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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '512.22886', 'ambient_counter': 22886, 'ambient_order': 512, 'ambient_tex': 'C_2^6.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': True, 'core_order': 256, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '512.22886.2.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': ['C1'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '256.29598', 'subgroup_hash': 29598, 'subgroup_order': 256, 'subgroup_tex': 'C_2^5:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '512.22886', 'aut_centralizer_order': None, 'aut_label': '2.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '32.i1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1'], 'contains': ['4.a1', '4.b1', '4.c1', '4.e1', '4.f1', '4.i1', '4.k1', '4.l1', '4.o1'], 'core': '2.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3462, 6519, 3467, 6675], 'generators': [4, 256, 2, 128, 200], 'label': '512.22886.2.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1', 'normal_contained_in': ['1.a1'], 'normal_contains': ['4.a1', '4.b1', '4.c1', '4.e1', '4.f1'], 'normalizer': '1.a1', 'old_label': '2.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2.a1', '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 168, 'aut_gen_orders': [14, 12], 'aut_gens': [[273313, 321413, 14129, 245383, 262963, 180013, 245615], [321425, 76321, 97079, 107441, 96775, 97063, 76547], [245671, 17575, 262955, 241925, 179725, 180013, 107513]], 'aut_group': None, 'aut_hash': 1860475951476847956, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 88080384, 'aut_permdeg': 72, 'aut_perms': [15409986402238812430922535011996385437504830040922932305549222329780380379875288834644405215645825961312, 24353470394232406435777715449406777896245313427838781858353143510008125060617872819134937121647180478009], 'aut_phi_ratio': 688128.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [2, 1, 8, 1], [2, 2, 56, 1], [2, 8, 2, 1], [4, 8, 14, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^{15}.C_2^4.\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 8483965898770251041, 'autcent_nilpotent': True, 'autcent_order': 524288, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{13}.C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '168.42', 'autcentquo_hash': 42, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PSL(2,7)', 'cc_stats': [[1, 1, 1], [2, 1, 15], [2, 2, 56], [2, 8, 2], [4, 8, 14]], 'center_label': '16.14', 'center_order': 16, 'central_product': True, 'central_quotient': '16.14', 'commutator_count': 1, 'commutator_label': '8.5', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 29598, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['128.1578', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [2, 2, 1, 56], [2, 8, 1, 2], [4, 8, 1, 14]], 'element_repr_type': 'GLZN', 'elementary': 2, 'eulerian_function': 3720, 'exponent': 4, 'exponents_of_order': [8], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '32.51', 'hash': 29598, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 2, 1, 1, 2], 'inner_gens': [[273313, 321413, 14129, 245383, 262963, 180013, 321497], [273313, 321413, 14129, 245383, 262963, 180013, 155597], [273313, 321413, 14129, 245383, 262963, 180013, 245615], [273313, 321413, 14129, 245383, 262963, 180013, 155885], [273313, 321413, 14129, 245383, 262963, 180013, 245615], [273313, 321413, 14129, 245383, 262963, 180013, 245615], [183175, 79511, 14129, 155821, 262963, 180013, 245615]], 'inner_hash': 14, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': False, 'inner_tex': 'C_2^4', 'inner_used': [1, 2, 4, 7], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 56]], 'label': '256.29598', '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': 'C2^5:D4', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 88, 'number_divisions': 88, 'number_normal_subgroups': 943, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 16445, 'number_subgroups': 32623, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 143], [4, 112]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [14, 4], 'outer_gen_pows': [13825, 13825], 'outer_gens': [[262663, 183175, 96791, 324875, 179725, 96775, 107501], [321709, 13829, 14129, 159275, 180013, 262963, 97133]], 'outer_group': None, 'outer_hash': 1790878771140052493, 'outer_nilpotent': False, 'outer_order': 5505024, 'outer_permdeg': 58, 'outer_perms': [901780755758347057444186394308658225432002081719583360966169697751807757125009, 637177896385242399314446046793464066393938767567929912027180532402659639550408], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^9.C_2^6.\\PSL(2,7)', 'pc_rank': 7, 'perfect': False, 'permutation_degree': None, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 56]], 'representations': {'PC': {'code': 4613981833200406592, 'gens': [1, 2, 3, 4, 5, 6, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 12550, 5390, 590, 166]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [1083311105, 1087522049, 2346386634, 30819648, 27215168, 2211644610, 1087570945, 22086720]}, 'GLZN': {'d': 2, 'p': 24, 'gens': [14113, 76325, 179725, 321485, 79799, 13829, 14047, 158987]}}, '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': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^5:D_4', 'transitive_degree': 32, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 2, 2], 'aut_gens': [[1, 4, 16, 32, 64, 128, 256], [113, 12, 32, 16, 64, 176, 368], [7, 236, 32, 16, 104, 152, 416], [293, 148, 32, 16, 88, 128, 416], [57, 28, 16, 32, 112, 128, 288], [27, 36, 24, 40, 64, 184, 450], [169, 20, 16, 32, 112, 176, 288], [161, 212, 32, 16, 64, 136, 320], [41, 60, 16, 32, 120, 184, 464], [195, 244, 16, 32, 72, 136, 496], [1, 4, 16, 32, 64, 128, 304], [9, 12, 16, 32, 64, 128, 304], [209, 4, 16, 32, 72, 136, 448], [225, 12, 16, 32, 72, 136, 448]], 'aut_group': '262144.m', 'aut_hash': 2792677175657778946, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 262144, 'aut_permdeg': 256, 'aut_perms': 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698045563426498057908561056654781644445053585060059053509291041555823572847550968289320388082472907105858486495201166975704671462671386177867078786218996613888484441696058684171840703259376539832428360031499201391428737225539106598193052238962291800902630875840775974877760094371579690600688684644596423263112588716458531381206398419990800340261689758042664951120323089736212609818100225554320715115459782414252539839964633458689939175772765920978944017131615676456871856281155575148492761768061949178044285], 'aut_phi_ratio': 1024.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 2, 4, 2], [2, 2, 8, 1], [2, 4, 2, 1], [2, 4, 4, 1], [2, 4, 16, 1], [2, 16, 1, 1], [4, 16, 1, 1], [4, 16, 2, 1], [4, 16, 4, 1], [4, 16, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^8.C_2^2\\wr C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '4096.brl', 'autcentquo_hash': 8293102879189356666, 'autcentquo_nilpotent': True, 'autcentquo_order': 4096, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^6.D_4^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 18], [2, 4, 22], [2, 16, 1], [4, 16, 23]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '128.170', 'commutator_count': 1, 'commutator_label': '32.51', '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'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 22886, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 18], [2, 4, 1, 22], [2, 16, 1, 1], [4, 16, 1, 7], [4, 16, 2, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 168, 'exponent': 4, 'exponents_of_order': [9], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '64.267', 'frattini_quotient': '8.5', 'hash': 22886, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 2, 2, 2, 2, 2, 2], 'inner_gens': [[1, 204, 32, 16, 112, 184, 344], [201, 4, 16, 32, 112, 176, 272], [49, 4, 16, 32, 64, 128, 256], [49, 4, 16, 32, 64, 128, 256], [49, 52, 16, 32, 64, 128, 256], [57, 52, 16, 32, 64, 128, 256], [89, 20, 16, 32, 64, 128, 256]], 'inner_hash': 170, 'inner_nilpotent': True, 'inner_order': 128, 'inner_split': False, 'inner_tex': 'C_2^4.D_4', 'inner_used': [1, 2, 3, 5, 7], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 28], [4, 24]], 'label': '512.22886', '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': 'C2^6.D4', 'ngens': 3, 'nilpotency_class': 3, 'nilpotent': True, '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': 15, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 68, 'number_divisions': 60, 'number_normal_subgroups': 166, 'number_subgroup_autclasses': 481, 'number_subgroup_classes': 9165, 'number_subgroups': 35450, 'old_label': None, 'order': 512, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 143], [4, 368]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 8, 'outer_gen_orders': [4, 4, 4, 4], 'outer_gen_pows': [0, 0, 0, 128], 'outer_gens': [[495, 84, 32, 16, 88, 128, 408], [339, 92, 40, 24, 72, 144, 338], [35, 52, 16, 32, 72, 128, 496], [325, 92, 24, 40, 96, 128, 274]], 'outer_group': '2048.cpg', 'outer_hash': 7841393102724273575, 'outer_nilpotent': True, 'outer_order': 2048, 'outer_permdeg': 20, 'outer_perms': [1717349696653822560, 123876524101122017, 1262860985941244047, 999339249144838336], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5.D_4^2', 'pc_rank': None, 'perfect': False, 'permutation_degree': None, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 24], [4, 28]], 'representations': {'PC': {'code': '178929658817847545385261450645281225666853381', 'gens': [1, 3, 5, 6, 7, 8, 9], 'pres': [9, 2, 2, 2, 2, 2, 2, 2, 2, 2, 18, 5510, 173, 74, 1444, 869, 7062, 1788, 13255, 3193, 27872, 5534]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^6.D_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
