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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '6912.he', 'ambient_counter': 187, 'ambient_order': 6912, 'ambient_tex': 'S_3^2:C_2\\wr A_4', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 2, 'counter': 1386, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6912.he.432.bt1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '432.bt1', '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': 432, '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': '16.11', 'subgroup_hash': 11, 'subgroup_order': 16, 'subgroup_tex': 'C_2\\times D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6912.he', 'aut_centralizer_order': None, 'aut_label': '432.bt1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '144.ca1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['144.de1', '144.df1', '216.f1', '216.h1', '216.bh1', '216.bz1', '216.cb1', '216.ce1', '216.ct1'], 'contains': ['864.k1', '864.n1', '864.bf1', '864.bg1', '864.bm1'], 'core': '3456.a1', 'coset_action_label': None, 'count': 36, 'diagramx': None, 'generators': [3776, 32140666579, 28003, 40319], 'label': '6912.he.432.bt1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '6.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.b1', 'old_label': '432.bt1', 'projective_image': '3456.dg', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '432.bt1', 'subgroup_fusion': None, 'weyl_group': '8.5'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2], 'aut_gens': [[126, 55, 289, 288], [127, 55, 289, 288], [126, 265, 1, 288], [54, 127, 289, 288], [414, 55, 289, 288], [127, 54, 289, 288], [414, 265, 289, 288]], 'aut_group': '64.138', 'aut_hash': 138, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 8, 'aut_perms': [2309, 526, 5329, 3043, 12316, 18498], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [4, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2', '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': 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, 3], [2, 2, 4], [4, 2, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, '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', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [4, 2, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 21, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 1], 'inner_gens': [[126, 265, 289, 288], [414, 55, 289, 288], [126, 55, 289, 288], [126, 55, 289, 288]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.11', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 8, 'linQ_dim': 3, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D4', 'ngens': 4, 'nilpotency_class': 2, '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': 10, 'number_divisions': 10, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 27, 'number_subgroups': 35, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 11], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 151, 0], 'outer_gens': [[54, 127, 289, 288], [127, 55, 289, 288], [415, 54, 1, 288], [127, 54, 289, 288]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2]], 'representations': {'PC': {'code': 8772, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 78, 34]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16322, 16432, 3198]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8912, 8156, 13286, 14044]}, 'Perm': {'d': 6, 'gens': [126, 55, 289, 288]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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\\times D_4', 'transitive_degree': 8, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 12, 12, 12, 12, 12, 6], 'aut_gens': [[4142, 18498, 28003, 25434265614, 414, 40319, 32140666579, 14493436958, 997930107, 3254, 26824105451], [36543, 28783, 35152, 87122939, 36177, 40319, 6267327917, 14493444337, 997930107, 16519, 26824097018], [3776, 28783, 23616, 87098136, 414, 40319, 6267328577, 14493436958, 997935493, 2676, 26824115901], [39905, 35152, 11536, 13412051736, 4142, 40319, 6267331041, 14493445420, 83482993, 33905, 13499138210], [39905, 35152, 28783, 2043384858, 4142, 40319, 19679375841, 26824111699, 7645306, 32341, 14493429198], [36543, 12316, 18498, 12940693916, 414, 40319, 33178508717, 13499145680, 1959947245, 10939, 26352710690], [414, 35152, 23616, 87105218, 3776, 40319, 6354422319, 13499148882, 25430651959, 3254, 14493460583], [3776, 11536, 35152, 1037875671, 36177, 40319, 33178494162, 26824106737, 1959915852, 32741, 14493443051]], 'aut_group': None, 'aut_hash': 6968525696234854860, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 55296, 'aut_permdeg': 92, 'aut_perms': [2892893949670740558252283757184440433863186963539856591817931077952586064887216903955442692907114771908457623168217046635571230648944148403254, 10408926593159511997865911384248771826147911965307093444563382626606555913109491539480480306297863027659010245356767134726142973000721836084934, 11713651913443785834406801916507921965581277132315176167332694684926658414237065765483664425766156212666717970802958369527628797016500890140951, 10717912946086468704929524397292609874123514352915340281088995489710877347549831932958923369924868629522754527484613797947614798265067978193933, 7315411579469089107487119062884021434378507923244479767996055921274636956461971904936189020416700287903071256666713898519956146748417351435772, 7261004707994402111329921384005082490822007248235473376376391657284030350605660200942484645992231708223984176046162381290233341929721543228921, 7043504835232474527186678311219836909650222864030059076593888423552668832014312642169630755830890632384473941254266606542983182157480288256782], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 6, 2, 1], [2, 9, 1, 2], [2, 12, 1, 1], [2, 24, 2, 1], [2, 36, 1, 1], [2, 36, 2, 1], [2, 54, 1, 1], [2, 108, 1, 1], [3, 4, 1, 2], [3, 16, 2, 1], [3, 64, 2, 2], [4, 12, 1, 1], [4, 36, 2, 1], [4, 72, 2, 1], [4, 108, 1, 1], [4, 144, 1, 1], [4, 432, 1, 1], [6, 4, 1, 2], [6, 12, 2, 1], [6, 16, 2, 1], [6, 24, 1, 2], [6, 24, 2, 1], [6, 48, 1, 1], [6, 48, 2, 1], [6, 64, 2, 2], [6, 72, 1, 1], [6, 72, 2, 1], [6, 96, 4, 2], [6, 144, 2, 2], [6, 192, 4, 2], [12, 24, 2, 1], [12, 48, 1, 1], [12, 72, 2, 1], [12, 288, 1, 1], [12, 288, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.(C_2\\times A_4).C_2^6', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '13824.fp', 'autcentquo_hash': 4107807662957182902, 'autcentquo_nilpotent': False, 'autcentquo_order': 13824, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_6^2.(C_2^2\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 3], [2, 9, 2], [2, 12, 1], [2, 24, 2], [2, 36, 3], [2, 54, 1], [2, 108, 1], [3, 4, 2], [3, 16, 2], [3, 64, 4], [4, 12, 1], [4, 36, 2], [4, 72, 2], [4, 108, 1], [4, 144, 1], [4, 432, 1], [6, 4, 2], [6, 12, 2], [6, 16, 2], [6, 24, 4], [6, 48, 3], [6, 64, 4], [6, 72, 3], [6, 96, 8], [6, 144, 4], [6, 192, 8], [12, 24, 2], [12, 48, 1], [12, 72, 2], [12, 288, 5]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '3456.dg', 'commutator_count': 1, 'commutator_label': '576.8639', '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', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 187, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 3], [2, 9, 1, 2], [2, 12, 1, 1], [2, 24, 1, 2], [2, 36, 1, 3], [2, 54, 1, 1], [2, 108, 1, 1], [3, 4, 1, 2], [3, 16, 2, 1], [3, 64, 2, 2], [4, 12, 1, 1], [4, 36, 1, 2], [4, 72, 1, 2], [4, 108, 1, 1], [4, 144, 1, 1], [4, 432, 1, 1], [6, 4, 1, 2], [6, 12, 1, 2], [6, 16, 2, 1], [6, 24, 1, 4], [6, 48, 1, 3], [6, 64, 2, 2], [6, 72, 1, 3], [6, 96, 2, 4], [6, 144, 2, 2], [6, 192, 2, 4], [12, 24, 1, 2], [12, 48, 1, 1], [12, 72, 1, 2], [12, 288, 1, 1], [12, 288, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 192, 'exponent': 12, 'exponents_of_order': [8, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 0, 8], [16, 1, 4]], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '864.4671', 'hash': 8860234625506843947, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 2, 6, 2, 1, 2, 6, 6, 3, 3], 'inner_gens': [[4142, 18498, 12316, 25434290939, 414, 40319, 32140666579, 14493457761, 997949728, 3194, 26824106119], [4142, 18498, 28003, 25434296843, 39905, 40319, 32140662700, 14493457761, 997957185, 25741, 26824122983], [36177, 18498, 28003, 25434288580, 39905, 40319, 32140662700, 14493436958, 997943842, 6620, 26824104662], [36543, 12316, 5329, 25434265614, 36177, 40319, 32140656702, 26352730579, 1959944608, 20933, 13499169383], [4142, 21821, 12316, 25434267178, 414, 40319, 32140662700, 14493436958, 997945765, 1032, 26824111117], [4142, 18498, 28003, 25434265614, 414, 40319, 32140666579, 14493436958, 997930107, 3254, 26824105451], [4142, 21821, 12316, 25434298071, 39905, 40319, 32140666579, 26352739041, 83501807, 10083, 26824122341], [36177, 21821, 28003, 13412064764, 414, 40319, 6267318700, 14493436958, 1959948501, 21517, 26824090632], [39905, 35152, 23616, 26911188778, 3776, 40319, 33178490960, 13499163437, 997930107, 32741, 26352709512], [414, 5329, 18498, 25434288382, 3776, 40319, 32140675041, 14493455297, 997944426, 3254, 26824097018], [39905, 34990, 21821, 1037857333, 36543, 40319, 32140675041, 14493455297, 1959921738, 29704, 26824105451]], 'inner_hash': 5688590992281673169, 'inner_nilpotent': False, 'inner_order': 3456, 'inner_split': True, 'inner_tex': 'D_6^2:(C_2\\times A_4)', 'inner_used': [1, 2, 3, 4, 7, 8, 9], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 12], [2, 3], [3, 4], [4, 24], [6, 9], [8, 3], [12, 12], [16, 12], [24, 2]], 'label': '6912.he', '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': 'S3^2:C2wrA4', 'ngens': 11, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 47, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 81, 'number_divisions': 63, 'number_normal_subgroups': 31, 'number_subgroup_autclasses': 1804, 'number_subgroup_classes': 3280, 'number_subgroups': 93214, 'old_label': None, 'order': 6912, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 367], [3, 296], [4, 912], [6, 3656], [12, 1680]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[4142, 18498, 28003, 25434265614, 414, 40319, 32140666579, 26352718238, 1959945092, 3254, 13499151851], [4142, 18498, 28003, 25434265614, 414, 40319, 32140662700, 26352718238, 1959924987, 3254, 13499151851], [4142, 18498, 28003, 25434265614, 414, 40319, 19679367379, 13499145758, 1959924987, 3254, 26352724331], [3776, 28003, 18498, 25434266778, 414, 40319, 32140657362, 14493436880, 12933063793, 1998, 13499162259]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [3, 4], [4, 9], [6, 9], [8, 9], [12, 12], [16, 5], [24, 2], [32, 4]], 'representations': {'PC': {'code': '3078616266128767002338408131127885413627421605058455737441233561878503117309305094465708998128717065271342018666690766945385847443865864799517609396754', 'gens': [1, 3, 4, 5, 8, 11], 'pres': [11, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 22, 103160, 85648, 272451, 98486, 80854, 147525, 16966, 16647, 158, 427685, 120400, 2667, 9806, 192, 177414, 2492, 81319, 57570, 46493, 18003, 3758, 260, 180584, 47550, 11932, 294, 21129, 42271, 10613, 17434, 322365, 52315, 26190]}, 'Perm': {'d': 14, 'gens': [4142, 18498, 28003, 25434265614, 414, 40319, 32140666579, 14493436958, 997930107, 3254, 26824105451]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3^2:C_2\\wr A_4', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
