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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1344.2788', 'ambient_counter': 2788, 'ambient_order': 1344, 'ambient_tex': 'D_{42}:Q_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 672, 'characteristic': True, 'core_order': 4, 'counter': 278, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1344.2788.336.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '336.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '336.161', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 161, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 336, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_6:D_{14}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.2788', 'aut_centralizer_order': 32256, 'aut_label': '336.c1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 32256, 'centralizer': '2.c1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['48.c1.a1', '112.c1.a1', '168.a1.a1', '168.i1.a1', '168.o1.a1', '168.p1.a1', '168.p1.b1', '168.q1.a1', '168.r1.a1'], 'contains': ['672.a1.a1'], 'core': '336.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4073, 6127, 5643, 5810, 3942, 6422, 4907, 5819], 'generators': [504], 'label': '1344.2788.336.c1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '336.c1.a1', 'normal_contained_in': ['48.c1.a1', '112.c1.a1', '168.a1.a1'], 'normal_contains': ['672.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '336.c1.a1', 'projective_image': '672.618', 'quotient_action_image': '2.1', 'quotient_action_kernel': '168.38', 'quotient_action_kernel_order': 168, 'quotient_fusion': None, 'short_label': '336.c1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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, 1], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, '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', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [12, 6, 6, 6, 6, 6, 6, 4, 12, 6], 'aut_gens': [[1, 2, 4, 672], [1065, 1234, 244, 672], [337, 482, 484, 672], [1233, 658, 1052, 672], [225, 178, 1052, 672], [449, 1218, 212, 672], [897, 130, 988, 672], [225, 1090, 764, 672], [841, 866, 1172, 672], [729, 418, 1076, 672], [1289, 66, 652, 672]], 'aut_group': None, 'aut_hash': 5255589997166976132, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 64512, 'aut_permdeg': 88, 'aut_perms': [24464803459692377847811079498860042459812824058211411888769582126410723597492267537784307871022570901648767566168294810408607128210315, 145268361272149041338615036341772067431979905236873635779036518116723626610344121162583521816046706596389006443405325666691107310446673, 83641373538073194234509863709360376384527718934791469077197249054432223695457772516156498238518562076005613847489045122528110622347405, 116150313644132414018254697783580662726454304905047519753037195769355865651338797954997382993183718237937046921647887093961162788826953, 38588417205980039029041221117933395145671598581934256094619140596225104719868747525161287216860452988216149025363248269148343984511335, 132697901458729685102967132351452704090159673725297845575494103884130601087641912752850293898526798502097452399147237914986366864147977, 117378975414990046869846692999035720132636674285361265953315833471645173463390083139932345841113900151913913882361976740303314087599267, 73568979698940160959419460166712961076270349930814995852557744602153654575438973200193061406982152735465797010341953434195534061520670, 173246613739053032637465035840041061279734039221937883567231223653155496827023100997403690151756315993892773384954232867599354039279024, 138862585347623973886790721673627426031149394754353240440703272664651272670105837100003034972222099336608241678401922501573950816111309], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 42, 2, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 12, 2, 1], [4, 24, 1, 1], [4, 28, 2, 1], [4, 56, 1, 1], [4, 84, 1, 1], [6, 2, 1, 3], [7, 2, 3, 1], [8, 4, 2, 1], [8, 84, 2, 1], [12, 4, 1, 2], [12, 56, 2, 2], [14, 2, 3, 3], [21, 4, 3, 1], [24, 4, 4, 1], [28, 4, 3, 2], [28, 24, 6, 2], [42, 4, 3, 3], [56, 4, 12, 1], [84, 4, 6, 2], [168, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_2^3', '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': 42, 'autcentquo_group': '1008.906', 'autcentquo_hash': 906, 'autcentquo_nilpotent': False, 'autcentquo_order': 1008, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 42, 2], [3, 2, 1], [4, 2, 2], [4, 12, 2], [4, 24, 1], [4, 28, 2], [4, 56, 1], [4, 84, 1], [6, 2, 3], [7, 2, 3], [8, 4, 2], [8, 84, 2], [12, 4, 2], [12, 56, 4], [14, 2, 9], [21, 4, 3], [24, 4, 4], [28, 4, 6], [28, 24, 12], [42, 4, 9], [56, 4, 12], [84, 4, 12], [168, 4, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '336.150', 'commutator_count': 1, 'commutator_label': '168.39', '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': 2788, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 42, 1, 2], [3, 2, 1, 1], [4, 2, 1, 2], [4, 12, 1, 2], [4, 24, 1, 1], [4, 28, 1, 2], [4, 56, 1, 1], [4, 84, 1, 1], [6, 2, 1, 3], [7, 2, 3, 1], [8, 4, 2, 1], [8, 84, 2, 1], [12, 4, 1, 2], [12, 56, 1, 2], [12, 56, 2, 1], [14, 2, 3, 3], [21, 4, 3, 1], [24, 4, 4, 1], [28, 4, 3, 2], [28, 24, 3, 2], [28, 24, 6, 1], [42, 4, 3, 3], [56, 4, 12, 1], [84, 4, 6, 2], [168, 4, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 10752, 'exponent': 168, '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': '8.2', 'frattini_quotient': '168.50', 'hash': 2788, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 84, 1], 'inner_gens': [[1, 2, 1292, 672], [1, 2, 1172, 672], [729, 850, 4, 672], [1, 2, 4, 672]], 'inner_hash': 150, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': None, 'inner_tex': 'C_{12}:D_{14}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 42], [4, 73]], 'label': '1344.2788', 'linC_count': 1248, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 228, 'linQ_dim': 16, 'linQ_dim_count': 16, 'linR_count': 216, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D42:Q16', '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': 39, 'number_characteristic_subgroups': 58, 'number_conjugacy_classes': 123, 'number_divisions': 44, 'number_normal_subgroups': 68, 'number_subgroup_autclasses': 232, 'number_subgroup_classes': 296, 'number_subgroups': 2676, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 87], [3, 2], [4, 248], [6, 6], [7, 6], [8, 176], [12, 232], [14, 18], [21, 12], [24, 16], [28, 312], [42, 36], [56, 48], [84, 48], [168, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 2, 6], 'outer_gen_pows': [392, 0, 192, 0, 64, 0], 'outer_gens': [[729, 1330, 52, 672], [897, 562, 956, 672], [561, 82, 956, 672], [113, 1122, 116, 672], [785, 66, 1180, 672], [225, 1090, 764, 672]], 'outer_group': '192.1543', 'outer_hash': 1543, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 192, 'outer_perms': [96164386935439260633268452222123627335463782810859825043164787179414745612060857249772005919743627267102350143947322386951212351546068493775892524864281756063615882075504515910885707902026401314180865838474636883671725112524383201911955444305580670120549750677723483080712668118573396529979102948215620182128528157973125440373185161975758085272000942099411, 61136948738452103738845290447799697105457518072685543645019820323968759365257877429719441759549419696501370795271909948254539222023670513512047450486977627738472176856188706968884867413524628078854519683281067187718556354455259388696967678012059787611288405119087617873564573692997138424785009314610631925555908846041547303168779166827111651001748053715019, 52773227444492499199211313005205808989139453541033303232350978403031517645764818645834072138415777734025643798285781885195407385732730868676407343292294350479344501076672142373707561006469334191352161436018455057236045903625918745674713776268791160802632803524473709005332808963006069173746254856597122589937725400953455522039073388939567246656345838282398, 64041233725527042783448252915775394185566151332388909126140744706317625350370349491576513716489772686017791107453244613407435248302679169195296669302883389070655777900170663592421177815531913286923865736928912319953982884353455107833300793532155157365841450683377006328198916994681796754947503654907188929797782415901000936530342832919664476745527175246190, 217168688821942171290312711436138814681962894217405664404793432259628708627784211793834193063340865594818568480468820425643898944731073685848232531524277111984399935141265404937011034091170496611048597698511255167469992104430079175157155037026560328565535530800890094889197025788374170777273411056995474453124631022522172384960248574862429354330185900869681, 13166983287848937866401044560333711761108374881762405442912639101011155115518719312118242201486635640289425213803311675520612157118043746163524248519535137472965924854333541141774825646045744428527368540866689800651382473923963948811357885070518597202006416986187296737497262860545943719225496840872535382771816939998357414572625149725234116064130361592411], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 7], [6, 4], [8, 2], [12, 6], [24, 5], [48, 2]], 'representations': {'PC': {'code': 26995454658134663840967837604762730363867878432880206699842, 'gens': [1, 2, 3, 8], 'pres': [8, -2, -2, -2, -2, -2, -3, -7, -2, 2688, 31010, 14074, 66, 18179, 5259, 91, 18564, 13132, 116, 12293, 15373, 189, 16142]}, 'Perm': {'d': 30, 'gens': [305292256521389608334116181161, 51335472094574341807, 399282193560917537, 527699650413560520, 527699650413560527, 621627883554148560, 1175091669949317120000, 9452345886333784620057231360000]}}, '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': 'D_{42}:Q_{16}', 'transitive_degree': 672, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [6, 6, 6, 6, 6, 42], 'aut_gens': [[1, 2, 4, 8], [149, 118, 4, 188], [293, 118, 4, 104], [49, 230, 4, 140], [97, 6, 4, 248], [101, 226, 4, 252], [289, 114, 4, 12]], 'aut_group': None, 'aut_hash': 5451351369529179689, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2016, 'aut_permdeg': 22, 'aut_perms': [1090696963343353579897, 908231706454761976456, 81683538447188867827, 664433072848664916826, 502337525864606593713, 708496207041020825526], 'aut_phi_ratio': 21.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 1], [2, 7, 2, 1], [2, 14, 1, 1], [2, 42, 1, 1], [3, 2, 1, 1], [4, 6, 1, 1], [4, 42, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 14, 2, 2], [7, 2, 3, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 12, 3, 1], [21, 4, 3, 1], [28, 12, 3, 1], [42, 4, 3, 1], [42, 4, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3\\times S_3\\times F_7', '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': 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, 1], [2, 2, 1], [2, 6, 1], [2, 7, 2], [2, 14, 1], [2, 42, 1], [3, 2, 1], [4, 6, 1], [4, 42, 1], [6, 2, 3], [6, 14, 4], [7, 2, 3], [14, 2, 3], [14, 4, 3], [14, 12, 3], [21, 4, 3], [28, 12, 3], [42, 4, 9]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '168.50', 'commutator_count': 1, 'commutator_label': '42.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 161, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['24.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 1], [2, 7, 1, 2], [2, 14, 1, 1], [2, 42, 1, 1], [3, 2, 1, 1], [4, 6, 1, 1], [4, 42, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 14, 1, 2], [6, 14, 2, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 12, 3, 1], [21, 4, 3, 1], [28, 12, 3, 1], [42, 4, 3, 1], [42, 4, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5376, 'exponent': 84, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[4, 0, 6]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '168.50', 'hash': 161, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 2, 1, 42], 'inner_gens': [[1, 2, 4, 104], [1, 2, 4, 236], [1, 2, 4, 8], [241, 118, 4, 8]], 'inner_hash': 50, 'inner_nilpotent': False, 'inner_order': 168, 'inner_split': True, 'inner_tex': 'S_3\\times D_{14}', 'inner_used': [1, 2, 4], 'irrC_degree': 4, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 22], [4, 15]], 'label': '336.161', 'linC_count': 54, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 40, 'linQ_dim': 10, 'linQ_dim_count': 40, 'linR_count': 144, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D6:D14', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 45, 'number_divisions': 24, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 96, 'number_subgroup_classes': 108, 'number_subgroups': 684, 'old_label': None, 'order': 336, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 79], [3, 2], [4, 48], [6, 62], [7, 6], [14, 54], [21, 12], [28, 36], [42, 36]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[5, 2, 4, 8], [1, 2, 4, 40]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 4, '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, 6], [4, 2], [6, 4], [12, 3], [24, 1]], 'representations': {'PC': {'code': 646130545326181344453391883, 'gens': [1, 2, 3, 4], 'pres': [6, -2, -2, -2, 2, -3, -7, 2499, 2841, 69, 6244, 1930, 118, 10373]}, 'GLZN': {'d': 2, 'p': 28, 'gens': [148770, 604085, 21965, 22065, 296771, 27658]}, 'Perm': {'d': 14, 'gens': [127, 6266977920, 997920000, 13412044800, 403200, 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': 'D_6:D_{14}', 'transitive_degree': 84, 'wreath_data': None, 'wreath_product': False}
