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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '128.1733', 'ambient_counter': 1733, 'ambient_order': 128, 'ambient_tex': 'C_2^4.D_4', 'central': True, 'central_factor': False, 'centralizer_order': 128, 'characteristic': True, 'core_order': 2, 'counter': 133, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '128.1733.64.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '64.d1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '64.253', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 253, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 64, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_8:C_2^2', 'simple': True, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': True, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '128.1733', 'aut_centralizer_order': None, 'aut_label': '64.d1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['32.c1', '32.d1', '32.q1', '32.r1', '32.s1'], 'contains': ['128.a1'], 'core': '64.d1', 'coset_action_label': None, 'count': 1, 'diagramx': [1731, 7272, 1711, 7274], 'generators': [1880081], 'label': '128.1733.64.d1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '64.d1', 'normal_contained_in': ['32.c1', '32.d1'], 'normal_contains': ['128.a1'], 'normalizer': '1.a1', 'old_label': '64.d1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '64.d1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', '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': [[1]], '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], [2, 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], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 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': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 4, 4, 2, 2, 4, 2, 2], 'aut_gens': [[774151, 775481, 2544799, 1880081, 1697607], [2544791, 1881679, 1881241, 1880081, 1696455], [774151, 2545513, 110633, 1880081, 1070721], [775303, 111071, 1880089, 1880081, 1733127], [774151, 1881103, 110633, 1880081, 1034049], [775303, 2544937, 775343, 1880081, 1696455], [2543639, 112223, 111785, 1880081, 1033473], [774151, 111071, 775343, 1880081, 1734279], [2543639, 2543785, 2543647, 1880081, 1697607]], 'aut_group': None, 'aut_hash': 2287880195645369898, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 40, 'aut_perms': [634226231818785630521016355193287485697460915267, 540591711839924292051394682011371089212142306039, 676709356761149776131474368341375004421947822094, 295155618499508986471022890146355025214531372926, 333350415952077172155380921386576886997904674687, 518205897929492719577931731516716123113674140156, 639579601117792364328833358532598514126561457440, 83813444553428263361284350136299754616538291323], 'aut_phi_ratio': 128.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 4, 1], [2, 4, 2, 1], [2, 4, 4, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 2, 4, 1], [4, 4, 4, 1], [4, 8, 2, 2], [8, 4, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^8.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 8092891664598236742, 'autcent_nilpotent': True, 'autcent_order': 2048, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8.C_2^3', 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': 4, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 4, 6], [4, 2, 8], [4, 4, 4], [4, 8, 4], [8, 4, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '16.11', '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': 1733, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['64.133', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 4, 1, 6], [4, 2, 1, 4], [4, 2, 2, 2], [4, 4, 1, 4], [4, 8, 1, 4], [8, 4, 2, 4]], 'element_repr_type': 'GLZN', 'elementary': 2, 'eulerian_function': 10080, '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': 1733, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [1, 2, 2, 1, 4], 'inner_gens': [[774151, 775481, 2544799, 1880081, 1697607], [774151, 775481, 2543647, 1880081, 1734279], [774151, 774329, 2544799, 1880081, 1733703], [774151, 775481, 2544799, 1880081, 1697607], [774151, 2545513, 775343, 1880081, 1697607]], 'inner_hash': 11, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': None, 'inner_tex': 'C_2\\times D_4', 'inner_used': [2, 3, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 20], [4, 2]], 'label': '128.1733', 'linC_count': None, 'linC_degree': None, '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': 'C2^4.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': 16, 'number_characteristic_subgroups': 28, 'number_conjugacy_classes': 38, 'number_divisions': 32, 'number_normal_subgroups': 116, 'number_subgroup_autclasses': 136, 'number_subgroup_classes': 380, 'number_subgroups': 716, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 31], [4, 64], [8, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 4, 4, 4, 4], 'outer_gen_pows': [110593, 110593, 110593, 110593, 110593], 'outer_gens': [[2544791, 1881103, 1881241, 1880081, 1070721], [2544791, 2544937, 111785, 1880081, 1070721], [2543639, 1882255, 774191, 1880081, 1697607], [775303, 775481, 110633, 1880081, 1697607], [2544791, 775481, 2543647, 1880081, 1070721]], 'outer_group': '512.554852', 'outer_hash': 342028529733335859, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 64, 'outer_perms': [58707105421296902703509374267941509585542385413250118021572354332821856033647875044420700, 100822950033395084507638143741714438911610306933790503816927472911375265846167202665261612, 50641123683238592790090254964654610634050885652525424759597174471258331798999373411044512, 26059419550755600841548325184513371516301663091001257426530242755314999978737644623764415, 62073394736892462411710092158760552028918459942971574128890641744482963522683262479492212], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4.C_2^5', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 22, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 2], [8, 2]], 'representations': {'PC': {'code': 36930082110863364144, 'gens': [1, 2, 3, 4, 5], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 723, 851, 228, 102, 2028, 124]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [111169, 1880081, 111745, 774151, 1734585, 111031, 1034049]}, 'Perm': {'d': 22, 'gens': [59275110081562139646, 114393868851270234505, 168863684306013854694, 222514552283369517649, 271026042451458739488, 288, 222514552283369517360]}}, '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_2^4.D_4', 'transitive_degree': 64, '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': 2, 'aut_exponent': 8, 'aut_gen_orders': [4, 4, 8, 8, 2], 'aut_gens': [[28679, 22555, 61575, 4129], [28679, 14365, 36993, 28839], [28807, 14461, 36993, 4193], [28679, 14429, 61575, 28903], [28807, 14429, 36993, 28903], [28679, 14429, 36993, 4321]], 'aut_group': '1024.dic', 'aut_hash': 6313253630026183879, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 1024, 'aut_permdeg': 20, 'aut_perms': [1153629239441056574, 816035011647252489, 1902789925529559240, 305317781866097418, 1416601845424130534], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 4, 4, 1], [4, 1, 4, 1], [4, 2, 1, 2], [4, 4, 4, 1], [8, 2, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4.D_4^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 4], [4, 1, 4], [4, 2, 2], [4, 4, 4], [8, 2, 8]], 'center_label': '8.2', 'center_order': 8, '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'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 253, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['32.42', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 4], [4, 1, 2, 2], [4, 2, 1, 2], [4, 4, 1, 4], [8, 2, 2, 4]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 5040, 'exponent': 8, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '16.14', 'hash': 253, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [1, 2, 2, 4], 'inner_gens': [[28679, 22555, 61575, 4129], [28679, 22555, 61447, 4321], [28679, 22683, 61575, 4129], [28679, 22747, 61575, 4129]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12]], 'label': '64.253', 'linC_count': None, 'linC_degree': None, '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': 'D8:C2^2', 'ngens': 4, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 11, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 28, 'number_divisions': 22, 'number_normal_subgroups': 81, 'number_subgroup_autclasses': 49, 'number_subgroup_classes': 133, 'number_subgroups': 201, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 23], [4, 24], [8, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 2, 4], 'outer_gen_pows': [4097, 4097, 61575, 4097, 61575], 'outer_gens': [[28679, 22555, 36993, 28903], [28679, 22555, 36865, 4129], [28679, 14493, 61447, 28775], [28679, 14461, 61447, 4257], [28807, 22651, 36993, 4129]], 'outer_group': '128.1135', 'outer_hash': 1135, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [167450520, 40723343, 168186383, 212340360, 41505138], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4], [8, 2]], 'representations': {'PC': {'code': 18421183945752, 'gens': [1, 2, 3, 4], 'pres': [6, -2, 2, 2, 2, -2, -2, 332, 681, 69, 730, 88]}, 'GLFq': {'d': 3, 'q': 9, 'gens': [61962567, 137858243, 374880322, 139331306, 86106566, 264279152]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [4129, 28679, 4161, 36865, 55323, 4225]}, 'Perm': {'d': 18, 'gens': [365311731826129, 816646503012841, 1203340966629127, 1581328363371295, 1914448859061120, 1581328363371294]}}, 'schur_multiplier': [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': 'D_8:C_2^2', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
