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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1024.ddn', 'ambient_counter': 2120, 'ambient_order': 1024, 'ambient_tex': '(C_2\\times D_4^2):D_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 16, 'counter': 891, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1024.ddn.16.SL', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '16.ro1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 16, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '64.215', 'subgroup_hash': 215, 'subgroup_order': 64, 'subgroup_tex': 'C_2^3:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1024.ddn', 'aut_centralizer_order': None, 'aut_label': '16.SL', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '128.G', 'complements': None, 'conjugacy_class_count': 8, 'contained_in': ['8.DJ', '8.IT', '8.IZ', '8.JH', '8.JK', '8.KB', '8.KD'], 'contains': ['32.FX', '32.FY', '32.GE', '32.GF', '32.HY', '32.HZ', '32.JH', '32.JJ', '32.KP', '32.KT', '32.KU', '32.OT', '32.PF', '32.XU', '32.YM'], 'core': '64.J', 'coset_action_label': None, 'count': 16, 'diagramx': [4109, -1, 4110, -1], 'generators': [40284840, 2789881476623, 1313988491527, 40279687, 1307674735926, 1313941673647], 'label': '1024.ddn.16.SL', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.I', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.C', 'old_label': '16.ro1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.SL', 'subgroup_fusion': None, 'weyl_group': None}
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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': [4, 2, 8], 'aut_gens': [[1, 2, 4, 16], [39, 58, 60, 56], [41, 10, 36, 56], [15, 18, 28, 49]], 'aut_group': '2048.coe', 'aut_hash': 7083753762161107245, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 2048, 'aut_permdeg': 16, 'aut_perms': [2972367177234, 4203240942724, 15366105380370], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 2, 4, 1], [2, 4, 4, 1], [4, 4, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr 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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', '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, 6], [2, 4, 4], [4, 4, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '16.14', 'commutator_count': 1, 'commutator_label': '4.2', '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': 215, '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, 6], [2, 4, 1, 4], [4, 4, 1, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 2520, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '16.14', 'hash': 215, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2, 2], 'inner_gens': [[1, 2, 4, 24], [1, 2, 12, 48], [1, 10, 4, 48], [9, 34, 36, 16]], 'inner_hash': 14, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': False, 'inner_tex': 'C_2^4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 4], [4, 2]], 'label': '64.215', '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': 'C2^3:D4', 'ngens': 4, 'nilpotency_class': 2, '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': 8, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 22, 'number_divisions': 22, 'number_normal_subgroups': 81, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 173, 'number_subgroups': 321, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 31], [4, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 8, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 6, 0, 0, 0, 0], 'outer_gens': [[7, 18, 20, 48], [33, 34, 4, 16], [1, 35, 37, 17], [33, 2, 4, 16], [1, 3, 5, 22], [1, 2, 4, 48], [1, 34, 36, 16]], 'outer_group': '128.928', 'outer_hash': 928, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 8, 'outer_perms': [5381, 2356, 13738, 12264, 37965, 2354, 12316], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\wr D_4', '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, 4], [4, 2]], 'representations': {'PC': {'code': 1144608855725332, 'gens': [1, 2, 3, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 116, 50, 724, 730, 376, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58301495115895688, 125055985569224674, 24992885291910908, 125101743041088268]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [11762753, 11040833, 7292993, 5128257, 31029334, 6210625]}, 'Perm': {'d': 12, 'gens': [41455445, 12782280, 94525327, 11290320, 132452760, 132452776]}}, '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': 'C_2^3:D_4', 'transitive_degree': 16, '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [4, 4, 8, 4, 8, 8], 'aut_gens': [[5167, 87102727, 302814840, 1313901388800, 5606321846392, 12526762728376], [4103693809943, 127387567, 19614852622078, 127387560, 12520495790657, 15316514841600], [16825370284657, 18126770083062, 1314028776367, 16825147303657, 14008840836601, 5612588778958], [19615028054278, 18126770083062, 1313941673640, 176158200, 12526762728376, 14008840836601], [4103693809943, 4103646986903, 1314164654647, 4103646992063, 9703614810942, 6920263514872], [16825370284657, 20916522950273, 2789705318416, 302814840, 8395940079943, 11212821059896], [18126857180862, 18126730524222, 5160, 16825107018817, 12520495790657, 14002613452807]], 'aut_group': '65536.z', 'aut_hash': 7116534239874182336, 'aut_nilpotency_class': 6, 'aut_nilpotent': True, 'aut_order': 65536, 'aut_permdeg': 64, 'aut_perms': [52591626948307115290803812964962180716499066940416715786308353773404458532969890448490811, 79733326356524296867543537387297126645539955715736747972024602866386158157742232478556858, 22751799738223641498436241376578862879180243805052763503992995566977967652642691723433401, 101169841853168296652318344727987780764917922908601714227056132654911963072060055399938585, 44846508443645655065913432885788396810722743632805870966401208460230441877411542988798238, 61413131909602486310467318809265549046621126430585046183456248624528609677139695243181437], 'aut_phi_ratio': 128.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 2, 2, 1], [2, 4, 2, 1], [2, 4, 4, 2], [2, 4, 8, 1], [2, 8, 2, 2], [2, 8, 4, 3], [2, 16, 2, 1], [4, 4, 2, 1], [4, 4, 4, 2], [4, 8, 1, 1], [4, 8, 2, 2], [4, 8, 4, 3], [4, 8, 8, 2], [4, 16, 2, 1], [4, 32, 2, 1], [4, 32, 8, 1], [8, 32, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9.C_2\\wr D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': None, 'autcentquo_hash': 7822646973264783822, 'autcentquo_nilpotent': True, 'autcentquo_order': 2048, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^5.D_4^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 4, 18], [2, 8, 16], [2, 16, 2], [4, 4, 10], [4, 8, 33], [4, 16, 2], [4, 32, 10], [8, 32, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '512.6536092', 'commutator_count': 1, 'commutator_label': '32.46', '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', '2.1', '2.1', '2.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 2120, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 4, 1, 18], [2, 8, 1, 16], [2, 16, 1, 2], [4, 4, 1, 10], [4, 8, 1, 33], [4, 16, 1, 2], [4, 32, 1, 10], [8, 32, 1, 4]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 5119672320, 'exponent': 8, 'exponents_of_order': [10], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[8, 1, 8]], 'familial': False, 'frattini_label': '32.46', 'frattini_quotient': '32.51', 'hash': 7003003021026862916, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 2, 4, 4], 'inner_gens': [[5167, 87102727, 263265840, 1313901388800, 5606321846392, 11212821054743], [5167, 87102727, 263265840, 1313901388800, 8395940079943, 15316514846767], [40279687, 127387567, 302814840, 1313901388800, 15310288183806, 9703614815857], [5167, 87102727, 302814840, 1313901388800, 6920263514872, 11212861334416], [5167, 2789705323576, 18126554366022, 40279680, 5606321846392, 12526722443536], [1313941668480, 2789745598096, 18126554366022, 5160, 5606362120912, 12526762728376]], 'inner_hash': 3725227854009754059, 'inner_nilpotent': True, 'inner_order': 512, 'inner_split': False, 'inner_tex': 'C_2^6:C_2^3', 'inner_used': [1, 2, 3, 4, 5, 6], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 32], [2, 56], [8, 12]], 'label': '1024.ddn', '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': False, 'metacyclic': False, 'monomial': True, 'name': '(C2*D4^2):D4', 'ngens': 6, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [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': 29, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 100, 'number_divisions': 100, 'number_normal_subgroups': 947, 'number_subgroup_autclasses': 3360, 'number_subgroup_classes': 22344, 'number_subgroups': 79669, 'old_label': None, 'order': 1024, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 239], [4, 656], [8, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 16819007022583, 0, 0, 0, 0, 0], 'outer_gens': [[4103734094783, 1314028776360, 18126594650862, 1314028776367, 6920263509719, 9703654364857], [20916435842633, 16825322735857, 4103646992063, 215717040, 6913996572238, 12520495790657], [5167, 1313988491520, 1314164654647, 1313901388800, 5612549225033, 11219087992462], [5167, 127387567, 1314164654647, 40284847, 5612549225033, 11219087992462], [2789792421136, 87102727, 20916299969273, 127387560, 6920263509719, 9703654369782], [5167, 87102727, 302814840, 40284847, 5606321846392, 12526762728376], [5167, 87102727, 1314164654647, 40284847, 5606321846392, 12526762728376]], 'outer_group': '128.1755', 'outer_hash': 1755, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 10, 'outer_perms': [126745, 1951327, 26334, 1, 2718174, 167280, 1275486], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 56], [8, 12]], 'representations': {'PC': {'code': '31859800216577785331555308478271677616527907748749314695626699018286266766861952945339170', 'gens': [1, 2, 4, 5, 6, 8, 9], 'pres': [10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 121, 51, 20803, 10413, 743, 29604, 2814, 3734, 5765, 2895, 755, 405, 175, 51207, 39057, 7077, 3247, 1657, 69128, 12978, 8678, 4368, 618, 268, 28819]}, 'Perm': {'d': 16, 'gens': [87102727, 1313901388800, 12526762728376, 302814840, 5167, 5606321846392]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times D_4^2):D_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
