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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '128.1883', 'ambient_counter': 1883, 'ambient_order': 128, 'ambient_tex': '\\OD_{16}:D_4', 'central': True, 'central_factor': False, 'centralizer_order': 128, 'characteristic': True, 'core_order': 4, 'counter': 74, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '128.1883.32.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '32.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '32.46', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 46, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 32, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2\\times D_4', 'simple': False, 'solvable': True, 'special_labels': ['Z', 'S', 'U2'], 'split': False, 'standard_generators': False, 'stem': True, 'subgroup': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '128.1883', 'aut_centralizer_order': 8192, 'aut_label': '32.b1', 'aut_quo_index': 48, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 8192, 'centralizer': '1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.a1', '16.b1', '16.c1', '16.d1', '16.e1', '16.m1', '16.n1', '16.o1'], 'contains': ['64.a1', '64.b1'], 'core': '32.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [4140, 7096, 4233, 7223], 'generators': [8, 64], 'label': '128.1883.32.b1', 'mobius_quo': 2, 'mobius_sub': 0, 'normal_closure': '32.b1', 'normal_contained_in': ['16.a1', '16.b1', '16.c1', '16.d1', '16.e1'], 'normal_contains': ['64.a1', '64.b1'], 'normalizer': '1.a1', 'old_label': '32.b1', 'projective_image': '32.46', 'quotient_action_image': '1.1', 'quotient_action_kernel': '32.46', 'quotient_action_kernel_order': 32, 'quotient_fusion': None, 'short_label': '32.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', '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]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 4, 2, 4, 4, 4, 2, 4, 2, 2], 'aut_gens': [[1, 2, 4, 16], [1, 10, 76, 80], [65, 99, 108, 89], [73, 10, 4, 88], [73, 2, 12, 124], [1, 42, 4, 80], [1, 102, 4, 88], [1, 74, 76, 48], [1, 47, 4, 61], [1, 74, 4, 80], [65, 2, 76, 112]], 'aut_group': '16384.kq', 'aut_hash': 2493808428860909963, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 16384, 'aut_permdeg': 48, 'aut_perms': [1652510121207743161409924688074226280480024548566306751754324, 9392922325414832238047741157822903398826634345083677533396643, 4699008264032556033411144581632659562853737801875776575443302, 8648281462645597129639625346494422337703756384312933304729096, 7991711708039276273987855665754461958709892450723265301627364, 10647358530203450047799927048720776094646047447572726698721174, 9512696022452453674642496652176802852992904709344439770786934, 11058685482776246362993525823232250435260070340662414111340269, 4888232796389600509231656066749574752068740190329268844711040, 3271645801186071356708289140735049727040521781083140718130033], 'aut_phi_ratio': 256.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 8, 2, 1], [2, 8, 4, 1], [4, 2, 2, 2], [4, 2, 4, 1], [4, 4, 2, 1], [4, 8, 2, 1], [8, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4.C_2^7:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '64.202', 'autcentquo_hash': 202, 'autcentquo_nilpotent': True, 'autcentquo_order': 64, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3:D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 8, 6], [4, 2, 8], [4, 4, 2], [4, 8, 2], [8, 4, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '32.46', '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', '2.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1883, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 8, 1, 6], [4, 2, 1, 8], [4, 4, 1, 2], [4, 8, 1, 2], [8, 4, 1, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 5040, 'exponent': 8, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '16.14', 'hash': 1883, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 4], 'inner_gens': [[1, 2, 4, 80], [1, 2, 12, 112], [1, 10, 4, 16], [65, 34, 4, 16]], 'inner_hash': 46, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'C_2^2\\times D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 4]], 'label': '128.1883', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'OD16:D4', 'ngens': 4, 'nilpotency_class': 3, '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': 12, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 32, 'number_divisions': 32, 'number_normal_subgroups': 112, 'number_subgroup_autclasses': 91, 'number_subgroup_classes': 328, 'number_subgroups': 772, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 55], [4, 40], [8, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 4, 2, 4, 2], 'outer_gen_pows': [96, 62, 0, 0, 100, 0, 101], 'outer_gens': [[1, 42, 4, 80], [9, 70, 100, 116], [65, 110, 100, 92], [65, 98, 36, 28], [1, 102, 12, 28], [1, 46, 68, 20], [73, 11, 76, 125]], 'outer_group': '512.7538482', 'outer_hash': 4576151551108898164, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 64, 'outer_perms': [6238161418628303083380937776806716302297773325053249153268953102793539810284356889640335, 62149592971615367387455809512344915638608418436649833542076066477916094462999577603031665, 46341271170391742445664750979861964394946743901598106038899088599373572049115000662111875, 62182594006534705281129652049956539535638012035889142815638236948237109573970000183358532, 79610230543909875834481931696662605198968564044441673367391319733970089943060888303093192, 34733864225370795763980673574612120901461892920273176695918688420524957291030032626170502, 11128856923838888994383537095367693700355837358779100965932831002900273847643268659377940], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3.C_2^6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 4]], 'representations': {'PC': {'code': 604419382499573027380, 'gens': [1, 2, 3, 5], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 135, 58, 2804, 1971, 102, 2028, 124]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58438776781765922, 25038638883202496, 125101749995522276, 25038645869518594]}, 'GLZN': {'d': 2, 'p': 24, 'gens': [14113, 235025, 101591, 13883, 184835, 18569, 13969]}, 'Perm': {'d': 12, 'gens': [15286445, 48763440, 99071409, 22182616, 134352720, 182610856, 182610840]}}, 'schur_multiplier': [2, 2, 2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\OD_{16}:D_4', 'transitive_degree': 32, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [12, 4], 'aut_gens': [[2565, 517, 1537, 2577], [517, 2593, 1553, 529], [2597, 2561, 1569, 529]], 'aut_group': '3072.dx', 'aut_hash': 2019763691454065160, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3072, 'aut_permdeg': 16, 'aut_perms': [11404422904581, 2678277315556], 'aut_phi_ratio': 192.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 2, 8, 1], [4, 2, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6:(C_2\\times S_4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '1536.408640869', 'autcent_hash': 4599629986281706675, 'autcent_nilpotent': False, 'autcent_order': 1536, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^6:S_4', '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, 7], [2, 2, 8], [4, 2, 4]], 'center_label': '8.5', 'center_order': 8, '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', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 46, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 8], [4, 2, 1, 4]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 105, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '16.14', 'hash': 46, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 1, 2, 2], 'inner_gens': [[2565, 517, 1537, 2577], [2565, 517, 1537, 2577], [2565, 517, 1537, 2609], [2565, 517, 1569, 2577]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 4]], 'label': '32.46', 'linC_count': 192, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 192, 'linQ_dim': 4, 'linQ_dim_count': 192, 'linR_count': 192, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*D4', 'ngens': 4, 'nilpotency_class': 2, '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 20, 'number_divisions': 20, 'number_normal_subgroups': 78, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 118, 'number_subgroups': 158, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 23], [4, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [513, 2577, 513, 513, 513, 513, 513, 513, 513], 'outer_gens': [[2561, 517, 1537, 2577], [2565, 517, 3601, 2577], [2561, 2565, 1537, 2577], [2597, 549, 3585, 2577], [2597, 517, 1541, 2577], [2597, 549, 1537, 529], [2597, 517, 1537, 2581], [2565, 517, 3585, 2577], [2565, 517, 1541, 2577]], 'outer_group': '768.1090070', 'outer_hash': 1090070, 'outer_nilpotent': False, 'outer_order': 768, 'outer_permdeg': 12, 'outer_perms': [41909165, 41141520, 727204, 182610863, 138989536, 135359400, 101693640, 182610840, 138989520], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^5:S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4]], 'representations': {'PC': {'code': 542728, 'gens': [1, 2, 3, 4], 'pres': [5, -2, 2, 2, 2, -2, 133, 58]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931074, 7233746, 7233586, 7115160]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [34876415, 29555524, 28816400, 14811013, 28233362]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [2561, 529, 2565, 1537, 545]}, 'Perm': {'d': 8, 'gens': [1567, 5160, 11536, 23, 11520]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_2^2\\times D_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
