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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1088.1448', 'ambient_counter': 1448, 'ambient_order': 1088, 'ambient_tex': 'D_{68}.C_8', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 8, 'counter': 34, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1088.1448.68.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '68.b1.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': 68, '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': '16.13', 'subgroup_hash': 13, 'subgroup_order': 16, 'subgroup_tex': 'D_4:C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1088.1448', 'aut_centralizer_order': 64, 'aut_label': '68.b1', 'aut_quo_index': None, 'aut_stab_index': 17, 'aut_weyl_group': '8.3', 'aut_weyl_index': 1088, 'centralizer': '136.f1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '34.a1.a1'], 'contains': ['136.a1.a1', '136.b1.a1', '136.c1.a1', '136.d1.a1', '136.h1.a1'], 'core': '136.a1.a1', 'coset_action_label': None, 'count': 17, 'diagramx': [3888, -1, 3891, -1, 3133, -1, 3123, -1], 'generators': [1, 8, 544], 'label': '1088.1448.68.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '17.a1.a1', 'old_label': '68.b1.a1', 'projective_image': '272.35', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '68.b1.a1', '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': '8.5', '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, 3, 2, 2], 'aut_gens': [[30, 377, 501], [120, 105, 65], [120, 377, 129], [65, 90, 120], [120, 253, 129], [120, 377, 501]], 'aut_group': '48.48', 'aut_hash': 48, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 6, 'aut_perms': [415, 450, 233, 403, 444], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [4, 1, 2, 1], [4, 2, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [4, 1, 2], [4, 2, 3]], 'center_label': '4.1', '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': 13, '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, 3], [4, 1, 2, 1], [4, 2, 1, 3]], 'element_repr_type': 'GLFp', 'elementary': 2, 'eulerian_function': 28, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 13, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[30, 253, 129], [120, 377, 501], [120, 377, 501]], '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': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.13', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4:C2', 'ngens': 3, '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': 9, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 20, 'number_subgroups': 23, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [377, 126], 'outer_gens': [[65, 377, 501], [65, 90, 30]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 1]], 'representations': {'PC': {'code': 8716, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 81, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [25793872, 22766861, 817574]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [30, 377, 501]}, 'Perm': {'d': 8, 'gens': [35278, 23616, 16582, 5167]}}, 'schur_multiplier': [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': 'D_4:C_2', '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': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 272, 'aut_gen_orders': [16, 8, 8, 16, 16], 'aut_gens': [[1, 2, 32], [1025, 114, 112], [449, 426, 624], [761, 282, 288], [273, 706, 864], [617, 1002, 944]], 'aut_group': None, 'aut_hash': 7933090346525517932, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 8704, 'aut_permdeg': 140, 'aut_perms': [3389336719308742368449396727943864306797710364322266170831931718580331272325366656822117322817095840487664179857968545152359824196989089811213821139591345403962679725412270628328445559415715910669539025493713385701771572090241254032829098062, 7262913310381324965050094246534523947982092852383469136251384321993294924354483627831732514942999793192463922918441447976636113785462032210336038649084998259619548786840626006166764329350724604788506321231412969009719449480258882828315723975, 775453524243633984262627305512315706246449447090469639836141763039508697461274036650353245292495522622468036961268360013444743924753722884235801925114128970449931084981548256669670179709456586154362423351617003339288406218849618965814427033, 12591628703725247665824464026982971685547051311210607477619563328762962586915726088633923699301771774767073427030518134104917473508513081464646196682597333542716957636195170501438071709264485573387657752408836633614693651155337710820313149376, 9587860147212261028614124878417722736342070392476028332198058637148211698291246572193591263408262091302109058481240297359612650930653820256639170961238688121608953736056443965912656508197308569472142816778185563214139094014991352452606077895], 'aut_phi_ratio': 17.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 68, 1, 1], [4, 1, 1, 2], [4, 2, 1, 1], [4, 68, 1, 1], [8, 4, 1, 2], [8, 17, 2, 2], [8, 34, 1, 2], [16, 34, 4, 2], [16, 68, 2, 2], [17, 4, 4, 1], [34, 4, 4, 1], [34, 8, 4, 1], [68, 4, 4, 2], [68, 8, 4, 1], [136, 8, 8, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_{17}:((C_2^2\\times C_8).C_2^4)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 272, 'autcentquo_group': '1088.1681', 'autcentquo_hash': 1681, 'autcentquo_nilpotent': False, 'autcentquo_order': 1088, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_{17}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 68, 1], [4, 1, 2], [4, 2, 1], [4, 68, 1], [8, 4, 2], [8, 17, 4], [8, 34, 2], [16, 34, 8], [16, 68, 4], [17, 4, 4], [34, 4, 4], [34, 8, 4], [68, 4, 8], [68, 8, 4], [136, 8, 16]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '272.35', 'commutator_count': 1, 'commutator_label': '68.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '17.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1448, '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, 68, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 68, 1, 1], [8, 4, 2, 1], [8, 17, 4, 1], [8, 34, 2, 1], [16, 34, 8, 1], [16, 68, 4, 1], [17, 4, 4, 1], [34, 4, 4, 1], [34, 8, 4, 1], [68, 4, 8, 1], [68, 8, 4, 1], [136, 8, 16, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 272, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 17], 'factors_of_order': [2, 17], 'faithful_reps': [[8, 0, 8]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '136.13', 'hash': 1448, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 68, 'inner_gen_orders': [2, 4, 34], 'inner_gens': [[1, 554, 1072], [569, 2, 672], [81, 450, 32]], 'inner_hash': 35, 'inner_nilpotent': False, 'inner_order': 272, 'inner_split': False, 'inner_tex': 'D_{34}:C_4', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 64, 'irrQ_dim': 64, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 12], [4, 32], [8, 8]], 'label': '1088.1448', 'linC_count': 256, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 32, 'linQ_degree_count': 2, 'linQ_dim': 32, 'linQ_dim_count': 2, 'linR_count': 64, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D68.C8', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 26, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 68, 'number_divisions': 18, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 56, 'number_subgroups': 562, 'old_label': None, 'order': 1088, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 71], [4, 72], [8, 144], [16, 544], [17, 16], [34, 48], [68, 64], [136, 128]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 8, 'outer_gen_orders': [4, 8], 'outer_gen_pows': [0, 572], 'outer_gens': [[1, 10, 1056], [1, 2, 928]], 'outer_group': '32.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 12, 'outer_perms': [87454080, 11825], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times C_8', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 49, 'pgroup': 0, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 3], [16, 3], [32, 3], [64, 1]], 'representations': {'PC': {'code': 1367995010600409478791775774400256928534394669367311, 'gens': [1, 2, 6], 'pres': [7, -2, -2, -2, -2, -2, 2, -17, 7757, 36, 422, 58, 80, 45029, 14124, 11107, 124, 50182, 6285, 12564]}, 'Perm': {'d': 49, 'gens': [264373090022696628905250554169108062063033696131149104165665, 533764986970353828095597167551886597221554139244221059053304, 26624211411226032876372090432265342, 533764986970353828095597184634368437798047069155472535024099, 43598893372724126282419778531685832, 52106203312720560795695319227113492, 13211809404964592863671159899388721000670316742091735040000000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{68}.C_8', 'transitive_degree': 544, 'wreath_data': None, 'wreath_product': False}
