-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1440.2534', 'ambient_counter': 2534, 'ambient_order': 1440, 'ambient_tex': 'C_{60}.S_4', 'central': False, 'central_factor': False, 'centralizer_order': 720, 'characteristic': True, 'core_order': 15, 'counter': 93, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1440.2534.96.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '96.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '96.65', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 65, 'quotient_metabelian': False, 'quotient_nilpotent': False, 'quotient_order': 96, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'A_4:C_8', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '15.1', 'subgroup_hash': 1, 'subgroup_order': 15, 'subgroup_tex': 'C_{15}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1440.2534', 'aut_centralizer_order': 4320, 'aut_label': '96.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '8.2', 'aut_weyl_index': 4320, 'centralizer': '2.a1.a1', 'complements': ['15.b1.a1', '15.b1.b1', '15.b1.c1'], 'conjugacy_class_count': 1, 'contained_in': ['32.a1.a1', '48.a1.a1', '48.c1.a1', '48.d1.a1'], 'contains': ['288.a1.a1', '480.a1.a1'], 'core': '96.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3544, 6035, 938, 5406, 3127, 4753, 3891, 4937], 'generators': [480, 24], 'label': '1440.2534.96.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '96.a1.a1', 'normal_contained_in': ['24.b1.a1', '48.a1.a1'], 'normal_contains': ['288.a1.a1', '480.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '96.a1.a1', 'projective_image': '1440.2534', 'quotient_action_image': '2.1', 'quotient_action_kernel': '48.31', 'quotient_action_kernel_order': 48, 'quotient_fusion': None, 'short_label': '96.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1], [11], [7]], 'aut_group': '8.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 6, 'aut_perms': [120, 9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [5, 1, 4, 1], [15, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', '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], [3, 1, 2], [5, 1, 4], [15, 1, 8]], 'center_label': '15.1', 'center_order': 15, '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': ['3.1', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [5, 1, 4, 1], [15, 1, 8, 1]], 'element_repr_type': 'PC', 'elementary': 15, 'eulerian_function': 1, 'exponent': 15, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [3, 5], 'faithful_reps': [[1, 0, 8]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '15.1', 'hash': 1, 'hyperelementary': 15, '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': 8, 'irrQ_dim': 8, 'irrR_degree': 2, 'irrep_stats': [[1, 15]], 'label': '15.1', 'linC_count': 8, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 4, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C15', 'ngens': 2, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 15, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 15, 'order_factorization_type': 11, 'order_stats': [[1, 1], [3, 2], [5, 4], [15, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[11], [7]], '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': 1, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [4, 1], [8, 1]], 'representations': {'PC': {'code': 87, 'gens': [1], 'pres': [2, -3, -5, 6]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58415890696506688]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [6351]}, 'Perm': {'d': 8, 'gens': [10080, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}', 'transitive_degree': 15, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [12, 4, 12, 12, 12, 30], 'aut_gens': [[1, 8, 120, 240], [549, 1264, 840, 1200], [79, 1024, 120, 360], [581, 824, 120, 240], [881, 296, 840, 600], [383, 464, 840, 1080], [501, 488, 840, 1080]], 'aut_group': None, 'aut_hash': 2220674056071093365, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 34560, 'aut_permdeg': 360, 'aut_perms': [2660447779644039484172277601220590345520998399088188769081733902161805715284205556940463541185635859051682878684298422483381470187566700681155137216307857275383992860179743559700549277073881764687521624230159409012821161436686795054880325299271984536305432346702101402317576794538878925735717204428299638372183049365134739621418332647177392357766999107320926188158029589974870373588842128906753987757077224815013078017226201748111956279940175567343545059737482835107957308808316760180945892809756047747808379582275327699050669244800017315244418128114384861760689752559605314992696304938332325081705653202153581241589775912377788642060869714637617494899237722439185437874296006847307413775489014019900182918752346308183392357962166698992338308175744538493709301882754, 2222995168385976857445875677335657103636261657635443798157226333220327736944849884344838870906701872741664177994513008006734581972734062161391591987743646106911144593608895387279665118627299635310545305210855567023686747273315393532099764227555982365554542440873784961017304060314927637760832027325994521311050779051446879007675458611336797212972449412193016951071799668417861536039912964253446315077897479365278826444643229778354279370869488252646268496855035250191878653554664308334473645246727084113754509824492292198013743254375535864047290190040209760233354052282294483980084283867599967967405920679322440473964365341407301561986048847127330216059224420301027204778764731878612271216341890610264211439243298820976891370759596460367465606209803403252771776196007, 2230071483208953790262599726985408979371476767161061135109999488012100535442749802913588823670121421634000206838653497610645811299106186167926247356893890019643645436074740020827779971262482078133297898469332418074057280734533961691283137207402729562492699585639112229384575178910478341074688897927048426019468444310005555209892020131729547011911219585371925382896671269033135589938979643822965918083077472503115399851851431864114146828090883058031214772646540388448584272019684645012083708954249165822065568741974219869716045493995585232801680062183106225172342988776774335205184621039168527049817868611695151623070496469913392756670283211963818956560550511008976756827045732980372120164797520160206410948589033365067247547900989232805888064000802628416753623452654, 448890847744104076124973418198047287371973918849700712511961681266485169199159109367916442241358639791273182658572784404693856681676323546139418737595156804082130065227842324904312981013598051990214306024184843574012438140832661491705376622927162585383700494302989453616345534689476308460078650615510138803568971329187967190694234922541507220557482503710062299430987726875235299459667491324117624054596661773295759646470206312568060600180976283282765795544318508989026967689062171731067136509396812491724463138833322845285969281886827029494811851940810328553373214472076216129954724607168186175304908454993317959739944372516655868672996509960367823656481757242115127973400256722026969802170634702938753199266032911891492824989733358796031773527114092913620005002584, 1701980819417655253430142407611644768663592847389703263278446246441585911665757098696429411389759939254922894745110382550679966169956043600226884585012344432433224315139407626580828857516325860206371882510458743035033056324209862268701734883188996398918069027211105137263435283284935217657073909378995007383698322870589104791064281737084008388523232786253872916418357184285647719654894779585858657145297929911297395437525169457403357575003946898539184578182334012397422531707714407000025219796796743043525571968875378090925144184275579080292370600729019576359659149665322522406455278295390203053528957918350747555700399772244670795862544242855376460979176295253131774580948250788529282119625421882940580051610690607216452139224804195677228967600224245235284826797847, 1409444480070792693853267250514774229796170658417984554819865762212335415963359754161400656882640622793441667697385777424901227383918001582709061749592027466179601692954935192397093065268684684514779796445521623814708870759802082795582964686643216183674759673751161716970779601996814286369565366200189703144113705392050352584539148010763563638399580426281127699671594360161737167620699148437108428204617861967869592838942833639308124893128754249232888407316208664752736543679477837467328306181295710779572568961496699503693541604847885822083734617223987498624377713194464451582180030996043626871806128848703368053167802906616539927713727741374973197125072466597788431906594799225392241087622158519238167456727730314561856634213080456017172700868026843280598799564021], 'aut_phi_ratio': 90.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [3, 8, 3, 1], [4, 1, 2, 1], [4, 3, 2, 1], [5, 2, 2, 1], [6, 2, 1, 1], [6, 6, 1, 2], [6, 8, 3, 1], [8, 90, 4, 2], [10, 2, 2, 1], [10, 6, 2, 2], [12, 2, 2, 1], [12, 6, 2, 1], [12, 8, 6, 1], [15, 2, 4, 1], [15, 8, 12, 1], [20, 2, 4, 1], [20, 6, 4, 1], [30, 2, 4, 1], [30, 6, 4, 2], [30, 8, 12, 1], [60, 2, 8, 1], [60, 6, 8, 1], [60, 8, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times F_5\\times C_3:S_3:S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': None, 'autcentquo_hash': 928336707834802966, 'autcentquo_nilpotent': False, 'autcentquo_order': 8640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times C_3:S_3:S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 2, 1], [3, 8, 3], [4, 1, 2], [4, 3, 2], [5, 2, 2], [6, 2, 1], [6, 6, 2], [6, 8, 3], [8, 90, 8], [10, 2, 2], [10, 6, 4], [12, 2, 2], [12, 6, 2], [12, 8, 6], [15, 2, 4], [15, 8, 12], [20, 2, 4], [20, 6, 4], [30, 2, 4], [30, 6, 8], [30, 8, 12], [60, 2, 8], [60, 6, 8], [60, 8, 24]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '360.141', 'commutator_count': 1, 'commutator_label': '180.31', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2534, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [3, 8, 1, 3], [4, 1, 2, 1], [4, 3, 2, 1], [5, 2, 2, 1], [6, 2, 1, 1], [6, 6, 1, 2], [6, 8, 1, 3], [8, 90, 4, 2], [10, 2, 2, 1], [10, 6, 2, 2], [12, 2, 2, 1], [12, 6, 2, 1], [12, 8, 2, 3], [15, 2, 4, 1], [15, 8, 4, 3], [20, 2, 4, 1], [20, 6, 4, 1], [30, 2, 4, 1], [30, 6, 4, 2], [30, 8, 4, 3], [60, 2, 8, 1], [60, 6, 8, 1], [60, 8, 8, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 40320, 'exponent': 120, 'exponents_of_order': [5, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 8]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '360.141', 'hash': 2534, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 15, 2, 6], 'inner_gens': [[1, 112, 120, 1320], [17, 8, 840, 1080], [1, 728, 120, 240], [601, 848, 120, 240]], 'inner_hash': 141, 'inner_nilpotent': False, 'inner_order': 360, 'inner_split': True, 'inner_tex': 'C_{15}:S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 6, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 88], [3, 8], [6, 28]], 'label': '1440.2534', 'linC_count': 288, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 12, 'linQ_dim': 11, 'linQ_dim_count': 6, 'linR_count': 72, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C60.S4', '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': 32, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 132, 'number_divisions': 44, 'number_normal_subgroups': 49, 'number_subgroup_autclasses': 104, 'number_subgroup_classes': 136, 'number_subgroups': 1032, 'old_label': None, 'order': 1440, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 7], [3, 26], [4, 8], [5, 4], [6, 38], [8, 720], [10, 28], [12, 64], [15, 104], [20, 32], [30, 152], [60, 256]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 4, 2, 6], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 88, 120, 360], [1, 104, 120, 240], [5, 8, 120, 240], [7, 1072, 120, 1320]], 'outer_group': '96.206', 'outer_hash': 206, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [362881, 5190, 289, 3996720], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [3, 2], [4, 7], [6, 3], [8, 9], [12, 4], [16, 4], [24, 3], [48, 1]], 'representations': {'PC': {'code': 599670727224064573816768091738821250788295105, 'gens': [1, 4, 6, 7], 'pres': [8, -2, -2, -2, -3, -5, -2, 2, -3, 16, 41, 3587, 123, 3844, 5069, 73926, 7590, 166, 61447]}, 'GLZN': {'d': 2, 'p': 100, 'gens': [21402081, 49000049, 51752576, 57000057, 1505001, 61414689, 51500051, 73329601]}, 'Perm': {'d': 20, 'gens': [13160522058128937, 12568, 18498, 6759455986483200, 1394852659200, 559198080, 122000787836928000, 256094948229120000]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{60}.S_4', 'transitive_degree': 360, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 6, 2, 2], 'aut_gens': [[10622, 53203, 38921, 6273], [8702, 53203, 38921, 6273], [10622, 12604, 37001, 6273], [15069, 53203, 6273, 37001], [43382, 18395, 38921, 6273], [10622, 51027, 38921, 6273]], 'aut_group': '96.226', 'aut_hash': 226, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [7, 1567, 1456, 16680, 11520], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 8, 1, 1], [4, 1, 2, 1], [4, 3, 2, 1], [6, 8, 1, 1], [8, 6, 4, 2], [12, 8, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 8, 1], [4, 1, 2], [4, 3, 2], [6, 8, 1], [8, 6, 8], [12, 8, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '24.12', 'commutator_count': 1, 'commutator_label': '12.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 65, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 8, 1, 1], [4, 1, 2, 1], [4, 3, 2, 1], [6, 8, 1, 1], [8, 6, 4, 2], [12, 8, 2, 1]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 36, 'exponent': 24, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 0, 4]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '24.12', 'hash': 65, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3, 2, 2], 'inner_gens': [[10622, 12604, 37001, 6273], [7935, 53203, 6273, 37001], [43382, 18395, 38921, 6273], [10622, 51027, 38921, 6273]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'S_4', 'inner_used': [1, 2, 3], 'irrC_degree': 3, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 6, 'irrep_stats': [[1, 8], [2, 4], [3, 8]], 'label': '96.65', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 4, 'linQ_dim': 7, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'A4:C8', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 20, 'number_divisions': 11, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 26, 'number_subgroups': 68, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [3, 8], [4, 8], [6, 8], [8, 48], [12, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [4097, 4097], 'outer_gens': [[27482, 53203, 38921, 6273], [8702, 53203, 38921, 6273]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [3, 2], [4, 2], [6, 1], [12, 1]], 'representations': {'PC': {'code': 1111536240760479940, 'gens': [1, 4, 5, 6], 'pres': [6, -2, -2, -2, -3, -2, 2, 12, 31, 387, 2164, 202, 347]}, 'GLFq': {'d': 3, 'q': 9, 'gens': [107079024, 87646943, 325511162, 344426264, 86106566, 133611126]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [36873, 8694, 51027, 12291, 39049, 4225]}, 'Perm': {'d': 12, 'gens': [3634857, 12568, 18498, 3991680, 40279680, 87091200]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_4:C_8', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
