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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '4374.ik', 'ambient_counter': 219, 'ambient_order': 4374, 'ambient_tex': 'C_9^2.(S_3\\times C_3^2)', 'central': False, 'central_factor': False, 'centralizer_order': 81, 'characteristic': False, 'core_order': 9, 'counter': 187, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4374.ik.162.t1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '162.t1', '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': 162, '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': '27.4', 'subgroup_hash': 4, 'subgroup_order': 27, 'subgroup_tex': 'C_9:C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4374.ik', 'aut_centralizer_order': 1458, 'aut_label': '162.t1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '18.3', 'aut_weyl_index': 13122, 'centralizer': '54.i1', 'complements': None, 'conjugacy_class_count': 3, 'contained_in': ['54.c1', '54.h1', '54.bz1', '54.bz2', '81.e1'], 'contains': ['486.b1', '486.d1', '486.k1'], 'core': '486.b1', 'coset_action_label': None, 'count': 9, 'diagramx': [3708, -1, 3134, -1], 'generators': [6, 1206], 'label': '4374.ik.162.t1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '54.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.c1', 'old_label': '162.t1', 'projective_image': '4374.ik', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '162.t1', 'subgroup_fusion': None, 'weyl_group': '18.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [3, 3, 3, 2], 'aut_gens': [[1, 3], [1, 5], [19, 21], [1, 12], [19, 24]], 'aut_group': '54.8', 'aut_hash': 8, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 54, 'aut_permdeg': 9, 'aut_perms': [232440, 53286, 123858, 3838], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 1, 2], [9, 3, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2:S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '9.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 9, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 3, 2], [9, 3, 6]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [9, 3, 2, 3]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 8, 'exponent': 9, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [[3, 0, 2]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '9.2', 'hash': 4, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3], 'inner_gens': [[1, 21], [10, 3]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': True, 'inner_tex': 'C_3^2', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 9], [3, 2]], 'label': '27.4', 'linC_count': 2, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C9:C3', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 11, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 8], [9, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [2, 0], 'outer_gens': [[1, 6], [1, 5]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 1]], 'representations': {'PC': {'code': 1766, 'gens': [1, 2], 'pres': [3, -3, 3, -3, 127, 22]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [52802554844085621, 108238938226410615]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [36709477, 10475581, 23068812]}, 'Perm': {'d': 9, 'gens': [357120, 243, 80884]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9:C_3', 'transitive_degree': 9, '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': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 18, 'aut_gen_orders': [6, 9, 6], 'aut_gens': [[1, 6, 18, 162, 486], [4093, 60, 3114, 1890, 1188], [3649, 1464, 1116, 1674, 1476], [4169, 6, 3114, 3132, 774]], 'aut_group': None, 'aut_hash': 6633038601780650679, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 236196, 'aut_permdeg': 405, 'aut_perms': [52899760818452712899894150357015491532877639927414926334631036931382241519401042494803334636577654837057042128441176466383194814344313385782356314014070672425231353615975770499898021630608469540026891693554429021437079306111314514888397531067822464138162518650482542376330523701725212942100703825878365032786250472302336853419617380382064852999646067740593263493147637232083444859457057194642594674316688533620161849723328688668611699632485868354378177528015317748825869806774366401925769767659761620443213586413508070587656377440440955085568236112556779300599065866026799681434543468112146461783262659905212182246106170482850138439112382975766708935303171937357209510390514750436695433580962685141269844783994837168931404314014835059455404672802207756258533969773683048372024466382153226738254813243842120276176029420365517017512776769424361097766268309639896408835217335770251056, 482861754197295963684250399084784407586032031173500159091855089181203138695520484001925304179044421031823845082733668396100680841893737843612429283138129925205380970923733942255961807177482749487786028952605425443290491025346789490937468092369455802231093466573093051273531936938102745352994275322341695186190934412233608305995458264263854069851145726176657501615200466675726023328070849198108101795950903284576032659894623809780140426568336612557790377775750453798056857617220926529914435773028431080188967150840628706366471518924643274977392855935272465178165757300252081420144463159778796052123083511081644192774803389873928671923096314174993794011905190288007222625672996451547423476359994246129154058911585029865020065658791905303337697511228720299235659650879269123936837435373301547159182380519597383320309327268281377423241033536834059449043175657421527909396085736974281962, 152452172449985640483062307604314503335886636006709962036932509847637222787276205046039566998926731337202226627878334572002128477646498275806594682179448145350489062614849131555565442556212041870572980245530632985951075674585729864735661332496446767030621796513297562404081631085968596751761523271962576320735072617133182218588156004039682420580967675968748043344493529220319364093771026334417965604932890049736449680942123206026262636216097359975113574749980864111975693398038981246415603904663309484234058609689678679981225956562842820002496607003143245438790160205154087869516791136718029926593700843826230129343697384370506235037374151605369566460634585780839500774029417959017827897414347195781738498746045594767621918827468295017456863048724483765473943906106312543943502426922648329192072573227455422743524347483347018609391864850718066214852759420912256917267310026523388534], 'aut_phi_ratio': 162.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 6, 3, 1], [3, 9, 1, 2], [3, 18, 1, 2], [3, 27, 6, 1], [3, 54, 6, 1], [6, 243, 1, 2], [6, 243, 6, 1], [9, 6, 3, 1], [9, 18, 1, 8], [9, 18, 9, 1], [9, 54, 3, 2], [9, 54, 6, 3]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3.C_3^3.C_3^3.C_6.C_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': 18, 'autcentquo_group': None, 'autcentquo_hash': 6633038601780650679, 'autcentquo_nilpotent': False, 'autcentquo_order': 236196, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^3.C_3^3.C_3^3.C_6.C_2', 'cc_stats': [[1, 1, 1], [2, 243, 1], [3, 2, 1], [3, 6, 4], [3, 9, 2], [3, 18, 2], [3, 27, 6], [3, 54, 6], [6, 243, 8], [9, 6, 3], [9, 18, 17], [9, 54, 24]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '4374.ik', 'commutator_count': 1, 'commutator_label': '243.31', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 219, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 4], [3, 9, 2, 1], [3, 18, 2, 1], [3, 27, 2, 3], [3, 54, 2, 3], [6, 243, 2, 4], [9, 6, 1, 3], [9, 18, 1, 11], [9, 18, 2, 3], [9, 54, 2, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 157248, 'exponent': 18, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[18, 1, 9]], 'familial': False, 'frattini_label': '27.2', 'frattini_quotient': '162.52', 'hash': 8906090938690754470, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 3, 9, 3, 9], 'inner_gens': [[1, 1518, 4320, 1836, 3798], [3025, 6, 126, 162, 3402], [721, 60, 18, 162, 486], [3349, 6, 18, 162, 486], [1711, 1464, 18, 162, 486]], 'inner_hash': 8906090938690754470, 'inner_nilpotent': False, 'inner_order': 4374, 'inner_split': True, 'inner_tex': 'C_9^2.(S_3\\times C_3^2)', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 18, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 18], [2, 36], [6, 9], [18, 12]], 'label': '4374.ik', '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': 'C9^2.(S3*C3^2)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 29, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 75, 'number_divisions': 48, 'number_normal_subgroups': 55, 'number_subgroup_autclasses': 263, 'number_subgroup_classes': 621, 'number_subgroups': 11804, 'old_label': None, 'order': 4374, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 243], [3, 566], [6, 1944], [9, 1620]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [6, 3, 3], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[125, 6, 18, 1890, 2574], [121, 6, 3096, 162, 2106], [1, 2976, 3672, 3186, 4194]], 'outer_group': '54.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 54, 'outer_permdeg': 9, 'outer_perms': [247052, 286792, 160204], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:C_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 12], [4, 16], [6, 3], [12, 3], [18, 12]], 'representations': {'PC': {'code': '317726314964626159778644978085961589643125593959095887935974697447934335', 'gens': [1, 3, 4, 6, 7], 'pres': [8, 2, 3, 3, 3, 3, 3, 3, 3, 16, 36434, 18874, 138243, 21323, 691, 123, 116644, 60492, 88133, 73885, 212694, 92750, 31774, 222, 96775, 1743]}, 'Perm': {'d': 27, 'gens': [10082234824127549464004956559, 7953528059260447158105322076, 2532183841820963904798787915]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9^2.(S_3\\times C_3^2)', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
