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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '960.10124', 'ambient_counter': 10124, 'ambient_order': 960, 'ambient_tex': 'S_3\\times D_4\\times C_{20}', 'central': False, 'central_factor': False, 'centralizer_order': 160, 'characteristic': False, 'core_order': 8, 'counter': 262, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.10124.60.t1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '60.t1', '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': 60, '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': '16.2', 'subgroup_hash': 2, 'subgroup_order': 16, 'subgroup_tex': 'C_4^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.10124', 'aut_centralizer_order': 256, 'aut_label': '60.t1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '16.14', 'aut_weyl_index': 768, 'centralizer': '6.h1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.t1', '20.t1', '30.g1', '30.h1', '30.k1'], 'contains': ['120.c1', '120.i1', '120.k1'], 'core': '120.c1', 'coset_action_label': None, 'count': 3, 'diagramx': [1357, -1, 2066, -1], 'generators': [5, 720], 'label': '960.10124.60.t1', 'mobius_quo': None, 'mobius_sub': 2, 'normal_closure': '20.t1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '60.t1', 'projective_image': '240.206', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '60.t1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 6, 2, 2], 'aut_gens': [[1, 4], [3, 5], [3, 12], [14, 13], [9, 6], [3, 14]], 'aut_group': '96.195', 'aut_hash': 195, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [134, 16, 1447, 11520, 5160], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [4, 1, 12, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,\\mathbb{Z}/4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '96.195', 'autcent_hash': 195, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,\\mathbb{Z}/4)', '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], [4, 1, 12]], 'center_label': '16.2', 'center_order': 16, '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', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', '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, 4], [1, 4]], '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, 16]], 'label': '16.2', 'linC_count': 48, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 12, 'linQ_dim': 4, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 16, 'number_divisions': 10, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 15, 'number_subgroups': 15, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[3, 5], [3, 12], [14, 13], [9, 6], [3, 14]], 'outer_group': '96.195', 'outer_hash': 195, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [134, 16, 1447, 11520, 5160], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,\\mathbb{Z}/4)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6]], 'representations': {'PC': {'code': 10245, 'gens': [1, 3], 'pres': [4, 2, 2, 2, 2, 8, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16917782, 35931238]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 377]}, 'Perm': {'d': 8, 'gens': [16560, 22, 5160, 7]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2', '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': '160.228', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 4, 2, 6, 4, 4, 4, 12, 4, 4], 'aut_gens': [[1, 2, 4, 16], [9, 490, 4, 952], [1, 242, 4, 600], [321, 10, 492, 464], [321, 10, 492, 496], [9, 730, 484, 664], [809, 490, 484, 760], [481, 2, 492, 752], [169, 250, 492, 112], [801, 722, 4, 952], [481, 482, 492, 696]], 'aut_group': None, 'aut_hash': 1392258107361899436, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 28, 'aut_perms': [66058601818253593448229498774, 6469211024505649589852391270, 238781099275363751501446466266, 229243358451835785706132189923, 68882119789793230956573963595, 249685488713601163574541629368, 135813009892194617304190770830, 156425520124506987159240796663, 203190472624117630117010058246, 141098351255142698216656964806], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 4, 1], [2, 3, 4, 1], [2, 6, 4, 1], [3, 2, 1, 1], [4, 1, 4, 1], [4, 2, 2, 2], [4, 2, 4, 1], [4, 3, 4, 1], [4, 6, 2, 2], [4, 6, 4, 1], [5, 1, 4, 1], [6, 2, 1, 3], [6, 4, 4, 1], [10, 1, 4, 3], [10, 2, 16, 1], [10, 3, 16, 1], [10, 6, 16, 1], [12, 2, 4, 1], [12, 4, 2, 2], [12, 4, 4, 1], [15, 2, 4, 1], [20, 1, 16, 1], [20, 2, 8, 2], [20, 2, 16, 1], [20, 3, 16, 1], [20, 6, 8, 2], [20, 6, 16, 1], [30, 2, 4, 3], [30, 4, 16, 1], [60, 2, 16, 1], [60, 4, 8, 2], [60, 4, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^7.C_2^5)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 1718285292446712972, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [2, 3, 4], [2, 6, 4], [3, 2, 1], [4, 1, 4], [4, 2, 8], [4, 3, 4], [4, 6, 8], [5, 1, 4], [6, 2, 3], [6, 4, 4], [10, 1, 12], [10, 2, 16], [10, 3, 16], [10, 6, 16], [12, 2, 4], [12, 4, 8], [15, 2, 4], [20, 1, 16], [20, 2, 32], [20, 3, 16], [20, 6, 32], [30, 2, 12], [30, 4, 16], [60, 2, 16], [60, 4, 32]], 'center_label': '40.9', 'center_order': 40, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10124, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1], ['6.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [2, 3, 1, 4], [2, 6, 1, 4], [3, 2, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 3], [4, 3, 2, 2], [4, 6, 1, 2], [4, 6, 2, 3], [5, 1, 4, 1], [6, 2, 1, 3], [6, 4, 1, 4], [10, 1, 4, 3], [10, 2, 4, 4], [10, 3, 4, 4], [10, 6, 4, 4], [12, 2, 2, 2], [12, 4, 1, 2], [12, 4, 2, 3], [15, 2, 4, 1], [20, 1, 8, 2], [20, 2, 4, 2], [20, 2, 8, 3], [20, 3, 8, 2], [20, 6, 4, 2], [20, 6, 8, 3], [30, 2, 4, 3], [30, 4, 4, 4], [60, 2, 8, 2], [60, 4, 4, 2], [60, 4, 8, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 20442240, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '240.206', 'hash': 10124, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 1, 6], 'inner_gens': [[1, 2, 4, 656], [1, 2, 4, 496], [1, 2, 4, 16], [321, 482, 4, 16]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 160], [2, 120], [4, 20]], 'label': '960.10124', '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': 'S3*D4*C20', 'ngens': 8, '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 48, 'number_characteristic_subgroups': 86, 'number_conjugacy_classes': 300, 'number_divisions': 90, 'number_normal_subgroups': 338, 'number_subgroup_autclasses': 352, 'number_subgroup_classes': 852, 'number_subgroups': 2064, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 47], [3, 2], [4, 80], [5, 4], [6, 22], [10, 188], [12, 40], [15, 8], [20, 320], [30, 88], [60, 160]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 4, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 128, 0, 0, 0], 'outer_gens': [[9, 730, 484, 472], [801, 722, 4, 952], [641, 2, 484, 656], [321, 10, 12, 16], [649, 490, 484, 176], [489, 482, 12, 184], [801, 490, 4, 376]], 'outer_group': '512.10493066', 'outer_hash': 4599981025227913554, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 256, 'outer_perms': [40279435487454189341872202983260337427752758438547925675676157283283061122474906192639549408173873194249196773526768230196473476444088573458154288431254942134916737230679701832633562202448856396478876445415819590282335126796220730549249337030295966595580207278768776455775354521594983490949628084420028378483319290503387229919064212463166709968306467546642198679292801176591505928329947005354495025162595391112313019056979652433131498738062607570302543846292305045017164230932059388821766483872320239995478, 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127758829636571905166708575319838716932788713540997468218688641082413837423561082326978483370323374474341004165728829948738975839571231543137987672605779486681352140719114735732802358500252826595579959373162096566738915813815285675256611888348723134392841019632434510535529744737358403488902282676692307659136436111007380566278552027633892317380121457278455607867738459458971401160319785994932171089762786732912385445437193285500662333959040399647037036201649752864996134891102487300768464651947023263929888], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2.C_2^5', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 24], [8, 21], [16, 8], [32, 1]], 'representations': {'PC': {'code': 37240525529039920624864889989110169607, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -5, 66, 26244, 9932, 116, 16901, 141, 39430, 222]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [792019, 189007, 40966, 297011, 27451, 27181, 630313, 513019]}, 'Perm': {'d': 16, 'gens': [40280400, 1313901388800, 180583603200, 87454080, 33, 40279680, 2789705318400, 5760]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times D_4\\times C_{20}', 'transitive_degree': 240, 'wreath_data': None, 'wreath_product': False}
