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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '6912.fa', 'ambient_counter': 131, 'ambient_order': 6912, 'ambient_tex': '(C_2^3\\times C_6^2):D_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 36, 'counter': 31, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6912.fa.12.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.c1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 12, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '576.8409', 'subgroup_hash': 8409, 'subgroup_order': 576, 'subgroup_tex': 'D_6^2.C_2^2', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6912.fa', 'aut_centralizer_order': None, 'aut_label': '12.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1728.b1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['6.c1'], 'contains': ['24.e1', '24.g1', '24.i1', '24.s1', '24.t1', '24.w1', '24.x1', '24.bz1', '24.ch1', '108.a1'], 'core': '192.a1', 'coset_action_label': None, 'count': 12, 'diagramx': None, 'generators': [12630, 2803676566609, 5693416658334, 414, 9335309747088, 1584, 1, 20677265671], 'label': '6912.fa.12.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.c1', 'old_label': '12.c1', 'projective_image': '3456.dd', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.c1', 'subgroup_fusion': None, 'weyl_group': '288.1031'}
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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': 3, 'aut_exponent': 12, 'aut_gen_orders': [4, 4, 4, 12, 12, 12, 12], 'aut_gens': [[1, 2, 4, 16, 96], [35, 458, 294, 216, 568], [9, 34, 438, 88, 120], [59, 10, 340, 464, 152], [217, 194, 206, 24, 568], [209, 202, 486, 24, 560], [265, 202, 494, 80, 120], [35, 34, 284, 472, 104]], 'aut_group': None, 'aut_hash': 2223990294858638248, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 18432, 'aut_permdeg': 72, 'aut_perms': [33951830505308993511308826675916599141505173378041866586224136609706689767752135935450227379711317505914, 59131930893868300944494610190203679365363608195272437098936648044698329342086288451551244094410018176170, 18344150745453666174066545469132679631829151061387242961729792681198698629770557164730670571179515258450, 22828682962395620945987906870184789525673630007871509059380820990253662328293045150540836599733717835743, 10487022345312686100271002187214050805363699563282238100928590587177427149430886204058666993341910993926, 48200106074687655249634324071447227814800095786162606508268439250807704711830181146032000146526786208547, 16094555936871717761700518319635915679701001191288422519172381763705567513338401105235292984111556821945], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 9, 2, 2], [2, 12, 2, 1], [2, 18, 2, 1], [3, 4, 1, 2], [4, 6, 8, 1], [4, 12, 2, 1], [4, 36, 4, 1], [6, 4, 1, 2], [6, 4, 2, 2], [6, 4, 4, 1], [6, 8, 2, 1], [6, 24, 2, 1], [12, 12, 8, 1], [12, 24, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.C_2^6.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '144.186', 'autcentquo_hash': 186, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^2:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 9, 4], [2, 12, 2], [2, 18, 2], [3, 4, 2], [4, 6, 8], [4, 12, 2], [4, 36, 4], [6, 4, 10], [6, 8, 2], [6, 24, 2], [12, 12, 8], [12, 24, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '144.186', 'commutator_count': 1, 'commutator_label': '36.13', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 8409, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['288.881', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 9, 1, 4], [2, 12, 1, 2], [2, 18, 1, 2], [3, 4, 1, 2], [4, 6, 2, 4], [4, 12, 1, 2], [4, 36, 1, 4], [6, 4, 1, 10], [6, 8, 1, 2], [6, 24, 1, 2], [12, 12, 2, 4], [12, 24, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1834560, 'exponent': 12, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '288.1031', 'hash': 8409, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 2, 3, 6], 'inner_gens': [[1, 2, 6, 16, 520], [1, 2, 4, 80, 480], [3, 2, 4, 464, 96], [1, 34, 228, 16, 96], [265, 194, 4, 16, 96]], 'inner_hash': 186, 'inner_nilpotent': False, 'inner_order': 144, 'inner_split': False, 'inner_tex': 'S_3^2:C_2^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 24], [8, 2]], 'label': '576.8409', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'D6^2.C2^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 54, 'number_divisions': 46, 'number_normal_subgroups': 97, 'number_subgroup_autclasses': 236, 'number_subgroup_classes': 578, 'number_subgroups': 3570, 'old_label': None, 'order': 576, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 103], [3, 8], [4, 216], [6, 104], [12, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 52, 16, 144], [49, 10, 292, 16, 96], [1, 2, 4, 88, 480], [49, 2, 4, 80, 480], [1, 2, 52, 80, 480], [1, 2, 14, 16, 512], [1, 2, 12, 80, 480]], 'outer_group': '128.1755', 'outer_hash': 1755, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 10, 'outer_perms': [375169, 768385, 1270440, 1, 847440, 1540134, 848928], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4], [4, 20], [8, 6]], 'representations': {'PC': {'code': 143067409401584715270586789041794112675730437, 'gens': [1, 2, 3, 5, 7], 'pres': [8, 2, 2, 2, 2, 2, 3, 2, 3, 146, 66, 1612, 4660, 116, 1549, 3093, 29126, 13454, 166, 28679, 12303]}, 'Perm': {'d': 16, 'gens': [1321086798384, 479861575, 1275487, 167280, 19680257886, 1275486, 2809384300800, 2896083590400]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_6^2.C_2^2', 'transitive_degree': 48, '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': 3, 'aut_exponent': 24, 'aut_gen_orders': [6, 8, 12, 4, 12], 'aut_gens': [[1495523197009, 2628844611967, 127382791], [1570646631048, 7864727656687, 20677276494], [114082577670, 6819153883926, 1321572695161], [374702630424, 6557532282006, 1321572680305], [3152349451728, 3140334495486, 87302045191], [1495523215129, 1495490187721, 355938108127]], 'aut_group': None, 'aut_hash': 6408924249735220887, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 110592, 'aut_permdeg': 288, 'aut_perms': [452607861457347432156723539432824757739815145011514219961072806199321702056919344278881590806133838221116547190393719993517735416584273202697969534745425583738210343803771934461217396052864167926682412013280159379858026933411401919841438890567457206636668222834261576300106941966679656271212870400713624306376429970068173806182031002234667592324797949294569626408548921805629751269160779991144760826840568664279209215030199678162398972283754189854559068876646407494008898294230479901166474503524686709386034579939152917839758724169327749128946602431451315747288345115837362332854091352, 642620112992599224239077406446806758672530025161358738309020117225408038520253863841871148068472166747517063057052411595232328638636460990373730534399305252338366346144958768980784171188390274053445228337708017497697043360676709619878209417883247619130554388524067426472631314923651935324938340904861456072884353292135357150625694384300888569792320772884280510856610739493070222809045433810004122490757318198332213654239952085585712877649208731840188849973262833752486685758998405089853577149294148527678054674063869693370692077282562314806278243691202966356083526759779465165611219616, 649946499542252777110612172790034657068939383181931955340459926076109680194297970003308078867575230745423158518997131859979478529863710153297067943667687273650428886489365647555975524173810455098364568610935904971036646987541212142745396648735337861613532284342487083202066636289079712208936856646526694696697163128114324540049301632062491140582218836302081760102151200136059586709593622623447995994943259142379309089399831376150049850984044750186452467261701881978497001133090351063242893286519193455876430744995033028028026137989826872249774004488742161342283916999871578997357435544, 502574829215299008165138921001346018465302514412119939843319569620001157458961627517976331579746214406296438705853592535485605813276250531315375652948777457889843058760066862688061635255123713132467447923494554723298899980968849581418680518990321264762324375581495678729846295689793313043360908422851637241929201937840736855076235856128864734332414129702164303506601203751495909379679608397972614806859517706054238198722613493747501068556038833870047045998133740750965224987107858423874961656065842949480741520914063879522681321736758132970258581551033974692508843550314086945159661381, 42443483987136443106285725916017559010001033686587628696135284454003698576491988779313264020973994666852427358215940018543970224873887006423205632672853852037498942909387264481622049465255451908384542166118777881260989064152899353008085747932282229556572200031675114908396689822853329834768604677320692125265533865769784549484493816087348691188845216870536522718499383370175389330810178495565360034801596667528562471302250483959552969367552184551194172689606813998567148313792222136950965733115640946121169747682869562523009776035923578786338543477562612307841511198401767198316552406], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 2, 2], [2, 9, 1, 2], [2, 27, 1, 2], [2, 54, 2, 2], [2, 72, 4, 1], [3, 4, 2, 1], [3, 32, 1, 1], [3, 128, 2, 1], [4, 72, 2, 1], [4, 72, 4, 1], [4, 72, 8, 1], [4, 216, 2, 1], [6, 4, 2, 1], [6, 12, 2, 2], [6, 12, 4, 2], [6, 24, 2, 2], [6, 32, 1, 1], [6, 128, 2, 1], [6, 144, 4, 1], [6, 288, 1, 2], [12, 144, 4, 1], [12, 144, 8, 1], [12, 288, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.(C_2\\times A_4).C_2^6.C_2', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 1178895800339485646, 'autcentquo_nilpotent': False, 'autcentquo_order': 27648, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(C_2^2\\times C_6^2).D_6:\\SD_{16}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [2, 6, 4], [2, 9, 2], [2, 27, 2], [2, 54, 4], [2, 72, 4], [3, 4, 2], [3, 32, 1], [3, 128, 2], [4, 72, 14], [4, 216, 2], [6, 4, 2], [6, 12, 12], [6, 24, 4], [6, 32, 1], [6, 128, 2], [6, 144, 4], [6, 288, 2], [12, 144, 12], [12, 288, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '3456.dd', 'commutator_count': 1, 'commutator_label': '864.4714', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 131, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3456.dd', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 1, 4], [2, 9, 1, 2], [2, 27, 1, 2], [2, 54, 1, 4], [2, 72, 1, 4], [3, 4, 1, 2], [3, 32, 1, 1], [3, 128, 1, 2], [4, 72, 1, 14], [4, 216, 1, 2], [6, 4, 1, 2], [6, 12, 1, 12], [6, 24, 1, 4], [6, 32, 1, 1], [6, 128, 1, 2], [6, 144, 1, 4], [6, 288, 1, 2], [12, 144, 1, 12], [12, 288, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 806400, 'exponent': 12, 'exponents_of_order': [8, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 8], [24, 1, 2]], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '1728.47870', 'hash': 190100191912175991, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 12, 6], 'inner_gens': [[1495523197009, 6906325285711, 9273079456831], [6887647853449, 2628844611967, 444121585945], [1515125611567, 2648403497089, 127382791]], 'inner_hash': 6847276218911204980, 'inner_nilpotent': False, 'inner_order': 3456, 'inner_split': False, 'inner_tex': '(C_2^2\\times C_6^2):D_{12}', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 10], [3, 8], [4, 8], [6, 18], [8, 4], [12, 24], [24, 4]], 'label': '6912.fa', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': None, 'name': '(C2^3*C6^2):D12', 'ngens': 3, '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': 35, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 84, 'number_divisions': 82, 'number_normal_subgroups': 48, 'number_subgroup_autclasses': 1961, 'number_subgroup_classes': 6042, 'number_subgroups': 189952, 'old_label': None, 'order': 6912, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 607], [3, 296], [4, 1440], [6, 1688], [12, 2880]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[6569506949329, 6569634320766, 9603107913990], [1495523208367, 2628844621494, 127398246], [2648523605166, 2628844621495, 4103606734566], [4029964228950, 2628844612128, 2789792417761]], 'outer_group': '32.27', 'outer_hash': 27, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [107, 82, 21821, 34938], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\wr C_2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [3, 8], [4, 10], [6, 18], [8, 4], [12, 24], [24, 4]], 'representations': {'PC': {'code': '281887555310655754646127905806530467087014652245384043370601358693251019974509686732632162000490574930236404730409783572101272525991837761190056', 'gens': [1, 2, 5, 7, 9, 10, 11], 'pres': [11, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3168, 11045, 56, 81446, 90, 152771, 347164, 20475, 90776, 9112, 158, 25349, 20608, 1611, 510054, 245801, 55468, 26373, 226, 29586, 12701, 235243, 96258, 58847, 285129, 23780, 166351, 59442, 261370, 313653, 169916, 32713]}, 'Perm': {'d': 16, 'gens': [1495523197009, 2628844611967, 127382791]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_2^3\\times C_6^2):D_{12}', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
