], [6, 168, 1, 1], [6, 252, 1, 4], [6, 504, 1, 2], [6, 756, 1, 2], [7, 216, 1, 1], [9, 168, 1, 3], [9, 336, 1, 3], [14, 648, 1, 1], [18, 504, 1, 3], [21, 432, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times {}^2G(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': 126, 'autcentquo_group': '9072.b', 'autcentquo_hash': 6831087576112572302, 'autcentquo_nilpotent': False, 'autcentquo_order': 9072, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times {}^2G(2,3)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 63, 1], [2, 189, 1], [3, 2, 1], [3, 56, 1], [3, 84, 2], [3, 112, 1], [3, 168, 2], [6, 126, 1], [6, 168, 1], [6, 252, 4], [6, 504, 2], [6, 756, 2], [7, 216, 1], [9, 168, 3], [9, 336, 3], [14, 648, 1], [18, 504, 3], [21, 432, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '9072.b', 'commutator_count': 1, 'commutator_label': '1512.780', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '504.156'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1512.779', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 63, 1, 1], [2, 189, 1, 1], [3, 2, 1, 1], [3, 56, 1, 1], [3, 84, 2, 1], [3, 112, 1, 1], [3, 168, 2, 1], [6, 126, 1, 1], [6, 168, 1, 1], [6, 252, 2, 2], [6, 504, 2, 1], [6, 756, 2, 1], [7, 216, 1, 1], [9, 168, 1, 1], [9, 168, 2, 1], [9, 336, 1, 1], [9, 336, 2, 1], [14, 648, 1, 1], [18, 504, 1, 1], [18, 504, 2, 1], [21, 432, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3408, 'exponent': 126, 'exponents_of_order': [4, 4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[14, 0, 2], [14, 1, 1], [16, 0, 2], [16, 1, 1], [42, 1, 1], [54, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '9072.b', 'hash': 6831087576112572302, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 126, 'inner_gen_orders': [2, 18], 'inner_gens': [[40764962, 255563309], [186240509, 88197241]], 'inner_hash': 6831087576112572302, 'inner_nilpotent': False, 'inner_order': 9072, 'inner_split': True, 'inner_tex': 'S_3\\times {}^2G(2,3)', 'inner_used': [1, 2], 'irrC_degree': 14, 'irrQ_degree': 14, 'irrQ_dim': 14, 'irrR_degree': 14, 'irrep_stats': [[1, 6], [2, 3], [7, 6], [8, 6], [14, 3], [16, 3], [21, 2], [27, 2], [42, 1], [54, 1]], 'label': '9072.b', 'linC_count': 18, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 2, 'linQ_dim': 9, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'S3*2G(2,3)', 'ngens': 2, '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, 0, 0, 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': 33, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 33, 'number_divisions': 24, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 144, 'number_subgroup_classes': 144, 'number_subgroups': 13956, 'old_label': None, 'order': 9072, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 255], [3, 674], [6, 3822], [7, 216], [9, 1512], [14, 648], [18, 1512], [21, 432]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': None, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [7, 2], [8, 2], [14, 3], [16, 3], [21, 2], [27, 2], [28, 1], [32, 1], [42, 1], [54, 1]], 'representations': {'Perm': {'d': 12, 'gens': [40764962, 88197241]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times {}^2G(2,3)', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
