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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1344.8700', 'ambient_counter': 8700, 'ambient_order': 1344, 'ambient_tex': '(C_2\\times C_{12}).D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 168, 'counter': 34, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1344.8700.8.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '8.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '168.56', 'subgroup_hash': 56, 'subgroup_order': 168, 'subgroup_tex': 'C_2\\times D_{42}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.8700', 'aut_centralizer_order': None, 'aut_label': '8.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '168.c1', 'complements': [], 'conjugacy_class_count': 2, 'contained_in': ['4.a1', '4.e1', '4.f1', '4.g1'], 'contains': ['16.a1', '16.e1', '16.i1', '24.f1', '56.f1'], 'core': '8.b1', 'coset_action_label': None, 'count': 2, 'diagramx': None, 'generators': [15, 672, 16, 4, 192], 'label': '1344.8700.8.b1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.b1', 'normal_contained_in': ['4.a1', '4.e1', '4.f1', '4.g1'], 'normal_contains': ['16.a1'], 'normalizer': '1.a1', 'old_label': '8.b1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.b1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, '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': 84, 'aut_gen_orders': [12, 2, 2, 42, 3, 2, 3, 12, 2], 'aut_gens': [[1, 2, 4], [85, 131, 52], [1, 114, 116], [1, 2, 5], [85, 46, 4], [85, 2, 89], [1, 86, 4], [1, 58, 4], [1, 134, 149], [1, 87, 4]], 'aut_group': '6048.bj', 'aut_hash': 3542203032238244068, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6048, 'aut_permdeg': 14, 'aut_perms': [13452369539, 766080, 479001600, 40324830, 6706022400, 6266937600, 403200, 7185026584, 13412044800], 'aut_phi_ratio': 126.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 21, 4, 1], [3, 2, 1, 1], [6, 2, 3, 1], [7, 2, 3, 1], [14, 2, 9, 1], [21, 2, 6, 1], [42, 2, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times S_4\\times F_7', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 21, 4], [3, 2, 1], [6, 2, 3], [7, 2, 3], [14, 2, 9], [21, 2, 6], [42, 2, 18]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '42.5', 'commutator_count': 1, 'commutator_label': '21.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 56, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['42.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 21, 1, 4], [3, 2, 1, 1], [6, 2, 1, 3], [7, 2, 3, 1], [14, 2, 3, 3], [21, 2, 6, 1], [42, 2, 6, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 224, 'exponent': 42, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '168.56', 'hash': 56, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [1, 2, 21], 'inner_gens': [[1, 2, 4], [1, 2, 164], [1, 10, 4]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 42, 'inner_split': True, 'inner_tex': 'D_{21}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 40]], 'label': '168.56', 'linC_count': 72, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 6, 'linQ_dim': 8, 'linQ_dim_count': 6, 'linR_count': 72, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D42', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 48, 'number_divisions': 20, 'number_normal_subgroups': 31, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 64, 'number_subgroups': 372, 'old_label': None, 'order': 168, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 87], [3, 2], [6, 6], [7, 6], [14, 18], [21, 12], [42, 36]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [6, 6, 6, 6, 3, 3], 'outer_gen_pows': [0, 0, 0, 0, 2, 2], 'outer_gens': [[1, 86, 20], [85, 86, 68], [1, 2, 76], [1, 87, 124], [84, 2, 69], [1, 2, 68]], 'outer_group': '144.188', 'outer_hash': 188, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 9, 'outer_perms': [41070, 769, 31, 90769, 80688, 48], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [6, 4], [12, 4]], 'representations': {'PC': {'code': 515363543722101899, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -3, -7, 1237, 42, 1608, 78, 1809]}, 'GLZN': {'d': 2, 'p': 21, 'gens': [9409, 9325, 120406, 74096, 120674]}, 'Perm': {'d': 14, 'gens': [482670847, 7983360, 11652480, 6706022400, 973]}}, 'schur_multiplier': [2, 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_2\\times D_{42}', 'transitive_degree': 84, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [12, 12, 12, 12, 28, 6, 12, 12], 'aut_gens': [[1, 2, 8, 48], [721, 710, 8, 532], [437, 34, 712, 148], [821, 2, 12, 1108], [937, 1034, 8, 240], [1061, 1046, 40, 720], [677, 18, 40, 820], [217, 1022, 8, 1200], [341, 1030, 712, 528]], 'aut_group': None, 'aut_hash': 6767643660662859756, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 516096, 'aut_permdeg': 160, 'aut_perms': [315276931375513320048481482728323649707306320542002357985266073865680498424693562020871615755110973085399451807474848289726373855212758433274568466222623356908798491924618505667681839972267368559596300144125453336347648258766918638561291892051873899282142494460693892595955926037460552, 34429799767629511732104795116297035067912543303321422319229992110633088006301184173252427098309231517533568970609463152445093128507649032958679767205514977168655021521328640705913454563627922971680778882387447820446650804220694251607118958545397159700460202462945488521378994594261961, 148661035156464259336655221077049529785011036109857718688544788022011383355871015854206430282043511126949060899601713252702941891725130204818355487705524791488285360745949742132647305546442185356598040209662279641078582985104329110459776970479482342376631054643267035340644659864434036, 203457735058636515961161925593364364416239803532893987506657693554736682108088120774346862257149749900762542431348640678684745587071693572035875629710291538294188170079628566720633027257427892284441806979607575889680075701826839450514423157271046769029013201443010157824010945891361953, 177004802990218584125178321829732373698003952747966637975454586842617238319939072626213807551077533293816350597356504343250148578279760439556301855740356806270079319480766357466180999013070776573347141604024797466325632760375592801024124052855223031679489326658743162147741348444296249, 67664165571904461273175325343515797558546504686890431467673377360719967217056030039733182986827856989249699087228118880668293675145997156630238289408426077566097189129442308646803690090571211040689632238501065001628195787948947619708558834501469790156873892705727668854123892652961124, 369409676339523380596466749699502116688605375744410946837627686814440630654568578905597968098913563845180693266973567225769973619691647702008703674743728323120515892087526265792889218160332065063470559925366009873123173383620908601423003471625341935161183431775490610913979089654694186, 61234277212940941027229814998230739495387017883919396873273426315989754926915858857444127142455675253807462107803889973540754212151239171415748647689564250708543660966668865505476558553986935242778249338724255613772332821187532506273327860714818830459171623121117529121113592062685579], 'aut_phi_ratio': 1344.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 84, 2, 1], [3, 2, 1, 1], [4, 2, 2, 2], [4, 12, 4, 1], [4, 28, 4, 1], [4, 84, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [7, 2, 3, 1], [12, 4, 2, 2], [12, 28, 8, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 4, 6, 1], [21, 4, 3, 1], [28, 2, 12, 1], [28, 4, 6, 1], [28, 12, 24, 1], [42, 4, 3, 1], [42, 4, 6, 1], [42, 4, 12, 1], [84, 4, 12, 2]], 'aut_supersolvable': True, 'aut_tex': '(C_2^5\\times C_{42}).C_6.C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', '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, 3], [2, 2, 2], [2, 84, 2], [3, 2, 1], [4, 2, 4], [4, 12, 4], [4, 28, 4], [4, 84, 2], [6, 2, 7], [7, 2, 3], [12, 4, 4], [12, 28, 8], [14, 2, 9], [14, 4, 6], [21, 4, 3], [28, 2, 12], [28, 4, 6], [28, 12, 24], [42, 4, 21], [84, 4, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '336.219', 'commutator_count': 1, 'commutator_label': '84.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 8700, '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, 2], [2, 84, 1, 2], [3, 2, 1, 1], [4, 2, 1, 4], [4, 12, 1, 4], [4, 28, 1, 4], [4, 84, 1, 2], [6, 2, 1, 3], [6, 2, 2, 2], [7, 2, 3, 1], [12, 4, 1, 4], [12, 28, 2, 4], [14, 2, 3, 3], [14, 4, 3, 2], [21, 4, 3, 1], [28, 2, 6, 2], [28, 4, 3, 2], [28, 12, 6, 4], [42, 4, 3, 3], [42, 4, 6, 2], [84, 4, 6, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1867320, 'exponent': 84, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '336.219', 'hash': 8700, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 2, 6, 14], 'inner_gens': [[1, 6, 8, 1296], [5, 2, 712, 48], [1, 690, 8, 48], [97, 2, 8, 48]], 'inner_hash': 219, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': False, 'inner_tex': 'D_6\\times D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 68], [4, 66]], 'label': '1344.8700', '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*C12).D28', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 36, 'number_conjugacy_classes': 150, 'number_divisions': 60, 'number_normal_subgroups': 160, 'number_subgroup_autclasses': 212, 'number_subgroup_classes': 540, 'number_subgroups': 4148, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 175], [3, 2], [4, 336], [6, 14], [7, 6], [12, 240], [14, 42], [21, 12], [28, 336], [42, 84], [84, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [12, 6, 12, 12, 12, 12], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[509, 10, 44, 912], [1057, 38, 12, 148], [697, 366, 12, 432], [101, 1010, 8, 148], [601, 46, 712, 1204], [721, 2, 44, 1204]], 'outer_group': '1536.554852', 'outer_hash': 8523979296117700338, 'outer_nilpotent': True, 'outer_order': 1536, 'outer_permdeg': 192, 'outer_perms': [238469582361962702667567358706034939483891125973820413740043794770932002721329256317015927804407424871388935399971757629145540958352769014454478028983759678671695752471160831704707394172454973307969294464927671533973191216551124295335431539923850636952257161853815196604520904141155841601914207806688145384546661908563690314010154787093501493104042258058908, 233695479773920732835320212943712972666378497024821835284532106227566030680525605379557461151175656596581836598460625444653188021445220037248819328447481753814471881756183000760360914936080933619194299877362456450420888083240339063210640699447247582163814780397969259787391569839821131652843295982216347123339931765557559624909884248446141110095904636500432, 288391015682706634214966819489564483082213520517016529080908250654594277231277432715102811032840469131270916442035338677991922142352648864127054499371455119708067629609729668003496925657404484471411148865126353403589345859552466612026688611180882164701220016929648439920746914143423552882283332020800281909262631511748432100165639312882441249234689075654381, 271881970211194746653550850893657525306142453865868200531362242729673026822410080385367978040982376828880574035917734759977140644560641377145713215562329413554473767185999591832741925559474842061643156921194165116514000107285608156099280233066233065259402304168316803017049590267763885871780452393564871701144368307260855892617071424447115785347183267391308, 240440287356956664316895597441933726626439360367990719971212815353797976985751022861471567059978144197578207920303030016277128472460159017292534755623467987015265747603555933356192671287993901416690365562886444908825680197556926519866877510543123625804980645365788055959236502344714327958729552614859535264602195470680287979637599139375453796132746681872088, 319679306885256671971498539898183920623161736041280316810102385575571185573484812569694411528482869576188936744391740152630372888107618493027423191666953496966291160558942386427164977101562364232355514315531346593666580448475675622150131643676097900931195463850973362723526308173250997735478252921368367200644939954041178050773393588693756440020290184038979], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3\\times C_2^4.C_2^5', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 6], [6, 8], [8, 2], [12, 12], [24, 2], [48, 2]], 'representations': {'PC': {'code': 31129462557865919142133806805682993545267887655901, 'gens': [1, 2, 4, 6], 'pres': [8, -2, -2, -2, -2, -3, -2, -2, -7, 5376, 97, 41, 11403, 91, 652, 62213, 141, 69894, 166, 73735]}, 'Perm': {'d': 30, 'gens': [305292256521392416386933689887, 51367887416382285733, 405492354738068423, 534771709046644440, 663245948300545463, 663245948300545440, 1175091669949317120000, 9452345886333784620057231360000]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_2\\times C_{12}).D_{28}', 'transitive_degree': 672, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 84, 'aut_gen_orders': [3, 3], 'aut_gens': [[1, 2, 4], [4, 5, 3], [2, 4, 1]], 'aut_group': '168.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 7, 'aut_perms': [4361, 244], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 168, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\PSL(2,7)', '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, 7]], 'center_label': '8.5', 'center_order': 8, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 8]], 'label': '8.5', 'linC_count': 28, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 28, 'linQ_dim': 3, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_divisions': 8, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 5, 3], [2, 4, 1]], 'outer_group': '168.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 168, 'outer_permdeg': 7, 'outer_perms': [4361, 244], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -2, 2, 2]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16482, 16322, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 13286, 13933]}, 'Perm': {'d': 6, 'gens': [120, 6, 1]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_2^3', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
