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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '972.798', 'ambient_counter': 798, 'ambient_order': 972, 'ambient_tex': 'C_3^3:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 81, 'characteristic': True, 'core_order': 81, 'counter': 39, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '972.798.12.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '12.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '12.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 12, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_6', 'simple': False, 'solvable': True, 'special_labels': ['D', 'L1', 'D1'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '81.15', 'subgroup_hash': 15, 'subgroup_order': 81, 'subgroup_tex': 'C_3^4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '972.798', 'aut_centralizer_order': 81, 'aut_label': '12.c1', 'aut_quo_index': 3, 'aut_stab_index': 1, 'aut_weyl_group': '96.209', 'aut_weyl_index': 81, 'centralizer': '12.c1.a1', 'complements': ['81.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '6.b1.a1', '6.d1.a1', '6.d1.b1'], 'contains': ['36.b1.a1', '36.b1.b1', '36.c1.a1', '36.g1.a1', '36.g1.b1', '36.l1.a1', '36.o1.a1', '36.z1.a1', '36.z1.b1', '36.bc1.a1', '36.bf1.a1'], 'core': '12.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4473, 4754, 6381, 5154, 4517, 4578, 5251, 5092], 'generators': [22843831, 734401, 33902551, 879733], 'label': '972.798.12.c1.a1', 'mobius_quo': 0, 'mobius_sub': -2, 'normal_closure': '12.c1.a1', 'normal_contained_in': ['4.a1.a1', '6.b1.a1', '6.d1.b1', '6.d1.a1'], 'normal_contains': ['36.b1.b1', '36.b1.a1', '36.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '12.c1.a1', 'projective_image': '972.798', 'quotient_action_image': '12.5', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '12.c1.a1', 'subgroup_fusion': None, 'weyl_group': '12.5'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '81.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 9360, 'aut_gen_orders': [2, 13], 'aut_gens': [[1, 3, 9, 27], [2, 3, 9, 27], [29, 2, 6, 18]], 'aut_group': None, 'aut_hash': 1045913143454417265, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24261120, 'aut_permdeg': 80, 'aut_perms': [894910464770715304112463402887197477343442096610712847651341310429022571428803639963392602465820278981730644545159864, 1812326269711177385513856665846033904155014172742552368476102969481397534672420762470589026159121368093405030544787282], 'aut_phi_ratio': 449280.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [3, 1, 80, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2.\\PSL(4,3).C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 9360, 'autcent_group': None, 'autcent_hash': 1045913143454417265, 'autcent_nilpotent': False, 'autcent_order': 24261120, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2.\\PSL(4,3).C_2', '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], [3, 1, 80]], 'center_label': '81.15', 'center_order': 81, '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': ['3.1', '3.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 4]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 40]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3, 5, 13], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '81.15', 'hash': 15, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 3, 9, 27], [1, 3, 9, 27], [1, 3, 9, 27], [1, 3, 9, 27]], '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, 81]], 'label': '81.15', 'linC_count': None, 'linC_degree': 4, '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': 'C3^4', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 81, 'number_divisions': 41, 'number_normal_subgroups': 212, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 212, 'number_subgroups': 212, 'old_label': None, 'order': 81, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 9360, 'outer_gen_orders': [2, 13], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 3, 9, 27], [29, 2, 6, 18]], 'outer_group': None, 'outer_hash': 1045913143454417265, 'outer_nilpotent': False, 'outer_order': 24261120, 'outer_permdeg': 80, 'outer_perms': [894910464770715304112463402887197477343442096610712847651341310429022571428803639963392602465820278981730644545159864, 1812326269711177385513856665846033904155014172742552368476102969481397534672420762470589026159121368093405030544787282], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2.\\PSL(4,3).C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 40]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4], 'pres': [4, -3, 3, 3, 3]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 24507118, 2939606, 36731395]}, 'Perm': {'d': 12, 'gens': [79833600, 80640, 240, 4]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3, 3], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4', 'transitive_degree': 81, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 12, 6, 6], 'aut_gens': [[44475821, 22843831, 732529, 33902551, 879733], [44833571, 23091331, 53436277, 33902551, 53803441], [31445717, 23088631, 20632672, 879733, 1097596], [31930757, 22849231, 53802127, 53803441, 1097596], [22968971, 23088631, 1242019, 33902551, 879733]], 'aut_group': '7776.is', 'aut_hash': 4994213685505676867, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7776, 'aut_permdeg': 60, 'aut_perms': [55851051313517162782594568271748998029500601472536474212235178164426538513183236, 1994699103317261193648898743843564535111242628738825076307092799132061320047942850, 28735516026232590182234210834959801048495348230471373775306067236249877625260489, 883251737826809002122264990504535656043576031599448350414097434256795980904054542], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 3, 2, 1], [3, 4, 1, 1], [3, 4, 2, 2], [3, 6, 1, 1], [3, 6, 2, 1], [3, 6, 4, 1], [3, 12, 2, 3], [3, 12, 4, 2], [6, 18, 1, 1], [6, 27, 2, 1], [6, 27, 4, 1], [6, 54, 1, 1], [6, 54, 2, 2], [6, 54, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4:(D_4\\times D_6)', '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': 12, 'autcentquo_group': '7776.is', 'autcentquo_hash': 4994213685505676867, 'autcentquo_nilpotent': False, 'autcentquo_order': 7776, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4:(D_4\\times D_6)', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 27, 2], [3, 2, 3], [3, 3, 2], [3, 4, 5], [3, 6, 7], [3, 12, 14], [6, 18, 1], [6, 27, 6], [6, 54, 9]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '972.798', 'commutator_count': 1, 'commutator_label': '81.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 798, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [3, 2, 1, 3], [3, 3, 2, 1], [3, 4, 1, 3], [3, 4, 2, 1], [3, 6, 1, 1], [3, 6, 2, 3], [3, 12, 1, 2], [3, 12, 2, 6], [6, 18, 1, 1], [6, 27, 2, 3], [6, 54, 1, 3], [6, 54, 2, 3]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 2]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '324.167', 'hash': 798, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [6, 3, 6, 3, 3], 'inner_gens': [[44475821, 23088631, 735229, 1097596, 879733], [1220891, 22843831, 732529, 33902551, 879733], [44470421, 22843831, 732529, 1097596, 53803441], [44836271, 22843831, 1101079, 33902551, 879733], [44475821, 22843831, 53802127, 33902551, 879733]], 'inner_hash': 798, 'inner_nilpotent': False, 'inner_order': 972, 'inner_split': True, 'inner_tex': 'C_3^3:S_3^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 12], [2, 18], [4, 15], [6, 2], [12, 4]], 'label': '972.798', 'linC_count': 90, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 10, 'linQ_dim': 10, 'linQ_dim_count': 10, 'linR_count': 10, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^3:S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 17, 'number_conjugacy_classes': 51, 'number_divisions': 34, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 203, 'number_subgroup_classes': 340, 'number_subgroups': 3204, 'old_label': None, 'order': 972, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 63], [3, 242], [6, 666]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [729001, 44469829, 729001], 'outer_gens': [[44475821, 22843831, 732529, 1097596, 879733], [44476649, 22849231, 732529, 879733, 1097596], [22605821, 22843831, 735229, 33902551, 53803441]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 9], [6, 2], [8, 6], [12, 2], [24, 1]], 'representations': {'PC': {'code': 13730692731309503197476112621618359, 'gens': [1, 3, 4, 6, 7], 'pres': [7, -2, -3, -3, -2, -3, -3, -3, 14, 1010, 450, 2523, 80, 2524, 9077, 530, 1791]}, 'GLZN': {'d': 2, 'p': 90, 'gens': [44469001, 1215931, 729829, 1096271, 879733, 1097596, 731701]}, 'Perm': {'d': 15, 'gens': [7, 13898309183, 19295141040, 325, 100602351360, 435, 194648877360]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3^2', 'transitive_degree': 36, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [6, 9], [6, 11]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 31], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [6, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.4', 'autcent_hash': 4, 'autcent_nilpotent': False, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_6', '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], [3, 1, 2], [6, 1, 6]], 'center_label': '12.5', 'center_order': 12, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [6, 1, 2, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '12.5', 'hash': 5, '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, 12]], 'label': '12.5', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 6, 'linQ_dim': 3, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C6', '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 12, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 9], [6, 11]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4]], 'representations': {'PC': {'code': 273, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16507]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 2064]}, 'Perm': {'d': 7, 'gens': [720, 24, 4]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], '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_6', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
