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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '19360.h', 'ambient_counter': 8, 'ambient_order': 19360, 'ambient_tex': 'C_{11}^2:(C_{10}\\times \\SD_{16})', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 242, 'counter': 83, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '19360.h.80.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '80.b1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '80.26', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 26, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 80, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5\\times \\SD_{16}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '242.4', 'subgroup_hash': 4, 'subgroup_order': 242, 'subgroup_tex': 'C_{11}:D_{11}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '19360.h', 'aut_centralizer_order': None, 'aut_label': '80.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '4840.a1.a1', 'complements': ['242.a1.b1', '242.a1.c1', '242.a1.a1', '242.a1.d1'], 'conjugacy_class_count': 1, 'contained_in': ['16.b1.b1', '40.a1.a1', '40.f1.b1'], 'contains': ['160.a1.a1', '880.g1.b1', '880.h1.b1', '880.i1.b1', '880.j1.b1'], 'core': '80.b1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [2000, 9363, 3474, 8518, 1807, 2807, 2441, 2603], 'generators': [4900, 1760, 8840], 'label': '19360.h.80.b1.b1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '80.b1.b1', 'normal_contained_in': ['16.b1.b1', '40.a1.a1'], 'normal_contains': ['160.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '80.b1.b1', 'projective_image': '19360.h', 'quotient_action_image': '40.10', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '80.b1.b1', 'subgroup_fusion': None, 'weyl_group': '4840.bt'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 1320, 'aut_gen_orders': [10, 3, 11], 'aut_gens': [[1, 2, 22], [75, 160, 12], [227, 126, 50], [117, 2, 22]], 'aut_group': '1597200.a', 'aut_hash': 6850908737282390520, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1597200, 'aut_permdeg': 121, 'aut_perms': [232524638634348476879640176713366780306142189790946628069270500874096531702540479557886513003054976483605317832960215362522825885711271804428799099152966581521199281541354164942085563464276900121104158, 561727268790064950291169461113923072571110178108121132054578075164262631327591686249445969298846644170970080224784116182098947415790270083843598886948617580223196725359384345720312473599696348594737264, 399780809018317232189649253538497917839684670390469125760693573962612287607315854961404166904367279524223472617204836814054235059502346760120040219833958349611474472624338221470091407276635504947264257], 'aut_phi_ratio': 14520.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 121, 1, 1], [11, 2, 60, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{11}^2.\\GL(2,11)', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1320, 'autcentquo_group': '1597200.a', 'autcentquo_hash': 6850908737282390520, 'autcentquo_nilpotent': False, 'autcentquo_order': 1597200, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{11}^2.\\GL(2,11)', 'cc_stats': [[1, 1, 1], [2, 121, 1], [11, 2, 60]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '242.4', 'commutator_count': 1, 'commutator_label': '121.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '11.1', '11.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 121, 1, 1], [11, 2, 5, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 7, 'exponent': 22, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '242.4', 'hash': 4, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [2, 11, 11], 'inner_gens': [[1, 20, 220], [5, 2, 22], [45, 2, 22]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 242, 'inner_split': False, 'inner_tex': 'C_{11}:D_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 60]], 'label': '242.4', 'linC_count': 1650, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 66, 'linQ_dim': 20, 'linQ_dim_count': 66, 'linR_count': 1650, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C11:D11', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 62, 'number_divisions': 14, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 28, 'number_subgroups': 268, 'old_label': None, 'order': 242, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 121], [11, 120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 660, 'outer_gen_orders': [10, 5], 'outer_gen_pows': [0, 1], 'outer_gens': [[1, 224, 198], [1, 134, 196]], 'outer_group': None, 'outer_hash': 1406648489472108391, 'outer_nilpotent': False, 'outer_order': 6600, 'outer_permdeg': 17, 'outer_perms': [88405066235553, 231585030120336], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_5.\\PSL(2,11).C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [10, 12]], 'representations': {'PC': {'code': 72876099, 'gens': [1, 2, 3], 'pres': [3, -2, -11, -11, 121, 1982]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [2143604792, 295918632, 418103038]}, 'Perm': {'d': 22, 'gens': [2439325392330397756, 55969592955894377761, 102425888546904322144]}}, 'schur_multiplier': [11], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}:D_{11}', 'transitive_degree': 121, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 220, 'aut_gen_orders': [20, 20, 20, 20, 10, 10], 'aut_gens': [[1, 10, 40, 440], [7441, 6970, 160, 16280], [8821, 11010, 5360, 13760], [4001, 14430, 10720, 12760], [14721, 11030, 2080, 1320], [1461, 3990, 120, 18440], [7361, 430, 16080, 2200]], 'aut_group': None, 'aut_hash': 6924539337684527549, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 154880, 'aut_permdeg': 297, 'aut_perms': [7981763939924546360764279891347510331416028051802311028847405748176021680524974958531783888904297415942642717507862049004320161971518192380606189829014126901419378151500691435705692825142189827977482899335345661026196356140681289702115466714016403813301679339142257995200908295976626190142752212711359289214332096802111076306118650912566705768746356007094827711799656106570118824436596745765730865798697708593311306919335153594543640844757055614989677436085366858493607917271772680313331884311406446131646366380326636633464951693243285444197902735317485948319399453823001895992766097361809504528490888641636, 7797365890094484635567463367269859991887348279547220125329318629411583540100741595692991023911141833845151056231714268299310125711668262631552991121631967976609011126022198090717474266640602644372342865375659358046153990934328856490775000909876154882609330544933236035392047605168536186463632634772952105063597859249474766122845428039087451000114601829336697053890456099492172549806130545991574857744662940736044726637720187202233799306409177178478454571532320414866489664107244676009738163736411832208882862625183076839171102221087548035311062575028833608863593234891007473165667028088811551088163196566534, 7774395186523272174222101450642361929796384303951529235881361372299167587561570868233534011630163741116098465267632161832832621211038825034169995471355890951752963983046706982386092735005729673638891143250676644805003730139400893284357727703957079448300046732554323177496455790501404620884122455772532905588958112101350309734256643178818425244058439282199421120116050468011286908779822859731660904866261368840804283786847526598505617831684232257152128246737759086420799857223671367486997845492632533290469065822109105415596392202995465003362137058207863140229592956710507365189625866236529047424060960638891, 829252450956721580879999891512276813257294288507975636575203368163787715681404733813811219125587317056307520481663358016293529034459148623076276642252404401062028581854462398323247017218907645062093693469959491687977389978878174960489335499140158627094974319731215270057947056034776960815417606036456449497948747852157011265254894816943839323315682986380142417512288823224947712310733339746205219139697807033630461428916831396185248102699897402885959181908261066251226104313615137998612431212244090292232864492141995513449832062024572917081355937428210078922492059485706660885807757251596036976288331400945, 10877110785860949840434608649598775219658896940717142729782485870223152081055582046264052872596708373605319430391674083255157081466170536402533285578400813783468898912906500336001761538492257072165398243953820599547690274178760186501098972478680199500558288145982845740397827958230377417585639101286949519432616444661757330972099231571603995282361575686710120728815039916250191718869867780606942950570727264143650911476726403562846934488338838533240953925814283572498044727322758105836665283390354057947984227052575969652866212153969377573059601298748094043040678047507414880593774431889870189322850646985821, 4528487603208450142208928240749216875254352526615846622728539469391732956610921856863191929995510893736018080708174787260357438820087951294837650884171020647201396201989689341020569605391713888502451083826846031749174321205036116711210119471668157155920776469795612701621045531899689667973711732493306703468866248897677646835477184907015452131406051423467689628612754050421159248079677351492143914191653418257635681378756101437770536229159843793562701282230478680712604135297644680047754965206528357568592921686771910084471337320841592081598968746811124875662137491860540518840629936683234362646415211890435], 'aut_phi_ratio': 22.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 44, 2, 1], [2, 121, 2, 1], [4, 2, 1, 1], [4, 44, 2, 1], [4, 242, 1, 1], [5, 121, 1, 4], [8, 242, 4, 1], [10, 121, 1, 4], [10, 121, 2, 4], [10, 484, 2, 4], [11, 20, 1, 2], [11, 40, 1, 2], [20, 242, 1, 8], [20, 484, 2, 4], [22, 20, 1, 2], [22, 40, 1, 2], [22, 440, 2, 1], [40, 242, 4, 4], [44, 40, 1, 2], [44, 40, 2, 2], [44, 440, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{11}^2.C_2^3.C_5.C_2^5', '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': 220, 'autcentquo_group': None, 'autcentquo_hash': 5713448921273139074, 'autcentquo_nilpotent': False, 'autcentquo_order': 19360, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{11}^2.C_{10}.C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 44, 2], [2, 121, 2], [4, 2, 1], [4, 44, 2], [4, 242, 1], [5, 121, 4], [8, 242, 4], [10, 121, 12], [10, 484, 8], [11, 20, 2], [11, 40, 2], [20, 242, 8], [20, 484, 8], [22, 20, 2], [22, 40, 2], [22, 440, 2], [40, 242, 16], [44, 40, 6], [44, 440, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '9680.ba', 'commutator_count': 1, 'commutator_label': '484.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '11.1', '11.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 44, 1, 2], [2, 121, 1, 2], [4, 2, 1, 1], [4, 44, 1, 2], [4, 242, 1, 1], [5, 121, 4, 1], [8, 242, 2, 2], [10, 121, 4, 3], [10, 484, 4, 2], [11, 20, 1, 2], [11, 40, 1, 2], [20, 242, 4, 2], [20, 484, 4, 2], [22, 20, 1, 2], [22, 40, 1, 2], [22, 440, 1, 2], [40, 242, 8, 2], [44, 40, 1, 2], [44, 40, 2, 2], [44, 440, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 15249024, 'exponent': 440, 'exponents_of_order': [5, 2, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[40, -1, 1], [40, 0, 4], [40, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '9680.ba', 'hash': 6161013968996120931, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 220, 'inner_gen_orders': [10, 4, 11, 22], 'inner_gens': [[1, 15870, 14440, 15400], [6181, 10, 12360, 9520], [5401, 7050, 40, 440], [4401, 10730, 40, 440]], 'inner_hash': 7887305892697447099, 'inner_nilpotent': False, 'inner_order': 9680, 'inner_split': True, 'inner_tex': 'C_2\\times D_{11}^2:C_{10}', 'inner_used': [1, 2, 4], 'irrC_degree': 40, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 40, 'irrep_stats': [[1, 40], [2, 30], [20, 8], [40, 10]], 'label': '19360.h', 'linC_count': 160, 'linC_degree': 22, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 16, 'linQ_dim': 24, 'linQ_dim_count': 16, 'linR_count': 80, 'linR_degree': 24, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C11^2:(C10*SD16)', 'ngens': 8, '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': 54, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 88, 'number_divisions': 40, 'number_normal_subgroups': 47, 'number_subgroup_autclasses': 182, 'number_subgroup_classes': 260, 'number_subgroups': 13840, 'old_label': None, 'order': 19360, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 331], [4, 332], [5, 484], [8, 968], [10, 5324], [11, 120], [20, 5808], [22, 1000], [40, 3872], [44, 1120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 4840, 0], 'outer_gens': [[4841, 4850, 40, 10120], [1, 9690, 40, 10120], [14521, 10, 40, 440], [12401, 16730, 400, 9240]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [6, 415, 127, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 10], [8, 2], [16, 2], [20, 8], [40, 6], [80, 2]], 'representations': {'PC': {'code': '237709070594190553847084917911028329208630051796201051475099379336692318935694701973809500934294955234405539276375703009009', 'gens': [1, 3, 5, 6], 'pres': [8, 2, 5, 2, 2, 11, 2, 2, 11, 16, 380882, 106210, 66, 86403, 203851, 15515, 577604, 3212, 49460, 828, 739205, 390733, 45717, 22205, 141, 640646, 369614, 104182, 51774, 166, 225287, 225295, 114199, 56351]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [10205032106821235, 10522401710269773, 3352440040187918, 28359274195767324, 16265167320216339, 39688262258321567, 40451685466516614, 41772741070013040]}, 'Perm': {'d': 30, 'gens': [10725617432828987907080673449809, 19932021658053053191376966091196, 29085808057853731426811931276437, 10192793457047840180283803040000, 36423478910311224388218174412800, 47009471610196414840525654366080, 56159299088347860887786997253751, 5329]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}^2:(C_{10}\\times \\SD_{16})', 'transitive_degree': 88, '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': '20.5', '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, 4, 4], 'aut_gens': [[1, 2], [1, 42], [1, 38], [1, 34], [61, 2]], 'aut_group': '64.196', 'aut_hash': 196, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 10, 'aut_perms': [368047, 126, 811447, 137], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [5, 1, 4, 1], [8, 2, 2, 1], [10, 1, 4, 1], [10, 4, 4, 1], [20, 2, 4, 1], [20, 4, 4, 1], [40, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4^2:C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '4.2', 'autcentquo_hash': 2, 'autcentquo_nilpotent': True, 'autcentquo_order': 4, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 1], [4, 2, 1], [4, 4, 1], [5, 1, 4], [8, 2, 2], [10, 1, 4], [10, 4, 4], [20, 2, 4], [20, 4, 4], [40, 2, 8]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '8.3', 'commutator_count': 1, 'commutator_label': '4.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 26, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.8', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [5, 1, 4, 1], [8, 2, 2, 1], [10, 1, 4, 1], [10, 4, 4, 1], [20, 2, 4, 1], [20, 4, 4, 1], [40, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 36, 'exponent': 40, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 0, 8]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '20.5', 'hash': 26, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4], 'inner_gens': [[1, 22], [61, 2]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': True, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 20], [2, 15]], 'label': '80.26', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*SD16', 'ngens': 5, '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': 12, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 35, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 20, 'number_subgroups': 30, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 5], [4, 6], [5, 4], [8, 4], [10, 20], [20, 24], [40, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 42], [1, 34]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 5], [8, 1], [16, 1]], 'representations': {'PC': {'code': 4904966904936451, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -2, -5, 221, 26, 662, 42, 58]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [4139, 13311]}, 'Perm': {'d': 13, 'gens': [2574270720, 3760560, 96, 1516929840, 482671440]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times \\SD_{16}', 'transitive_degree': 40, 'wreath_data': None, 'wreath_product': False}
