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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '8192.wy', 'ambient_counter': 597, 'ambient_order': 8192, 'ambient_tex': 'C_2^{10}.D_4', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': True, 'core_order': 256, 'counter': 1525, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '8192.wy.32._.OH', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '32.OH', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '32.51', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': None, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 32, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '256.55643', 'subgroup_hash': None, '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': '8192.wy', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [945036115450381694, 494447461208680350, 3314457, 4863547740, 776196453793973340, 507231460158158244, 3408417444, 249695279010334470], 'label': '8192.wy.32._.OH', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '32.OH', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32._.OH', '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': '64.267', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [28, 8], 'aut_gens': [[152075, 317975, 317963, 262945, 152077, 235025, 302181], [262663, 14125, 69137, 235021, 152065, 14113, 127071], [69125, 262663, 235301, 96773, 13843, 235025, 210003]], 'aut_group': None, 'aut_hash': 8650529136949381026, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2113929216, 'aut_permdeg': 128, 'aut_perms': [220169487809954371984339352394953405340960121060815288061142886235517232454221398258938703368400363260616337901068588991702955276282126625582693112978097976811349658357780681874971155600989899466982149665882355372262, 279418557677782068272332483880183061623237599022907729218502211613646199544183792787424098527955364392380248083777977624525662549521902831639221982793264520664868704262669342446122561256748155206148845475953852267998], 'aut_phi_ratio': 16515072.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 1, 28, 1], [2, 2, 48, 1], [2, 4, 8, 1], [4, 4, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^{12}.C_2^6.C_3.C_2^4.\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': None, 'autcent_hash': 1217713355939891613, 'autcent_nilpotent': False, 'autcent_order': 352321536, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^9.C_2^6.C_2^6.\\PSL(2,7)', '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, 31], [2, 2, 48], [2, 4, 8], [4, 4, 24]], 'center_label': '32.51', 'center_order': 32, 'central_product': True, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', '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': 55643, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['32.27', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 31], [2, 2, 1, 48], [2, 4, 1, 8], [4, 4, 1, 24]], 'element_repr_type': 'GLZN', 'elementary': 2, 'eulerian_function': 39060, 'exponent': 4, 'exponents_of_order': [8], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '64.267', 'hash': 55643, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 1, 1, 2, 2, 1, 2], 'inner_gens': [[152075, 317975, 317963, 262945, 152077, 235025, 302181], [152075, 317975, 317963, 262945, 152077, 235025, 302181], [152075, 317975, 317963, 262945, 152077, 235025, 302181], [152075, 317975, 317963, 262945, 152077, 235025, 301893], [152075, 317975, 317963, 262945, 152077, 235025, 292869], [152075, 317975, 317963, 262945, 152077, 235025, 302181], [152075, 317975, 317963, 262657, 262949, 235025, 302181]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [4, 5, 7], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 64], [2, 48]], 'label': '256.55643', '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': 6, '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': 112, 'number_divisions': 112, 'number_normal_subgroups': 3951, 'number_subgroup_autclasses': 76, 'number_subgroup_classes': 21431, 'number_subgroups': 39727, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 159], [4, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [8, 4], 'outer_gen_pows': [13825, 13825], 'outer_gens': [[179725, 317975, 262951, 235315, 152081, 235025, 292665], [262675, 152087, 318251, 262967, 96773, 14113, 292947]], 'outer_group': None, 'outer_hash': 107269122420418331, 'outer_nilpotent': False, 'outer_order': 264241152, 'outer_permdeg': 56, 'outer_perms': [463941551914395841906068647998027240012038880263846902110004437143998550870, 363964828237844027049701066718063149096289864964181610126980930464133177850], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^9.C_2^6.S_3.C_2^3.\\PSL(2,7)', 'pc_rank': 7, 'perfect': False, 'permutation_degree': None, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 6, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 64], [2, 48]], 'representations': {'PC': {'code': 11267830729146432, 'gens': [1, 2, 3, 4, 5, 6, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 1374, 822, 166]}, 'GLZN': {'d': 2, 'p': 24, 'gens': [14113, 53235, 179725, 13831, 69125, 317957, 152075, 13829]}}, 'schur_multiplier': [2, 2, 2, 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, 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': 64, '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': False, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[3415701612, 1372384777867763196, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 1099504540658219364, 13160704529065404, 2227251], [776196452329421113, 2091831012752434770, 4836323119, 269008239319142400, 494447461210502713, 45237780972039621, 776196453778350109, 776196452359635934, 750167282679811650, 1827603218452219963, 737383278877670451, 13160703092503774], [288173604925795057, 558932552770641272, 513717263489289690, 1978453934, 26072418278542582, 507231458718732579, 249695277050419200, 26072418255206400, 1959915270, 1076860262189383486, 507231460166529618, 249695279012158395], [45237782413726395, 70932026830578274, 288173603484914979, 507231458728194734, 256097827041804579, 513717264937529562, 533259328411020184, 750167282672559043, 288173604945133534, 691924263117993248, 256097827063965538, 513717263496936018], [494447459789961134, 1759698473423541210, 494447462648555365, 776196449449344000, 776196453796197564, 45237780985141787, 507231460168029381, 507231458719819785, 513717263491147311, 1827603218931989101, 737383282275566967, 776196453795060546], [737383280855351644, 2104760860979487282, 763412540547355218, 494447457822796880, 269008240779402980, 288173603484914979, 494447461219924644, 507231457279948880, 507231460146913950, 1038550199122895648, 513717264937529562, 269008240780132444], [750167279803007078, 294703613783433976, 45237780994241782, 776196452341425711, 45237781006215658, 249695281886742319, 494447462646697603, 13160703115758058, 737383282273709323, 1038550197645772088, 26072418289418926, 737383283710633603]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 6442450944, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 1572864.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 1, 4, 1], [2, 2, 3, 2], [2, 2, 6, 1], [2, 2, 16, 1], [2, 4, 12, 1], [2, 4, 96, 1], [2, 8, 6, 1], [2, 8, 12, 1], [2, 8, 16, 1], [2, 8, 48, 1], [2, 16, 2, 1], [2, 16, 24, 1], [2, 32, 1, 1], [2, 64, 1, 1], [4, 8, 48, 1], [4, 16, 6, 1], [4, 16, 24, 1], [4, 16, 48, 1], [4, 32, 3, 1], [4, 32, 12, 2], [4, 64, 3, 1], [4, 64, 4, 1], [4, 128, 12, 1], [4, 256, 2, 1], [8, 256, 6, 1]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 28], [2, 4, 108], [2, 8, 82], [2, 16, 26], [2, 32, 1], [2, 64, 1], [4, 8, 48], [4, 16, 78], [4, 32, 27], [4, 64, 7], [4, 128, 12], [4, 256, 2], [8, 256, 6]], 'center_label': '8.5', 'center_order': 8, 'central_product': None, 'central_quotient': '1024.dia', 'commutator_count': 1, 'commutator_label': '256.55643', 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 13, 'conjugacy_classes_known': True, 'counter': 597, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': None, 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 28], [2, 4, 1, 108], [2, 8, 1, 82], [2, 16, 1, 26], [2, 32, 1, 1], [2, 64, 1, 1], [4, 8, 1, 48], [4, 16, 1, 78], [4, 32, 1, 27], [4, 64, 1, 7], [4, 128, 1, 12], [4, 256, 1, 2], [8, 256, 1, 6]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': None, 'exponent': 8, 'exponents_of_order': [13], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '256.55643', 'frattini_quotient': '32.51', 'hash': 583603832557243529, 'hyperelementary': 2, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2], 'inner_gens': [[3415701612, 2104760860978762236, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 1827603220889318550, 13160704529065404, 2227251], [494447457833715128, 1372384777867763196, 13160704529065404, 763412539119896678, 1134021, 494447461227208812, 256097827032728070, 26072421151989612, 249695279017955004, 647232745756754432, 256097825603020880, 2227251], [3415701612, 886867527468659076, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 2231996450286268230, 13160704529065404, 2227251], [3415701612, 2104760860978762236, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 1465504277843187564, 13160704529065404, 2227251], [3415701612, 1372384777867763196, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 1099504540660038090, 13160704529065404, 2227251], [3415701612, 1372384777867763196, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 2219149944505612950, 13160704529065404, 2227251], [3415701612, 1372384777867763196, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 1865996713148574324, 13160704529065404, 2227251], [3415701612, 2104760860978762236, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 2231996448377493444, 13160704529065404, 2227251], [3415701612, 1372384777867763196, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 2206449459931795230, 13160704529065404, 2227251], [13160704521444870, 294703613781617294, 494447461227208812, 288173606372218356, 2227251, 13160704529065404, 513717260621273528, 249695279017955004, 26072421151989612, 1099504540658219364, 513717263477320350, 1134021], [3415701612, 886867527468659076, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 1491051264288750750, 13160704529065404, 2227251], [3415701612, 1372384777867763196, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 1099504540660038090, 13160704529065404, 2227251]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 1024, 'inner_split': None, 'inner_tex': 'C_2^7:D_4', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 32], [2, 120], [4, 216], [8, 66]], 'label': '8192.wy', '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': 'C2^10.D4', 'ngens': 12, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [1, 7, 35, 55, 119, 135, 239, 447, 771, 451, 275, 155, 31, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 29, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 434, 'number_divisions': 434, 'number_normal_subgroups': 2722, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 8192, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 1663], [4, 4992], [8, 1536]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': None, 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 6291456, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': None, 'perfect': False, 'permutation_degree': None, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': False, 'ratrep_stats': [[1, 32], [2, 120], [4, 216], [8, 66]], 'representations': {'PC': {'code': '29727309218498437082027581886779596488827248641415259081989795200570909623097363789108595848626321662267209454257825968463238486223287752683534', 'gens': [1, 2, 4, 6, 7, 8, 9, 10, 11, 12, 13], 'pres': [13, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 66301, 66, 84398, 10012, 1251, 62624, 13653, 146, 51497, 165989, 127938, 71791, 35924, 189299, 236295, 143124, 17505, 8782, 31832, 107661, 53386, 26723, 347369, 248582, 101955, 51008, 17483, 455544, 62437, 31250]}, 'Perm': {'d': 20, 'gens': [3415701612, 1372384777867763196, 256097825603020880, 269008243663771740, 1134021, 494447461227208812, 256097827032728070, 513717260621273528, 249695279017955004, 1099504540658219364, 13160704529065404, 2227251]}}, 'schur_multiplier': [2, 2, 2, 2, 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': 5, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 4, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^{10}.D_4', 'transitive_degree': None, 'wreath_data': None, 'wreath_product': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 26040, 'aut_gen_orders': [2, 5], 'aut_gens': [[1, 2, 4, 8, 16], [5, 2, 4, 12, 16], [11, 8, 27, 26, 13]], 'aut_group': '9999360.a', 'aut_hash': 2505043529366670169, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9999360, 'aut_permdeg': 31, 'aut_perms': [2450617681096434660778009755366, 945758899833610526331495647332850], 'aut_phi_ratio': 624960.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 31, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(5,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 26040, 'autcent_group': '9999360.a', 'autcent_hash': 2505043529366670169, 'autcent_nilpotent': False, 'autcent_order': 9999360, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(5,2)', '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, 31]], 'center_label': '32.51', 'center_order': 32, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 51, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 5]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 31]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3, 5, 7, 31], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '32.51', 'hash': 51, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8, 16], [1, 2, 4, 8, 16], [1, 2, 4, 8, 16], [1, 2, 4, 8, 16], [1, 2, 4, 8, 16]], '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, 32]], 'label': '32.51', 'linC_count': None, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, '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', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 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': 32, 'number_divisions': 32, 'number_normal_subgroups': 374, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 374, 'number_subgroups': 374, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 31]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 26040, 'outer_gen_orders': [2, 5], 'outer_gen_pows': [0, 0], 'outer_gens': [[5, 2, 4, 12, 16], [11, 8, 27, 26, 13]], 'outer_group': '9999360.a', 'outer_hash': 2505043529366670169, 'outer_nilpotent': False, 'outer_order': 9999360, 'outer_permdeg': 31, 'outer_perms': [2450617681096434660778009755366, 945758899833610526331495647332850], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(5,2)', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4, 5], 'pres': [5, -2, 2, 2, 2, 2]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 705686952884, 706461793860, 706460730980, 706461792404]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [5226561, 7292993, 5128257, 20225089, 6210625]}, 'Perm': {'d': 10, 'gens': [362880, 5040, 120, 6, 1]}}, 'schur_multiplier': [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': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^5', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
