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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '672.974', 'ambient_counter': 974, 'ambient_order': 672, 'ambient_tex': 'C_4.D_{84}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 56, 'counter': 23, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '672.974.6.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.c1.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': 6, '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': '112.27', 'subgroup_hash': 27, 'subgroup_order': 112, 'subgroup_tex': 'C_{14}:Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '672.974', 'aut_centralizer_order': 4, 'aut_label': '6.c1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '1344.11444', 'aut_weyl_index': 12, 'centralizer': '168.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1', '3.a1.a1'], 'contains': ['12.a1.a1', '12.d1.a1', '12.g1.a1', '12.h1.a1', '12.h1.b1', '42.c1.a1'], 'core': '12.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [8112, -1, 3961, -1, 5810, -1, 4971, -1], 'generators': [197, 168, 336, 2, 96], 'label': '672.974.6.c1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '6.c1.a1', 'projective_image': '336.196', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.c1.a1', 'subgroup_fusion': None, 'weyl_group': '56.5'}
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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': 84, 'aut_gen_orders': [6, 4, 2, 2, 28], 'aut_gens': [[1, 2, 56], [1, 106, 56], [85, 110, 84], [57, 2, 56], [1, 26, 56], [63, 2, 56]], 'aut_group': '2688.dd', 'aut_hash': 8872731775984099151, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2688, 'aut_permdeg': 15, 'aut_perms': [199308211658, 543272043452, 199348168320, 88745933372, 294154886209], 'aut_phi_ratio': 56.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 2, 1], [4, 14, 4, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 2, 6, 1], [28, 2, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2\\times F_7', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '84.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 84, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 14, 4], [7, 2, 3], [14, 2, 9], [28, 2, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '28.3', 'commutator_count': 1, 'commutator_label': '14.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 27, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['56.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 14, 1, 4], [7, 2, 3, 1], [14, 2, 3, 3], [28, 2, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 168, 'exponent': 28, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '56.12', 'hash': 27, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 14, 1], 'inner_gens': [[1, 54, 56], [5, 2, 56], [1, 2, 56]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 28, 'inner_split': False, 'inner_tex': 'D_{14}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 26]], 'label': '112.27', 'linC_count': 48, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 10, 'linQ_dim_count': 4, 'linR_count': 48, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C14:Q8', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 34, 'number_divisions': 16, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 38, 'number_subgroups': 104, 'old_label': None, 'order': 112, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 60], [7, 6], [14, 18], [28, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [0, 14, 0, 0], 'outer_gens': [[1, 2, 84], [15, 2, 56], [85, 2, 56], [1, 94, 84]], 'outer_group': '96.167', 'outer_hash': 167, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [1144200, 767640, 19441464, 4092484], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [6, 4], [12, 2]], 'representations': {'PC': {'code': 22697803233104895678, 'gens': [1, 2, 5], 'pres': [5, -2, -2, -2, -7, -2, 140, 541, 26, 782, 42, 963]}, 'GLZN': {'d': 2, 'p': 21, 'gens': [9325, 120406, 61776, 80557, 74096]}, 'Perm': {'d': 17, 'gens': [23906015522047, 40320, 45968909051520, 68436399225600, 973]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{14}:Q_8', 'transitive_degree': 112, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 6, 6, 12, 6, 12], 'aut_gens': [[1, 2, 4], [481, 2, 404], [369, 2, 292], [161, 338, 220], [321, 338, 212], [585, 338, 54], [593, 338, 150], [345, 338, 548]], 'aut_group': None, 'aut_hash': 7631554720823566362, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 92, 'aut_perms': [4423552128490991366342491448316008146872578157583845378339388771820121774793912396838270995483779350663411959752201331659894182650224054057665, 3550174424493415060380163219087310486737962774809299948162974112219884833721921220112100957133674780042304412874925536175616568407153952641961, 12142337508592083676911610384889449537674003437385204477927758403445782138466402524747828656836559109803058464582180539772445025732186783656009, 1928526626194222942109578246098614026815444288223793685078925354250551818951590635718421528125493843069451230170610787929598130320169803471453, 8147931604509215487667592176025170275133831603649953689762409977575964853577317264215476794801198786036556478673311710295751985905849453781824, 9841902057178323443281243730802131249169640812511198709650293692609262552646437804422833899531299371476647932192155910964841726553529550923130, 2084701219361354605062425335987590979682899343879259872876320065376995045709960292859431437566607983894505973686190469000801103886714291911986], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 84, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 84, 1, 1], [4, 84, 2, 1], [6, 2, 1, 1], [6, 4, 1, 1], [7, 2, 3, 1], [8, 4, 2, 1], [12, 2, 2, 1], [12, 4, 1, 1], [14, 2, 3, 1], [14, 4, 3, 1], [21, 2, 6, 1], [24, 4, 4, 1], [28, 2, 6, 1], [28, 4, 3, 1], [42, 2, 6, 1], [42, 4, 6, 1], [56, 4, 12, 1], [84, 2, 12, 1], [84, 4, 6, 1], [168, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': None, 'autcentquo_hash': 5451351369529179689, 'autcentquo_nilpotent': False, 'autcentquo_order': 2016, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3\\times S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 84, 1], [3, 2, 1], [4, 2, 2], [4, 84, 3], [6, 2, 1], [6, 4, 1], [7, 2, 3], [8, 4, 2], [12, 2, 2], [12, 4, 1], [14, 2, 3], [14, 4, 3], [21, 2, 6], [24, 4, 4], [28, 2, 6], [28, 4, 3], [42, 2, 6], [42, 4, 6], [56, 4, 12], [84, 2, 12], [84, 4, 6], [168, 4, 24]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '336.196', 'commutator_count': 1, 'commutator_label': '84.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 974, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 84, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 84, 1, 3], [6, 2, 1, 1], [6, 4, 1, 1], [7, 2, 3, 1], [8, 4, 1, 2], [12, 2, 2, 1], [12, 4, 1, 1], [14, 2, 3, 1], [14, 4, 3, 1], [21, 2, 6, 1], [24, 4, 2, 2], [28, 2, 6, 1], [28, 4, 3, 1], [42, 2, 6, 1], [42, 4, 6, 1], [56, 4, 6, 2], [84, 2, 12, 1], [84, 4, 6, 1], [168, 4, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5376, 'exponent': 168, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[4, -1, 12]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '168.56', 'hash': 974, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 84], 'inner_gens': [[1, 2, 332], [1, 2, 340], [345, 338, 4]], 'inner_hash': 196, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': True, 'inner_tex': 'C_2\\times D_{84}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 48, 'irrQ_dim': 96, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 82], [4, 21]], 'label': '672.974', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 16, 'linQ_dim': 16, 'linQ_dim_count': 16, 'linR_count': 12, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4.D84', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 27, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 111, 'number_divisions': 32, 'number_normal_subgroups': 47, 'number_subgroup_autclasses': 100, 'number_subgroup_classes': 120, 'number_subgroups': 1036, 'old_label': None, 'order': 672, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 87], [3, 2], [4, 256], [6, 6], [7, 6], [8, 8], [12, 8], [14, 18], [21, 12], [24, 16], [28, 24], [42, 36], [56, 48], [84, 48], [168, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 2], 'outer_gens': [[1, 2, 452], [1, 338, 4], [1, 2, 620], [1, 2, 102]], 'outer_group': '48.52', 'outer_hash': 52, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [744, 3628800, 24, 40323], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 3], [6, 4], [8, 1], [12, 6], [24, 3], [48, 1]], 'representations': {'PC': {'code': 21879011117004926129730341233241950296347915, 'gens': [1, 2, 3], 'pres': [7, -2, -2, -2, -2, -2, -3, -7, 2360, 6974, 3579, 58, 18595, 80, 22964, 102, 26885, 166, 28230]}, 'Perm': {'d': 26, 'gens': [621574835720048964849643, 3079988947639, 4491426195058, 5670545596524, 7177921676593, 6758061133824000, 16754357281155028254720000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4.D_{84}', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
