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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1536.201107024', 'ambient_counter': 201107024, 'ambient_order': 1536, 'ambient_tex': 'S_3\\times C_2^5:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 192, 'counter': 50, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1536.201107024.4.BK', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '4.bk1', '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': 4, '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': '384.9328', 'subgroup_hash': 9328, 'subgroup_order': 384, 'subgroup_tex': 'S_3\\times C_2^3:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1536.201107024', 'aut_centralizer_order': None, 'aut_label': '4.BK', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '192.A', 'complements': None, 'conjugacy_class_count': 6, 'contained_in': ['2.A'], 'contains': ['8.E', '8.CE', '8.CG', '8.CI', '8.CJ', '8.CK', '8.CL', '8.CN', '8.CQ', '8.CX', '8.CZ', '8.DG', '8.DH', '8.DM', '8.DN', '12.N'], 'core': '8.E', 'coset_action_label': None, 'count': 12, 'diagramx': None, 'generators': [1, 24, 1152, 6, 192, 4, 768, 108], 'label': '1536.201107024.4.BK', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.A', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.A', 'old_label': '4.bk1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.BK', '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': 12, 'aut_gen_orders': [12, 6, 6, 6, 2, 12], 'aut_gens': [[13903, 69133, 69557, 266185, 97255], [155813, 69613, 24427, 69557, 180205], [152141, 235297, 24427, 245471, 262867], [318329, 318247, 266473, 79859, 263059], [69347, 152083, 190303, 321929, 262867], [14191, 69229, 152219, 266473, 96871], [79787, 318055, 24715, 235169, 180205]], 'aut_group': None, 'aut_hash': 2776734662927589858, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 147456, 'aut_permdeg': 48, 'aut_perms': [7263970475724668788450749568740431631923535270163496892886729, 4649014806612517493609007474753352709951819569043775879799612, 9799051000951023568212403395912459834463319216588472863649705, 3245078251436274858624688449958924809845053605727463152623732, 8584767042967323047907588219582038038246523002357099612142028, 2190974095308718327480657004066090051525831912395648259236644], 'aut_phi_ratio': 1152.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 3, 2], [2, 3, 8, 1], [2, 4, 6, 1], [2, 12, 6, 1], [3, 2, 1, 1], [4, 4, 2, 1], [4, 4, 6, 1], [4, 12, 2, 1], [4, 12, 6, 1], [6, 2, 1, 1], [6, 2, 3, 2], [6, 8, 6, 1], [12, 8, 2, 1], [12, 8, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^8.C_3.D_6.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4096.bri', 'autcent_hash': 3677823439335518020, 'autcent_nilpotent': True, 'autcent_order': 4096, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{12}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 3, 8], [2, 4, 6], [2, 12, 6], [3, 2, 1], [4, 4, 8], [4, 12, 8], [6, 2, 7], [6, 8, 6], [12, 8, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '48.51', 'commutator_count': 1, 'commutator_label': '24.15', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 9328, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['6.1', 1], ['64.73', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 3, 1, 8], [2, 4, 1, 6], [2, 12, 1, 6], [3, 2, 1, 1], [4, 4, 1, 6], [4, 4, 2, 1], [4, 12, 1, 6], [4, 12, 2, 1], [6, 2, 1, 7], [6, 8, 1, 6], [12, 8, 1, 6], [12, 8, 2, 1]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 43680, 'exponent': 12, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '48.51', 'hash': 9328, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 2, 2, 3], 'inner_gens': [[13903, 69133, 69269, 190303, 97255], [13903, 69133, 69557, 266185, 96871], [14191, 69133, 69557, 100597, 97255], [96985, 69133, 235457, 266185, 97255], [13903, 69517, 69557, 266185, 97255]], 'inner_hash': 51, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': False, 'inner_tex': 'C_2^2\\times D_6', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 36], [4, 14]], 'label': '384.9328', '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*C2^3:D4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 66, 'number_divisions': 63, 'number_normal_subgroups': 223, 'number_subgroup_autclasses': 216, 'number_subgroup_classes': 1812, 'number_subgroups': 6998, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 127], [3, 2], [4, 128], [6, 62], [12, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 6, 4, 2], 'outer_gen_pows': [13825, 13825, 13825, 13825], 'outer_gens': [[263173, 69133, 152507, 266185, 97255], [321713, 152083, 79859, 152507, 180205], [152141, 235297, 152507, 245471, 262867], [96985, 152371, 152219, 266473, 97255]], 'outer_group': '3072.bbo', 'outer_hash': 8324277783353456216, 'outer_nilpotent': False, 'outer_order': 3072, 'outer_permdeg': 18, 'outer_perms': [6267312023, 1068369959024648, 1108907864103589, 753233848029496], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^7.S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 32], [4, 14], [8, 1]], 'representations': {'PC': {'code': 108323344119670297593651950605313, 'gens': [1, 2, 3, 5, 7], 'pres': [8, -2, -2, -2, -2, -2, -2, -2, -3, 290, 66, 8324, 2420, 116, 8974, 166, 8207]}, 'GLZN': {'d': 2, 'p': 24, 'gens': [155665, 13903, 14017, 69269, 155741, 96775, 14113, 179725]}, 'Perm': {'d': 15, 'gens': [87657298129, 186810626309, 479004643, 375196, 12316, 18498, 87657292800, 3991680]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_2^3:D_4', 'transitive_degree': 96, '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': 4, 'aut_exponent': 12, 'aut_gen_orders': [6, 6, 6, 6, 4, 4, 2, 4, 6], 'aut_gens': [[1, 2, 12, 48, 96, 192, 768], [409, 98, 516, 1392, 864, 162, 120], [9, 1250, 876, 1050, 1272, 570, 504], [389, 122, 108, 984, 1152, 336, 864], [393, 98, 108, 432, 96, 336, 864], [389, 514, 132, 168, 96, 1026, 888], [409, 490, 492, 1032, 864, 210, 1176], [25, 2, 12, 1392, 1248, 576, 768], [29, 1162, 804, 618, 792, 72, 480], [413, 410, 420, 984, 768, 744, 1248]], 'aut_group': None, 'aut_hash': 5880632343620133947, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1179648, 'aut_permdeg': 76, 'aut_perms': [1204740986926411398435257670137427718907397229792369578292528047547528246845823045707513655082708182105114443652, 538696661063890829301974011494706408513894734519484672399960324362482865791646181562581480747296261666446939049, 1138922672864654588440075441533485818757407705294953252468196263255171288755177250063890214878269206399736734560, 857907530192965126370474103272472259216867120873348037639506141259187431269473677880688690561174165181006393835, 548765016905116519127851809289701114459537195571005071416435449278920422063979818947396878026131938668395078731, 338079443493836541625705124224997200327877893337798569520252749973987328111241911129515213084861710726023301551, 1358778471218876732992062868238327349540354426897914658635153553799454575410598837068488380768149043905070448629, 294904617639700828976131988397211387977627637630742109276784214277492500118321047670209321955225164202703822941, 1128006671983303088327714299536183530824783427511624309336518873515855059618133991100364260216867259457418900888], 'aut_phi_ratio': 2304.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 2, 6, 1], [2, 3, 4, 1], [2, 4, 3, 1], [2, 4, 12, 1], [2, 6, 2, 1], [2, 6, 6, 1], [2, 8, 2, 1], [2, 12, 3, 1], [2, 12, 12, 1], [2, 24, 2, 1], [3, 2, 1, 1], [4, 4, 12, 1], [4, 8, 2, 1], [4, 12, 12, 1], [4, 16, 6, 1], [4, 24, 2, 1], [4, 48, 6, 1], [6, 2, 1, 3], [6, 4, 2, 1], [6, 4, 6, 1], [6, 8, 3, 1], [6, 8, 12, 1], [6, 16, 2, 1], [12, 8, 12, 1], [12, 16, 2, 1], [12, 32, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times C_2^7.A_4.C_2^5\\times S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '1024.djt', 'autcent_hash': 1957374279625110839, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '1152.157864', 'autcentquo_hash': 8514248756003452164, 'autcentquo_nilpotent': False, 'autcentquo_order': 1152, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^5:S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 8], [2, 3, 4], [2, 4, 15], [2, 6, 8], [2, 8, 2], [2, 12, 15], [2, 24, 2], [3, 2, 1], [4, 4, 12], [4, 8, 2], [4, 12, 12], [4, 16, 6], [4, 24, 2], [4, 48, 6], [6, 2, 3], [6, 4, 8], [6, 8, 15], [6, 16, 2], [12, 8, 12], [12, 16, 2], [12, 32, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '384.18491', 'commutator_count': 1, 'commutator_label': '48.52', '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', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 201107024, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['256.25886', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 8], [2, 3, 1, 4], [2, 4, 1, 15], [2, 6, 1, 8], [2, 8, 1, 2], [2, 12, 1, 15], [2, 24, 1, 2], [3, 2, 1, 1], [4, 4, 1, 12], [4, 8, 1, 2], [4, 12, 1, 12], [4, 16, 1, 6], [4, 24, 1, 2], [4, 48, 1, 6], [6, 2, 1, 3], [6, 4, 1, 8], [6, 8, 1, 15], [6, 16, 1, 2], [12, 8, 1, 12], [12, 16, 1, 2], [12, 32, 1, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2133196800, 'exponent': 12, 'exponents_of_order': [9, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.14', 'frattini_quotient': '96.230', 'hash': 201107024, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 6, 2, 2, 2, 2, 2], 'inner_gens': [[1, 10, 12, 48, 96, 192, 768], [5, 2, 36, 144, 96, 960, 768], [1, 26, 12, 144, 96, 960, 768], [1, 98, 108, 48, 96, 192, 1152], [1, 2, 12, 48, 96, 576, 768], [1, 770, 780, 48, 480, 192, 768], [1, 2, 12, 432, 96, 192, 768]], 'inner_hash': 18491, 'inner_nilpotent': False, 'inner_order': 384, 'inner_split': None, 'inner_tex': 'C_2^5:D_6', 'inner_used': [1, 2, 3, 4, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 72], [4, 36], [8, 6], [16, 1]], 'label': '1536.201107024', '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*C2^5:D4', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 33, 'number_characteristic_subgroups': 43, 'number_conjugacy_classes': 147, 'number_divisions': 147, 'number_normal_subgroups': 1147, 'number_subgroup_autclasses': 1988, 'number_subgroup_classes': 39956, 'number_subgroups': 198282, 'old_label': None, 'order': 1536, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 383], [3, 2], [4, 640], [6, 190], [12, 320]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [6, 4, 4, 6, 2, 4, 6], 'outer_gen_pows': [0, 0, 0, 0, 0, 8, 0], 'outer_gens': [[29, 394, 396, 1032, 1248, 546, 120], [1, 1186, 1164, 1002, 792, 528, 480], [385, 1162, 780, 216, 768, 474, 504], [33, 410, 12, 1026, 1272, 528, 96], [25, 2, 12, 432, 96, 378, 1272], [389, 1274, 900, 984, 1152, 144, 480], [389, 874, 900, 354, 1272, 456, 480]], 'outer_group': '2592.od', 'outer_hash': 7149489729732426729, 'outer_nilpotent': False, 'outer_order': 3072, 'outer_permdeg': 256, 'outer_perms': 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408614575988484185012232517989940062466245244431324653328981332235190977947211148834480384855454586954959856745460073711557571488975459525267200325686101934659726487148945178779605742219467851466837606336640257674811674008315495722547447526042706052453172660816720173912911799720908134759393194711048409415720279760993159922879843131425874481883326156765043304523170975331374927305640060264128029740079710163961924519351696918693309579543585705360056499938143308776901419662775610312790088896586823116869681], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_3^2:D_6\\times S_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': None, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 72], [4, 36], [8, 6], [16, 1]], 'representations': {'PC': {'code': '6366885661907235155562152500026202126400407890380803713152', 'gens': [1, 2, 4, 6, 7, 8, 10], 'pres': [10, -2, -2, -3, -2, -2, -2, -2, -2, -2, 2, 201, 51, 242, 733, 113, 4335, 755, 535, 38417, 6437, 547, 237, 2459]}}, 'schur_multiplier': [2, 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': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_2^5:D_4', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
