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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1088.31', 'ambient_counter': 31, 'ambient_order': 1088, 'ambient_tex': '\\OD_{32}:C_{34}', 'central': True, 'central_factor': False, 'centralizer_order': 1088, 'characteristic': True, 'core_order': 136, 'counter': 14, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1088.31.8.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '8.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.3', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4', 'simple': False, 'solvable': True, 'special_labels': ['Z', 'U2'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '136.2', 'subgroup_hash': 2, 'subgroup_order': 136, 'subgroup_tex': 'C_{136}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1088.31', 'aut_centralizer_order': 16, 'aut_label': '8.c1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '64.183', 'aut_weyl_index': 16, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '4.d1.a1', '4.f1.a1'], 'contains': ['16.b1.a1', '136.c1.a1'], 'core': '8.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3333, 7889, 7801, 8121, 7580, 2144, 7580, 2144], 'generators': [548, 64, 16, 8], 'label': '1088.31.8.c1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '8.c1.a1', 'normal_contained_in': ['4.b1.a1'], 'normal_contains': ['16.b1.a1', '136.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.c1.a1', 'projective_image': '8.3', 'quotient_action_image': '1.1', 'quotient_action_kernel': '8.3', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '8.c1.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': '136.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 16, 'aut_gen_orders': [2, 2, 16], 'aut_gens': [[1], [35], [69], [105]], 'aut_group': '64.183', 'aut_hash': 183, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 20, 'aut_perms': [121645100408832000, 355687428096000, 1401602636313], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1], [17, 1, 16, 1], [34, 1, 16, 1], [68, 1, 32, 1], [136, 1, 64, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_{16}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '64.183', 'autcent_hash': 183, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{16}', '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], [8, 1, 4], [17, 1, 16], [34, 1, 16], [68, 1, 32], [136, 1, 64]], 'center_label': '136.2', 'center_order': 136, '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', '17.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1], [17, 1, 16, 1], [34, 1, 16, 1], [68, 1, 32, 1], [136, 1, 64, 1]], 'element_repr_type': 'PC', 'elementary': 34, 'eulerian_function': 1, 'exponent': 136, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 17], 'faithful_reps': [[1, 0, 64]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '34.2', 'hash': 2, 'hyperelementary': 34, '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': 64, 'irrQ_dim': 64, 'irrR_degree': 2, 'irrep_stats': [[1, 136]], 'label': '136.2', 'linC_count': 64, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 2, 'linQ_dim': 20, 'linQ_dim_count': 2, 'linR_count': 32, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C136', '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': 136, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 136, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [4, 2], [8, 4], [17, 16], [34, 16], [68, 32], [136, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [2, 2, 16], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[35], [69], [105]], 'outer_group': '64.183', 'outer_hash': 183, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 20, 'outer_perms': [121645100408832000, 355687428096000, 1401602636313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{16}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [8, 17], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1], [16, 2], [32, 1], [64, 1]], 'representations': {'PC': {'code': 699341930511, 'gens': [1], 'pres': [4, -2, -2, -2, -17, 8, 21, 34]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [4931, 44226, 78624]}, 'Perm': {'d': 25, 'gens': [4502951279245783203840000, 334764638208000, 1913056780673713618944000, 621574841765398929408000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [136], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{136}', 'transitive_degree': 136, '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': '272.23', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 16, 'aut_gen_orders': [4, 4, 2, 2, 4, 16], 'aut_gens': [[1, 2, 32], [1, 6, 32], [1, 10, 32], [1, 18, 32], [1, 570, 48], [553, 2, 32], [1, 2, 96]], 'aut_group': None, 'aut_hash': 3160242281144326426, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 1024, 'aut_permdeg': 36, 'aut_perms': [4527719934047319568571000801902773756367, 6370163911027060794751887178517703420366, 295767283552125810520977087869421159966, 6092297747225080530201311677815711765295, 46359276964533400625, 248299763933550219904649672502352683552720], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 4, 1, 1], [8, 1, 4, 1], [8, 2, 2, 1], [8, 4, 2, 1], [16, 2, 8, 1], [16, 4, 4, 1], [17, 1, 16, 1], [34, 1, 16, 1], [34, 2, 16, 1], [34, 4, 16, 1], [68, 1, 32, 1], [68, 2, 16, 1], [68, 4, 16, 1], [136, 1, 64, 1], [136, 2, 32, 1], [136, 4, 32, 1], [272, 2, 128, 1], [272, 4, 64, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2^2\\times C_8).C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '256.26308', 'autcent_hash': 26308, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4\\times C_{16}', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '4.2', 'autcentquo_hash': 2, 'autcentquo_nilpotent': True, 'autcentquo_order': 4, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [4, 1, 2], [4, 2, 1], [4, 4, 1], [8, 1, 4], [8, 2, 2], [8, 4, 2], [16, 2, 8], [16, 4, 4], [17, 1, 16], [34, 1, 16], [34, 2, 16], [34, 4, 16], [68, 1, 32], [68, 2, 16], [68, 4, 16], [136, 1, 64], [136, 2, 32], [136, 4, 32], [272, 2, 128], [272, 4, 64]], 'center_label': '136.2', 'center_order': 136, 'central_product': True, 'central_quotient': '8.3', 'commutator_count': 1, 'commutator_label': '4.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '17.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['64.31', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 4, 1, 1], [8, 1, 4, 1], [8, 2, 2, 1], [8, 4, 2, 1], [16, 2, 8, 1], [16, 4, 4, 1], [17, 1, 16, 1], [34, 1, 16, 1], [34, 2, 16, 1], [34, 4, 16, 1], [68, 1, 32, 1], [68, 2, 16, 1], [68, 4, 16, 1], [136, 1, 64, 1], [136, 2, 32, 1], [136, 4, 32, 1], [272, 2, 128, 1], [272, 4, 64, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 432, 'exponent': 272, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 17], 'faithful_reps': [[2, 0, 128]], 'familial': False, 'frattini_label': '16.5', 'frattini_quotient': '68.5', 'hash': 31, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2], 'inner_gens': [[1, 570, 48], [553, 2, 32], [17, 2, 32]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': True, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 256, 'irrQ_dim': 256, 'irrR_degree': 4, 'irrep_stats': [[1, 272], [2, 204]], 'label': '1088.31', 'linC_count': 128, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 32, 'linQ_degree_count': 4, 'linQ_dim': 32, 'linQ_dim_count': 4, 'linR_count': 64, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'OD32:C34', 'ngens': 7, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 34, 'number_conjugacy_classes': 476, 'number_divisions': 24, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 56, 'number_subgroups': 82, 'old_label': None, 'order': 1088, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 8], [8, 16], [16, 32], [17, 16], [34, 112], [68, 128], [136, 256], [272, 512]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [2, 4, 16], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 14, 32], [1, 10, 32], [1, 2, 96]], 'outer_group': '128.837', 'outer_hash': 837, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 22, 'outer_perms': [51090942171709440000, 51219345333252096000, 1401602636313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4\\times C_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 49, 'pgroup': 0, 'primary_abelian_invariants': [2, 8, 17], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 3], [16, 5], [32, 4], [64, 3], [256, 1]], 'representations': {'PC': {'code': 2033772657527364677074124176910974991, 'gens': [1, 2, 6], 'pres': [7, -2, -2, -2, -2, -2, 2, -17, 7981, 36, 422, 58, 80, 2021, 124]}, 'Perm': {'d': 49, 'gens': [291322258351212076880766110991504914417332182149989662720000000, 101425310388425719334706507440860125259268239818494701412352000, 25350585729035464156108997278353269116201955850462397906944000, 88471637574347248606003823257811855068144196862018811158528000, 37759001356278293408717324428246400867136348998925911506944000, 12419420874373282104691941482291922192923886086820530577408000, 334764638208000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 136], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\OD_{32}:C_{34}', 'transitive_degree': 544, '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': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
