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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1088.154', 'ambient_counter': 154, 'ambient_order': 1088, 'ambient_tex': 'C_{136}.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 544, 'characteristic': True, 'core_order': 68, 'counter': 41, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1088.154.16.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '16.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '16.11', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 11, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times D_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '68.2', 'subgroup_hash': 2, 'subgroup_order': 68, 'subgroup_tex': 'C_{68}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1088.154', 'aut_centralizer_order': 128, 'aut_label': '16.b1', 'aut_quo_index': 8, 'aut_stab_index': 1, 'aut_weyl_group': '32.16', 'aut_weyl_index': 128, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '8.c1.a1', '8.d1.a1', '8.h1.a1', '8.i1.a1', '8.i1.b1', '8.j1.a1'], 'contains': ['32.a1.a1', '272.b1.a1'], 'core': '16.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2284, 8285, 1531, 713, 6795, 6846, 7977, 6615], 'generators': [816, 64, 544], 'label': '1088.154.16.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '16.b1.a1', 'normal_contained_in': ['8.a1.a1', '8.c1.a1', '8.d1.a1'], 'normal_contains': ['32.a1.a1', '272.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '16.b1.a1', 'projective_image': '32.28', 'quotient_action_image': '2.1', 'quotient_action_kernel': '8.5', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '16.b1.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': '68.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 16, 'aut_gen_orders': [2, 16], 'aut_gens': [[1], [35], [37]], 'aut_group': '32.16', 'aut_hash': 16, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 18, 'aut_perms': [355687428096000, 1401602636313], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [17, 1, 16, 1], [34, 1, 16, 1], [68, 1, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{16}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '32.16', 'autcent_hash': 16, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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], [17, 1, 16], [34, 1, 16], [68, 1, 32]], 'center_label': '68.2', 'center_order': 68, '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', '17.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [17, 1, 16, 1], [34, 1, 16, 1], [68, 1, 32, 1]], 'element_repr_type': 'PC', 'elementary': 34, 'eulerian_function': 1, 'exponent': 68, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 17], 'faithful_reps': [[1, 0, 32]], 'familial': True, 'frattini_label': '2.1', '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': 32, 'irrQ_dim': 32, 'irrR_degree': 2, 'irrep_stats': [[1, 68]], 'label': '68.2', 'linC_count': 32, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 2, 'linQ_dim': 18, 'linQ_dim_count': 2, 'linR_count': 16, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C68', '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': 68, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 68, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [17, 16], [34, 16], [68, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [2, 16], 'outer_gen_pows': [0, 0], 'outer_gens': [[35], [37]], 'outer_group': '32.16', 'outer_hash': 16, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 18, 'outer_perms': [355687428096000, 1401602636313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{16}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [4, 17], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [16, 2], [32, 1]], 'representations': {'PC': {'code': 11153423, 'gens': [1], 'pres': [3, -2, -2, -17, 6, 16]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [4931, 63882, 78624]}, 'Perm': {'d': 21, 'gens': [7541996225347584000, 334764638208000, 2439304381882368000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [68], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{68}', 'transitive_degree': 68, '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': '136.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 16, 'aut_gen_orders': [8, 2, 2, 2, 4, 4, 16], 'aut_gens': [[1, 2, 8], [1, 138, 8], [1, 2, 824], [1, 2, 552], [1, 6, 8], [549, 2, 824], [1, 274, 8], [1, 2, 840]], 'aut_group': None, 'aut_hash': 8154153396567325628, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 36, 'aut_perms': [8829913627211761792558109532929197271160, 599148915698358366629039754365003138046, 15511210043395149206323326, 7517438241352511418593062398211614359328, 599280759502468219771307341968081640446, 5231334709789732267624229383678248962040, 248299738964940088005284074813290481138864], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 8, 2, 1], [8, 2, 2, 1], [8, 4, 1, 1], [8, 4, 2, 1], [8, 8, 2, 1], [17, 1, 16, 1], [34, 1, 16, 1], [34, 2, 16, 1], [34, 4, 16, 1], [68, 2, 16, 2], [68, 4, 16, 1], [68, 8, 32, 1], [136, 2, 32, 1], [136, 4, 16, 1], [136, 4, 32, 1], [136, 8, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{16}\\times C_4.C_2^5.C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '128.2136', 'autcent_hash': 2136, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_{16}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '32.46', 'autcentquo_hash': 46, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [4, 2, 2], [4, 4, 1], [4, 8, 2], [8, 2, 2], [8, 4, 3], [8, 8, 2], [17, 1, 16], [34, 1, 16], [34, 2, 16], [34, 4, 16], [68, 2, 32], [68, 4, 16], [68, 8, 32], [136, 2, 32], [136, 4, 48], [136, 8, 32]], 'center_label': '34.2', 'center_order': 34, 'central_product': True, 'central_quotient': '32.28', 'commutator_count': 1, 'commutator_label': '8.2', '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': 154, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['64.154', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 8, 1, 2], [8, 2, 2, 1], [8, 4, 1, 1], [8, 4, 2, 1], [8, 8, 1, 2], [17, 1, 16, 1], [34, 1, 16, 1], [34, 2, 16, 1], [34, 4, 16, 1], [68, 2, 16, 2], [68, 4, 16, 1], [68, 8, 16, 2], [136, 2, 32, 1], [136, 4, 16, 1], [136, 4, 32, 1], [136, 8, 16, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 103152, 'exponent': 136, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 17], 'faithful_reps': [[4, 0, 32]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '136.15', 'hash': 154, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 4], 'inner_gens': [[1, 6, 8], [549, 2, 824], [1, 274, 8]], 'inner_hash': 28, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'C_4:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 128, 'irrQ_dim': 128, 'irrR_degree': 8, 'irrep_stats': [[1, 136], [2, 102], [4, 34]], 'label': '1088.154', 'linC_count': 32, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 8, 'linQ_dim': 32, 'linQ_dim_count': 8, 'linR_count': 16, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C136.D4', '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': 272, 'number_divisions': 28, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 78, 'number_subgroup_classes': 100, 'number_subgroups': 162, 'old_label': None, 'order': 1088, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 24], [8, 32], [17, 16], [34, 112], [68, 384], [136, 512]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [2, 2, 2, 16], 'outer_gen_pows': [8, 0, 0, 0], 'outer_gens': [[1, 138, 8], [1, 2, 824], [1, 2, 552], [1, 2, 840]], 'outer_group': '128.2136', 'outer_hash': 2136, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 22, 'outer_perms': [51090942171709440000, 121645100408832000, 355687428096000, 1401602636313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 49, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 17], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 1], [8, 1], [16, 8], [32, 4], [64, 1], [128, 1]], 'representations': {'PC': {'code': 1772662430702322147541926687519647966691343, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -2, -2, -2, -2, -17, 85, 36, 11510, 2872, 11546, 80, 9811, 102, 124]}, 'Perm': {'d': 49, 'gens': [215270041651238110379454230844937855408378102903493099520000000, 139184311744701524200673277893036046963437588005359841124352000, 50712644313715021082405462925974397276814001301185229471744000, 25350585729034648763476669016095062210546921700937795026944000, 37758998699411724737649737926307286770937154910095410315264000, 12419420874373282104691941482291922192923886086820530577408000, 334764638208000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 34], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{136}.D_4', '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2], 'aut_gens': [[126, 55, 289, 288], [127, 55, 289, 288], [126, 265, 1, 288], [54, 127, 289, 288], [414, 55, 289, 288], [127, 54, 289, 288], [414, 265, 289, 288]], 'aut_group': '64.138', 'aut_hash': 138, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 8, 'aut_perms': [2309, 526, 5329, 3043, 12316, 18498], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [4, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_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, 3], [2, 2, 4], [4, 2, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, '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', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [4, 2, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 21, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 1], 'inner_gens': [[126, 265, 289, 288], [414, 55, 289, 288], [126, 55, 289, 288], [126, 55, 289, 288]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.11', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 8, 'linQ_dim': 3, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 10, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 27, 'number_subgroups': 35, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 11], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 151, 0], 'outer_gens': [[54, 127, 289, 288], [127, 55, 289, 288], [415, 54, 1, 288], [127, 54, 289, 288]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2]], 'representations': {'PC': {'code': 8772, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 78, 34]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16322, 16432, 3198]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8912, 8156, 13286, 14044]}, 'Perm': {'d': 6, 'gens': [126, 55, 289, 288]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times D_4', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
