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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1760.355', 'ambient_counter': 355, 'ambient_order': 1760, 'ambient_tex': 'C_{11}:(C_{10}\\times Q_{16})', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 110, 'counter': 37, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1760.355.16.a1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '16.a1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '16.9', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 9, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'Q_{16}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '110.2', 'subgroup_hash': 2, 'subgroup_order': 110, 'subgroup_tex': 'C_{11}:C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.355', 'aut_centralizer_order': 64, 'aut_label': '16.a1', 'aut_quo_index': 2, 'aut_stab_index': 2, 'aut_weyl_group': '110.1', 'aut_weyl_index': 128, 'centralizer': '110.a1.a1', 'complements': ['110.d1.b1', '110.d1.c1', '110.d1.a1', '110.d1.d1'], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1'], 'contains': ['32.a1.a1', '80.a1.b1', '176.a1.b1'], 'core': '16.a1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [9698, 9474, 9339, 1381, 9011, 1125, 360, 2485], 'generators': [920, 160, 16], 'label': '1760.355.16.a1.b1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '16.a1.b1', 'normal_contained_in': ['8.a1.a1'], 'normal_contains': ['32.a1.a1', '80.a1.b1'], 'normalizer': '1.a1.a1', 'old_label': '16.a1.b1', 'projective_image': '880.21', 'quotient_action_image': '2.1', 'quotient_action_kernel': '8.4', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '16.a1.b1', 'subgroup_fusion': None, 'weyl_group': '110.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 110, 'aut_gen_orders': [10, 11], 'aut_gens': [[1, 5], [1, 35], [81, 5]], 'aut_group': '110.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 110, 'aut_permdeg': 11, 'aut_perms': [1673202, 31767604], 'aut_phi_ratio': 2.75, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 11, 1, 4], [10, 11, 1, 4], [11, 5, 2, 1], [22, 5, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_{11}', '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': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [5, 11, 4], [10, 11, 4], [11, 5, 2], [22, 5, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '55.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '11.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 11, 4, 1], [10, 11, 4, 1], [11, 5, 2, 1], [22, 5, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 110, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[5, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '110.2', 'hash': 2, 'hyperelementary': 5, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 55, 'inner_gen_orders': [5, 11], 'inner_gens': [[1, 75], [41, 5]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 55, 'inner_split': True, 'inner_tex': 'C_{11}:C_5', 'inner_used': [1, 2], 'irrC_degree': 5, 'irrQ_degree': 10, 'irrQ_dim': 10, 'irrR_degree': 10, 'irrep_stats': [[1, 10], [5, 4]], 'label': '110.2', 'linC_count': 2, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C11:C10', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, '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': 6, 'number_conjugacy_classes': 14, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 28, 'old_label': None, 'order': 110, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [5, 44], [10, 44], [11, 10], [22, 10]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [4], 'outer_gens': [[1, 85]], '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': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [10, 2]], 'representations': {'PC': {'code': 1186286309, 'gens': [1, 2], 'pres': [3, -5, -2, -11, 451, 16, 362]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [1343, 13320, 1335]}, 'Perm': {'d': 13, 'gens': [1, 48041406, 567358848]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}:C_{10}', 'transitive_degree': 22, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [20, 10, 10, 20, 2, 20, 10], 'aut_gens': [[1, 2, 80], [61, 282, 720], [941, 1422, 400], [1, 1502, 720], [21, 442, 1080], [881, 1742, 1680], [1, 1682, 1080], [881, 142, 1200]], 'aut_group': None, 'aut_hash': 7334205941972959822, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14080, 'aut_permdeg': 96, 'aut_perms': [402301515854980344273039731273618828084532219813227417501016350773943541538588479664034157340463337246411283725958455992927834663284197934316384992880, 298621250106876200512934236503965152758111579125071179554251779469351985599499767371954373192322854035625104117028028493688217916783731743793559115435, 6044558758337461569571566677812299837529616810608108355112787755083491318052125286287491660498443850161175022558895054962235301152817258721904982691, 802508515378447568840914662796644686038244437888428520391663633213306426905538379680707227252507309430628113878474140015020086037003225119504906474397, 885754732321393763631474161739599936532668060924131793046620852745219216637466388109599214691211636738816204902792005426739241186957502590148933066770, 8923443750600565961172060726528262672276327494903409727708839262882844243734180276248402536597813881024725680419648409349299733730630928884230007003, 883197386786200241515328008509731750667113940251450569163158375846125611286677775562850403706911028893334902255316332410564785446000898803786213782841], 'aut_phi_ratio': 22.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 1, 2], [4, 4, 2, 1], [4, 44, 2, 1], [5, 11, 1, 4], [8, 22, 4, 1], [10, 11, 1, 4], [10, 11, 2, 4], [11, 10, 1, 1], [20, 22, 1, 8], [20, 44, 2, 8], [22, 10, 1, 1], [22, 10, 2, 1], [40, 22, 4, 4], [44, 20, 1, 2], [44, 20, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{11}.(C_{10}\\times D_4).C_2^4', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '440.42', 'autcentquo_hash': 42, 'autcentquo_nilpotent': False, 'autcentquo_order': 440, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 4, 2], [4, 44, 2], [5, 11, 4], [8, 22, 4], [10, 11, 12], [11, 10, 1], [20, 22, 8], [20, 44, 16], [22, 10, 3], [40, 22, 16], [44, 20, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '440.11', 'commutator_count': 1, 'commutator_label': '44.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 355, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['880.21', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 4, 1, 2], [4, 44, 1, 2], [5, 11, 4, 1], [8, 22, 2, 2], [10, 11, 4, 3], [11, 10, 1, 1], [20, 22, 4, 2], [20, 44, 4, 4], [22, 10, 1, 3], [40, 22, 8, 2], [44, 20, 1, 2], [44, 20, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 124992, 'exponent': 440, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '440.42', 'hash': 355, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 220, 'inner_gen_orders': [2, 20, 11], 'inner_gens': [[1, 62, 80], [21, 2, 1360], [1, 482, 80]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 440, 'inner_split': True, 'inner_tex': 'D_{22}:C_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 30], [10, 8], [20, 2]], 'label': '1760.355', 'linC_count': 80, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 4, 'linQ_dim': 18, 'linQ_dim_count': 4, 'linR_count': 24, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C11:(C10*Q16)', '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 46, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 80, 'number_divisions': 32, 'number_normal_subgroups': 55, 'number_subgroup_autclasses': 84, 'number_subgroup_classes': 120, 'number_subgroups': 722, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [4, 100], [5, 44], [8, 88], [10, 132], [11, 10], [20, 880], [22, 30], [40, 352], [44, 120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 20, 0], 'outer_gens': [[41, 2, 1680], [881, 42, 1680], [921, 42, 120], [921, 922, 120]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [7, 16, 120, 11640], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 29, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 10], [8, 2], [10, 4], [16, 2], [20, 4]], 'representations': {'PC': {'code': 11032761968854402741268738980442961394330848045287016790000309, 'gens': [1, 2, 6], 'pres': [7, -2, -2, -2, -2, -5, -2, -11, 280, 869, 36, 926, 58, 80, 28572, 2539, 3806, 3183, 124, 23533, 5900, 8847, 1994]}, 'GLZN': {'d': 2, 'p': 77, 'gens': [15722750, 29218127, 457073, 30983305, 5807055, 19630962, 15522156]}, 'Perm': {'d': 29, 'gens': [10904406619526537425340565991, 1, 799819538072785, 22616640318319234590154752000, 1181150107543566, 1559624222783880, 337556895292183262877941760000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}:(C_{10}\\times Q_{16})', 'transitive_degree': 352, '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': 8, 'aut_gen_orders': [2, 2, 8], 'aut_gens': [[1, 2], [1, 10], [1, 14], [3, 2]], 'aut_group': '32.43', 'aut_hash': 43, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [10824, 18366, 7679], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 4, 2, 1], [8, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_8:C_2', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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, 1], [4, 2, 1], [4, 4, 2], [8, 2, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, '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'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 4, 1, 2], [8, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 8, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, -1, 2]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '4.2', 'hash': 9, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4], 'inner_gens': [[1, 14], [5, 2]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 3]], 'label': '16.9', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'Q16', 'ngens': 2, 'nilpotency_class': 3, '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': 7, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 7, 'number_subgroup_classes': 9, 'number_subgroups': 11, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 10], [8, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 2], 'outer_gens': [[1, 10], [3, 2]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1]], 'representations': {'PC': {'code': 39993887, 'gens': [1, 2], 'pres': [4, -2, 2, -2, -2, 32, 113, 21, 146, 34]}, 'Lie': [{'d': 2, 'q': 9, 'gens': [5109, 720, 1460, 3648], 'family': 'CSOPlus'}, {'d': 2, 'q': 7, 'gens': [2125, 2064, 1452, 675], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 7, 'gens': [1690, 2042]}, 'Perm': {'d': 16, 'gens': [12526762711680, 6920136155398, 4097467152142, 1313941673647]}}, 'schur_multiplier': [], 'semidirect_product': False, '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': 'Q_{16}', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
