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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '51840.bb', 'ambient_counter': 28, 'ambient_order': 51840, 'ambient_tex': 'F_9:S_6', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 72, 'counter': 72, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '51840.bb.40.j1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '40.j1.a1', 'outer_equivalence': False, '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': 40, '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': '1296.3513', 'subgroup_hash': 3513, 'subgroup_order': 1296, 'subgroup_tex': 'C_3^4:\\SD_{16}', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '51840.bb', 'aut_centralizer_order': None, 'aut_label': '40.j1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '17280.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['20.a1.a1'], 'contains': ['80.f1.a1', '80.h1.a1', '80.i1.a1', '120.l1.a1', '120.o1.a1', '360.j1.a1'], 'core': '720.a1.a1', 'coset_action_label': None, 'count': 20, 'diagramx': None, 'generators': [193117491546, 224189, 362608165062, 275505928124, 321987, 166153, 80884, 157281], 'label': '51840.bb.40.j1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '20.a1.a1', 'old_label': '40.j1.a1', 'projective_image': '51840.bb', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '40.j1.a1', 'subgroup_fusion': None, 'weyl_group': '864.4662'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [3, 6, 3, 6, 2, 2, 2, 4, 4], 'aut_gens': [[1, 6, 144, 432], [145, 438, 144, 432], [145, 642, 144, 864], [1, 582, 144, 432], [53, 582, 144, 432], [293, 726, 288, 864], [5, 6, 144, 432], [1, 102, 288, 864], [1121, 1266, 720, 576], [1193, 546, 576, 1008]], 'aut_group': '1728.47860', 'aut_hash': 47860, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1728, 'aut_permdeg': 56, 'aut_perms': [5899746938309844084571946529470876729715781592974422756653187576812275942, 5986051588959757439694253050201929821213135597501901477203568449958513276, 128491637230925286720910843580901641886609335869812740213184144897383778, 52543316616065065026120338281545635435361601223254501901612369029105557978, 58181322873863424704257038376835348536012542132631264329205216181037444932, 51020812653431576053775365807339341222821228403663362687238144000000000000, 202052265270140321548489841122194987730586082949624562149601863997867065, 62442898036624301329923241890089317466128317309537582361388537473807622644, 52233106368122732672719022779518627311210840036133037698950802443866218250], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 36, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [3, 16, 1, 1], [3, 16, 2, 1], [4, 18, 1, 1], [4, 108, 1, 1], [6, 9, 2, 1], [6, 18, 1, 1], [6, 18, 2, 1], [6, 36, 2, 1], [6, 72, 1, 1], [6, 72, 2, 1], [8, 18, 1, 2], [12, 18, 2, 2], [12, 18, 4, 1], [12, 108, 2, 1], [24, 18, 2, 4], [24, 18, 4, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times F_9:D_6', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '864.4662', 'autcentquo_hash': 4662, 'autcentquo_nilpotent': False, 'autcentquo_order': 864, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9:D_6', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 36, 1], [3, 1, 2], [3, 2, 3], [3, 8, 3], [3, 16, 3], [4, 18, 1], [4, 108, 1], [6, 9, 2], [6, 18, 3], [6, 36, 2], [6, 72, 3], [8, 18, 2], [12, 18, 8], [12, 108, 2], [24, 18, 16]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '432.740', 'commutator_count': 1, 'commutator_label': '108.36', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3513, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['432.740', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 36, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [3, 16, 1, 1], [3, 16, 2, 1], [4, 18, 1, 1], [4, 108, 1, 1], [6, 9, 2, 1], [6, 18, 1, 1], [6, 18, 2, 1], [6, 36, 2, 1], [6, 72, 1, 1], [6, 72, 2, 1], [8, 18, 2, 1], [12, 18, 2, 2], [12, 18, 4, 1], [12, 108, 2, 1], [24, 18, 4, 2], [24, 18, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 192, 'exponent': 24, 'exponents_of_order': [4, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1296.3513', 'hash': 3513, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 24, 3, 3], 'inner_gens': [[1, 66, 288, 432], [85, 6, 1152, 1008], [289, 726, 144, 432], [1, 1158, 144, 432]], 'inner_hash': 740, 'inner_nilpotent': False, 'inner_order': 432, 'inner_split': True, 'inner_tex': 'F_9:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': 16, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 32, 'irrep_stats': [[1, 12], [2, 33], [8, 6], [16, 3]], 'label': '1296.3513', 'linC_count': 128, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 20, 'linQ_dim': 12, 'linQ_dim_count': 20, 'linR_count': 48, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^4: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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 26, 'number_conjugacy_classes': 54, 'number_divisions': 26, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 133, 'number_subgroup_classes': 136, 'number_subgroups': 1648, 'old_label': None, 'order': 1296, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 45], [3, 80], [4, 126], [6, 360], [8, 36], [12, 360], [24, 288]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 108], 'outer_gens': [[5, 102, 144, 432], [37, 102, 576, 720]], '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': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 7], [4, 5], [8, 5], [16, 4], [32, 1]], 'representations': {'PC': {'code': 332479808716779663092689543522142568138236271799694204231, 'gens': [1, 3, 7, 8], 'pres': [8, -2, -3, -2, -2, -2, -3, -3, 3, 16, 1586, 66, 4227, 91, 4804, 116, 4613, 16134, 10774, 4062, 710, 10775, 799, 2343]}, 'Perm': {'d': 15, 'gens': [12944373256, 21247752960, 325, 24922679040, 13898309040, 101556408240, 207614615040, 435]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:\\SD_{16}', '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [6, 8, 12, 24, 2], 'aut_gens': [[224189, 45507, 63975, 219574, 93924338623, 3812640, 257443], [281995, 80884, 68681, 138690, 461761276636, 998277573, 232179], [224189, 321987, 63975, 184197, 287484530623, 375138106335, 257443], [40892, 357364, 298904, 321987, 7188891606, 442641064349, 188355], [317372, 184197, 342806, 219574, 354944007276, 462320092201, 254656], [40892, 80884, 152824, 321987, 105859617779, 1480550998, 349603]], 'aut_group': None, 'aut_hash': 4648263275506320880, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 207360, 'aut_permdeg': 378, 'aut_perms': [64651589012243086384211735481935523247498367359348841224668525226069569399295919488492273640351869624950783491082707127135635752506822898403746159772142078604082281582076781527893892535026712967487939953809946198931129102767574188616092460888414236923247618206531139851697192209628283743915799418782415656024639241209285972471581177335076628048757237722752795309446156252810671675607691977426203546966044665463016200481813909751461194559057202080807632334804293401935196992667636009417179620173552140728980074404029079547910492320762953001301184979910611571839134228611775630711843703172845700668808616173578933530069728668591036047964507768852650710485671940816730773370697056236427984265182335084326703058498718525646336431558441888565674120843945345396033914643871217140924694490197475227033151602948930647199, 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41069697848656119108396398973307149552850334353582540624009600762298919340081400904729613780030472080088942567283366452805404911429335486476280253372436466228500486600699170141310844792163758863990274286208194814510914151499397722487620709676685697060778031962701236796454340171382002054785323358788972871411939949226358468317364097552012738822430577351414488003676389301964651709982090264800003072894834981527207106308919201925143772985717191057338111577365843872809584411126793868837242211264073826058249238914067437602603136289745396648578120316021005499045955460746929654968045955564366954126688820320835301077931740829206092824161100296931767605633147014929924152561448016497747078024331090675437050673036255109892179151155350273982001629475508400842457677991868383757930982927556973060931327753951225961855], 'aut_phi_ratio': 15.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 45, 1, 1], [2, 180, 2, 1], [2, 405, 1, 1], [3, 8, 1, 1], [3, 40, 2, 1], [3, 320, 2, 1], [4, 18, 1, 1], [4, 90, 1, 1], [4, 540, 2, 1], [4, 810, 1, 2], [4, 1080, 1, 1], [4, 1620, 1, 1], [4, 3240, 1, 1], [5, 144, 1, 1], [6, 360, 1, 1], [6, 360, 2, 2], [6, 1440, 2, 1], [6, 2880, 2, 1], [8, 18, 1, 2], [8, 810, 1, 2], [8, 1620, 1, 2], [10, 1296, 1, 1], [12, 720, 1, 1], [12, 720, 2, 1], [12, 2160, 1, 1], [12, 4320, 2, 1], [15, 1152, 1, 1], [20, 1296, 2, 1], [24, 720, 2, 2], [40, 1296, 2, 2]], 'aut_supersolvable': False, 'aut_tex': '\\PSU(3,2):C_2.A_6.C_2^2', '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': 120, 'autcentquo_group': None, 'autcentquo_hash': 4648263275506320880, 'autcentquo_nilpotent': False, 'autcentquo_order': 207360, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PSU(3,2):C_2.A_6.C_2^2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 45, 1], [2, 180, 2], [2, 405, 1], [3, 8, 1], [3, 40, 2], [3, 320, 2], [4, 18, 1], [4, 90, 1], [4, 540, 2], [4, 810, 2], [4, 1080, 1], [4, 1620, 1], [4, 3240, 1], [5, 144, 1], [6, 360, 5], [6, 1440, 2], [6, 2880, 2], [8, 18, 2], [8, 810, 2], [8, 1620, 2], [10, 1296, 1], [12, 720, 3], [12, 2160, 1], [12, 4320, 2], [15, 1152, 1], [20, 1296, 2], [24, 720, 4], [40, 1296, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '51840.bb', 'commutator_count': 1, 'commutator_label': '12960.bb', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '360.118'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 28, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 45, 1, 1], [2, 180, 1, 2], [2, 405, 1, 1], [3, 8, 1, 1], [3, 40, 1, 2], [3, 320, 1, 2], [4, 18, 1, 1], [4, 90, 1, 1], [4, 540, 1, 2], [4, 810, 1, 2], [4, 1080, 1, 1], [4, 1620, 1, 1], [4, 3240, 1, 1], [5, 144, 1, 1], [6, 360, 1, 5], [6, 1440, 1, 2], [6, 2880, 1, 2], [8, 18, 2, 1], [8, 810, 2, 1], [8, 1620, 2, 1], [10, 1296, 1, 1], [12, 720, 1, 3], [12, 2160, 1, 1], [12, 4320, 1, 2], [15, 1152, 1, 1], [20, 1296, 2, 1], [24, 720, 2, 2], [40, 1296, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2544, 'exponent': 120, 'exponents_of_order': [7, 4, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[40, 1, 4], [72, 1, 2], [80, 1, 2], [128, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '51840.bb', 'hash': 7027567246814722481, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 120, 'inner_gen_orders': [2, 3, 4, 3, 40, 2, 8], 'inner_gens': [[224189, 80884, 63975, 321987, 93924338623, 3812640, 257443], [178682, 45507, 294198, 219574, 93924435998, 3812640, 73250], [224189, 184197, 63975, 138690, 93924338623, 3669149, 257443], [5515, 45507, 20073, 219574, 93924263725, 3629398, 27320], [224189, 321987, 63975, 184197, 93924338623, 442641043935, 257443], [224189, 45507, 166153, 276480, 93968111559, 3812640, 157281], [224189, 276480, 63975, 80884, 93924338623, 3838993, 257443]], 'inner_hash': 7027567246814722481, 'inner_nilpotent': False, 'inner_order': 51840, 'inner_split': True, 'inner_tex': 'F_9:S_6', 'inner_used': [1, 2, 3, 5, 6], 'irrC_degree': 40, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 40, 'irrep_stats': [[1, 4], [2, 3], [5, 8], [8, 2], [9, 4], [10, 10], [16, 8], [18, 3], [20, 3], [40, 4], [72, 2], [80, 2], [128, 1]], 'label': '51840.bb', 'linC_count': 16, 'linC_degree': 13, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 13, 'linQ_degree_count': 16, 'linQ_dim': 13, 'linQ_dim_count': 16, 'linR_count': 16, 'linR_degree': 13, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'F9:S6', 'ngens': 7, '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, 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, 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': 13, 'number_conjugacy_classes': 54, 'number_divisions': 45, 'number_normal_subgroups': 13, 'number_subgroup_autclasses': 732, 'number_subgroup_classes': 1156, 'number_subgroups': 320286, 'old_label': None, 'order': 51840, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 819], [3, 728], [4, 8748], [5, 144], [6, 10440], [8, 4896], [10, 1296], [12, 12960], [15, 1152], [20, 2592], [24, 2880], [40, 5184]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [181545401353, 0], 'outer_gens': [[224189, 321987, 63975, 184197, 374622905023, 287883610335, 257443], [224189, 80884, 166153, 138690, 93924565959, 3812640, 157281]], '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': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1], [5, 8], [8, 2], [9, 4], [10, 6], [16, 2], [18, 1], [20, 3], [32, 1], [36, 1], [40, 5], [64, 1], [72, 2], [80, 2], [128, 1]], 'representations': {'Perm': {'d': 15, 'gens': [224189, 45507, 63975, 219574, 93924338623, 3812640, 257443]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'F_9:S_6', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
