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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '4608.v', 'ambient_counter': 22, 'ambient_order': 4608, 'ambient_tex': 'D_{24}.C_{96}', 'central': False, 'central_factor': False, 'centralizer_order': 192, 'characteristic': False, 'core_order': 4, 'counter': 462, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4608.v.576._.K', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '576.i1.a1', 'outer_equivalence': False, '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': 576, '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': '8.3', 'subgroup_hash': 3, 'subgroup_order': 8, 'subgroup_tex': 'D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4608.v', 'aut_centralizer_order': 256, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '8.3', 'aut_weyl_index': 1536, 'centralizer': '24._.O', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['192._.C', '192._.Z', '288._.F', '288._.N', '288._.P'], 'contains': ['1152._.C', '1152._.D'], 'core': '1152._.C', 'coset_action_label': None, 'count': 3, 'diagramx': [9119, -1, 8055, -1, 9042, -1, 9040, -1], 'generators': [805174465, 1380299136, 2084593], 'label': '4608.v.576._.K', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '192._.C', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3._.B', 'old_label': '576.i1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '576._.K', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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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': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], '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': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, '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': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '384.5547', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 48, 'aut_gen_orders': [16, 24, 24, 16, 8, 48, 48, 8], 'aut_gens': [[37442, 395398321, 35945290], [6882766, 115025093, 913010366], [481149, 395398321, 158159276], [287377, 992089873, 833930728], [37442, 992089873, 71890580], [343540, 1301219333, 1250896092], [4039104, 395398321, 862686960], [3187202, 395398321, 1049602468], [5622862, 1272463101, 589502756]], 'aut_group': None, 'aut_hash': 2675128990628703738, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 58, 'aut_perms': [1209836495979212660927431096201921996793240735553015761880256400263328622580693, 460404172012635887940176008295548619537204047757590778225083279146513706895115, 174652649647282533900418686626609112483306293175539093391243115546255472235550, 1597639767924934956743718311830340876443831605492568598712822068395741430687228, 1765178233114510148290555356882976171987611967425069352141110052884293477525307, 1074154375750739599809226010366626104094520919021880789240330648703660255467695, 2331923544139373031937616687423484820025328957153845510685169469375855377795222, 460087216947899050909888636323091739061296835940936754624644615139315293135174], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 4, 1], [4, 12, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 3], [6, 2, 4, 1], [6, 12, 4, 1], [8, 1, 4, 1], [8, 2, 2, 3], [8, 12, 4, 1], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 2], [12, 2, 8, 2], [12, 2, 16, 1], [12, 12, 4, 1], [16, 1, 8, 1], [16, 2, 4, 1], [16, 2, 8, 1], [16, 12, 8, 1], [24, 1, 8, 1], [24, 2, 4, 7], [24, 2, 8, 4], [24, 12, 8, 1], [32, 1, 16, 1], [32, 2, 8, 1], [32, 2, 16, 1], [32, 12, 16, 1], [48, 1, 16, 1], [48, 2, 8, 3], [48, 2, 16, 4], [48, 2, 32, 1], [48, 12, 16, 1], [64, 1, 32, 1], [64, 2, 16, 1], [64, 2, 32, 1], [64, 12, 32, 1], [96, 1, 32, 1], [96, 2, 16, 3], [96, 2, 32, 4], [96, 2, 64, 1], [96, 12, 32, 1], [192, 1, 64, 1], [192, 2, 32, 3], [192, 2, 64, 4], [192, 2, 128, 1], [192, 12, 64, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:((C_4\\times C_8).C_2^6.C_2)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '256.55608', 'autcent_hash': 55608, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\times C_{16}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '48.38', 'autcentquo_hash': 38, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 12, 2], [3, 1, 2], [3, 2, 3], [4, 1, 2], [4, 2, 5], [4, 12, 2], [6, 1, 2], [6, 2, 11], [6, 12, 4], [8, 1, 4], [8, 2, 6], [8, 12, 4], [12, 1, 4], [12, 2, 46], [12, 12, 4], [16, 1, 8], [16, 2, 12], [16, 12, 8], [24, 1, 8], [24, 2, 60], [24, 12, 8], [32, 1, 16], [32, 2, 24], [32, 12, 16], [48, 1, 16], [48, 2, 120], [48, 12, 16], [64, 1, 32], [64, 2, 48], [64, 12, 32], [96, 1, 32], [96, 2, 240], [96, 12, 32], [192, 1, 64], [192, 2, 480], [192, 12, 64]], 'center_label': '192.2', 'center_order': 192, 'central_product': True, 'central_quotient': '24.6', 'commutator_count': 1, 'commutator_label': '12.2', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 22, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1536.10843710', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 2, 2], [4, 12, 1, 2], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 5], [6, 12, 2, 2], [8, 1, 4, 1], [8, 2, 2, 3], [8, 12, 2, 2], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 10], [12, 12, 2, 2], [16, 1, 8, 1], [16, 2, 4, 1], [16, 2, 8, 1], [16, 12, 4, 2], [24, 1, 8, 1], [24, 2, 4, 7], [24, 2, 8, 4], [24, 12, 4, 2], [32, 1, 16, 1], [32, 2, 8, 1], [32, 2, 16, 1], [32, 12, 8, 2], [48, 1, 16, 1], [48, 2, 8, 3], [48, 2, 16, 6], [48, 12, 8, 2], [64, 1, 32, 1], [64, 2, 16, 1], [64, 2, 32, 1], [64, 12, 16, 2], [96, 1, 32, 1], [96, 2, 16, 3], [96, 2, 32, 6], [96, 12, 16, 2], [192, 1, 64, 1], [192, 2, 32, 3], [192, 2, 64, 6], [192, 12, 32, 2]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 2236416, 'exponent': 192, 'exponents_of_order': [9, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 256]], 'familial': False, 'frattini_label': '64.50', 'frattini_quotient': '72.48', 'hash': 1280522339947909588, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 12, 1], 'inner_gens': [[37442, 1337164657, 35945290], [1850291, 395398321, 35945290], [37442, 395398321, 35945290]], 'inner_hash': 6, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': False, 'inner_tex': 'D_{12}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 128, 'irrQ_dim': 128, 'irrR_degree': None, 'irrep_stats': [[1, 384], [2, 1056]], 'label': '4608.v', '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': 'D24.C96', 'ngens': 3, '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 88, 'number_characteristic_subgroups': 138, 'number_conjugacy_classes': 1440, 'number_divisions': 110, 'number_normal_subgroups': 202, 'number_subgroup_autclasses': 351, 'number_subgroup_classes': 495, 'number_subgroups': 1178, 'old_label': None, 'order': 4608, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 27], [3, 8], [4, 36], [6, 72], [8, 64], [12, 144], [16, 128], [24, 224], [32, 256], [48, 448], [64, 512], [96, 896], [192, 1792]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [16, 8, 16, 8, 4, 8], 'outer_gen_pows': [776418240, 7189058, 7189058, 776418240, 611069953, 7189058], 'outer_gens': [[4039104, 992089873, 366641958], [5622862, 1272463101, 589502756], [4851827, 1301219333, 381020074], [630917, 1272463101, 323507610], [2084593, 86268861, 1200572686], [7188864, 115025093, 1171816454]], 'outer_group': '512.10493039', 'outer_hash': 3046548263559564634, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 512, 'outer_perms': 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'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\times C_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 134, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 32, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [4, 19], [8, 19], [16, 17], [32, 13], [64, 8], [128, 8]], 'representations': {'PC': {'code': '329680630424262361155756325310135021435345469787175764357129', 'gens': [1, 2, 9], 'pres': [11, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 56, 90, 124, 158, 192, 226, 446696, 294, 443529, 328, 371722]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [37442, 395398321, 35945290]}, 'Perm': {'d': 134, 'gens': [15436099512715775457938543278326507815845689574710838830107688810017577763733977450011984164779589723803741620601164288034107911637534346484885657452980049119193810696159407465763620684673469402907159176922987348319091426457781, 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