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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '512.7530050', 'ambient_counter': 7530050, 'ambient_order': 512, 'ambient_tex': 'D_4^2:C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 32, 'counter': 59, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '512.7530050.8.bc1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.bc1', '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': 8, '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': '64.90', 'subgroup_hash': 90, 'subgroup_order': 64, 'subgroup_tex': 'C_2^4:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '512.7530050', 'aut_centralizer_order': None, 'aut_label': '8.bc1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '64.c1', 'complements': None, 'conjugacy_class_count': 24, 'contained_in': ['4.a1', '4.l1', '4.u1'], 'contains': ['16.b1', '16.bc1', '16.bd1', '16.ed1'], 'core': '16.b1', 'coset_action_label': None, 'count': 48, 'diagramx': [4308, -1, 4317, -1], 'generators': [83629560, 207290280, 94525320, 18669007, 94525336, 139725360], 'label': '512.7530050.8.bc1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1', 'old_label': '8.bc1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.bc1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[385206, 849655, 1, 167280, 26334, 1275486], [1549495, 849655, 1, 167280, 188166, 1275486], [455167, 849654, 1, 1154310, 1260120, 1275486], [1834944, 2799055, 1275487, 1154310, 1260120, 1275486], [1956366, 2718175, 1275487, 167280, 26334, 1275486], [1951560, 849655, 1275487, 167280, 26334, 1275486], [1829527, 849655, 1, 167280, 1260120, 1275486], [466200, 849655, 1, 167280, 1260120, 1275486], [1951560, 2799055, 1, 167280, 26334, 1275486], [1951560, 849655, 1, 167280, 26334, 1275486]], 'aut_group': '512.6407070', 'aut_hash': 3144020168943847438, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 16, 'aut_perms': [1329869992338, 20553834452565, 6100576858833, 38151148566, 4720369622400, 12933144595414, 14990504486051, 4047851006169, 4751527303459], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 1, 2], [2, 2, 2, 2], [2, 4, 2, 1], [4, 4, 2, 1], [4, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^6:C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '16.11', 'autcentquo_hash': 11, 'autcentquo_nilpotent': True, 'autcentquo_order': 16, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 6], [2, 4, 2], [4, 4, 10]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '16.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 90, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['32.6', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 6], [2, 4, 1, 2], [4, 4, 1, 2], [4, 4, 2, 4]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 168, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '8.5', 'hash': 90, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 2, 1, 2, 2, 1], 'inner_gens': [[385206, 930649, 1, 1154310, 26334, 1275486], [1829526, 849655, 1, 167280, 1260120, 1275486], [385206, 849655, 1, 167280, 26334, 1275486], [1951560, 849655, 1, 167280, 26334, 1275486], [385206, 2799055, 1, 167280, 26334, 1275486], [385206, 849655, 1, 167280, 26334, 1275486]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': False, 'inner_tex': 'C_2^2:C_4', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 4], [4, 2]], 'label': '64.90', 'linC_count': 16, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 8, 'linQ_dim': 5, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4:C4', 'ngens': 6, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 22, 'number_divisions': 18, 'number_normal_subgroups': 41, 'number_subgroup_autclasses': 55, 'number_subgroup_classes': 105, 'number_subgroups': 217, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 23], [4, 40]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 849655, 0], 'outer_gens': [[385206, 849655, 1275487, 167280, 26334, 1275486], [374784, 849655, 1275487, 1154310, 26334, 1275486], [1539529, 849655, 1275487, 1154310, 1134264, 1275486], [2447310, 2799054, 1275487, 1154310, 1134264, 1275486]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [23, 7, 1560, 5896], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 2]], 'representations': {'PC': {'code': 18696068175890, 'gens': [1, 2, 4, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 31, 489, 370, 376, 88]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706304316028, 706461441036]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [11040833, 29499457, 7292993, 17109059, 5128257, 18848835]}, 'Perm': {'d': 10, 'gens': [385206, 849655, 1, 167280, 26334, 1275486]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4:C_4', 'transitive_degree': 16, '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 6, 4, 4, 4, 4, 4, 2, 4, 2, 4, 2, 4], 'aut_gens': [[41141527, 90882847, 94399200, 94525336, 94525327, 139725360, 177960, 83629560, 94525320], [41141527, 90882847, 94399223, 94525327, 94525336, 139725360, 177960, 83629560, 94525320], [251522647, 90882863, 94399207, 23, 94525336, 139725360, 15049560, 83629560, 94525320], [251522647, 94399207, 90882840, 94525336, 94525327, 15049560, 139725360, 14928480, 94525320], [172046887, 83640367, 136095240, 94525336, 94525327, 15049560, 302376360, 83629560, 94525320], [172046887, 98155447, 94399200, 94525336, 94525327, 132452760, 15049560, 83629560, 94525320], [172046896, 98155447, 94399223, 94525336, 94525327, 132452760, 15049560, 83629560, 94525320], [172046880, 11290320, 167167, 16, 94525327, 132452760, 15049560, 83629560, 94525320], [207290296, 18558007, 94399200, 16, 7, 139725360, 177960, 83629560, 94525320], [41141527, 18558000, 94399216, 16, 94525327, 139725360, 177960, 83629560, 94525320], [207290296, 90882847, 167160, 94525336, 94525327, 139725360, 177960, 83629560, 94525320], [262897943, 98155463, 94399216, 94525336, 94525327, 132452760, 15049560, 83629560, 94525320], [207290280, 90882856, 167183, 94525336, 94525327, 139725360, 177960, 83629560, 94525320], [41141520, 18558000, 94399223, 94525336, 7, 139725360, 177960, 83629560, 94525320]], 'aut_group': '786432.ie', 'aut_hash': 5057471148814902162, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 786432, 'aut_permdeg': 96, 'aut_perms': [870845827671069201049378893821313739173996491645553911600909292736130827781933627267579486886151733766821430509791516176744320924563810731, 812597144008958114614617998564704522112343265782877262858207256368633343063847016545991428185530571801915609318785554308026404465696328325862874039561, 283273545640630142422918720186555967903610640649385790089321670910674444842702013809638201058323602708802610378992401654672172182959932295716027359933, 41081621266568949207908333677354880246052116257221869252949055169436905846081162150606912143305064322417604931182667298475625389571080420655951250102, 557938669782256584498878476704753707796982932457750486137166538221482498359542904545601145996486054305057649742723120732252585952646384911022343362274, 105861841575627349340060401826933645313187184724355310869522687530585016333915042568875346804603640636012845624941634110765742797290776512488062939069, 606869860515750396906850537214577644495957154665704503471627531774011959937889148349515849827997977945890034665394727826577264841765699166770348026553, 803675829417437467684468445740089268900404723528091442149732014396963097681308544613708648842981391257569205776438772004773483289951532252197673582420, 3022678104356530442361611382461857667261660942936018656484086261104904979705213581468267383729316845405681719525926730199103876374393843899469455022, 803675829290697771042062411236884350506671029609680547832406420772434200298864669655939665434442217991260167305469074785985359845416680580320012657728, 98225104409240297462316937124631441731871075096333298751311250704374652442089434132474319724161100848825172204410610370113122764495926346457672162659, 303541177792401941630643930337027271537835552093502374060443229023675147121977789813785685595601026324592989203141519849555749199767020525141848129129, 519165637686038656986436358826679955211366611826235899478622769598062514996708555714302970702826551422754127037819222043449258516292911785489129172144], 'aut_phi_ratio': 3072.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 2, 1, 1], [2, 2, 3, 1], [2, 4, 2, 1], [2, 4, 6, 1], [2, 4, 16, 1], [2, 8, 4, 2], [4, 4, 1, 1], [4, 4, 3, 1], [4, 4, 8, 1], [4, 8, 4, 1], [4, 8, 8, 1], [4, 16, 8, 1], [8, 16, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^{12}.(D_4\\times S_4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': None, 'autcent_hash': 4161364993999926539, 'autcent_nilpotent': False, 'autcent_order': 12288, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^7.C_2^4.S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': None, 'autcentquo_hash': 138, 'autcentquo_nilpotent': True, 'autcentquo_order': 64, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\wr C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 4], [2, 4, 24], [2, 8, 8], [4, 4, 12], [4, 8, 12], [4, 16, 8], [8, 16, 4]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '64.138', 'commutator_count': 1, 'commutator_label': '16.11', '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', '2.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 7530050, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['128.928', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 4], [2, 4, 1, 24], [2, 8, 1, 8], [4, 4, 1, 12], [4, 8, 1, 12], [4, 16, 1, 8], [8, 16, 1, 4]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 13332480, 'exponent': 8, 'exponents_of_order': [9], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '32.51', 'hash': 7530050, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 1, 1, 2, 2, 2, 1], 'inner_gens': [[41141527, 298746847, 94520280, 94525336, 94525327, 139725360, 177960, 14928480, 94525320], [251522647, 90882847, 94399200, 94525336, 94525327, 139725360, 15049560, 83629560, 94525320], [41091247, 90882847, 94399200, 94525336, 94525327, 302376360, 177960, 83629560, 94525320], [41141527, 90882847, 94399200, 94525336, 94525327, 139725360, 177960, 83629560, 94525320], [41141527, 90882847, 94399200, 94525336, 94525327, 139725360, 177960, 83629560, 94525320], [41141527, 90882847, 15054600, 94525336, 94525327, 139725360, 94394160, 83629560, 94525320], [41141527, 98155447, 94399200, 94525336, 94525327, 295108680, 177960, 83629560, 94525320], [207290287, 90882847, 94399200, 94525336, 94525327, 139725360, 177960, 83629560, 94525320], [41141527, 90882847, 94399200, 94525336, 94525327, 139725360, 177960, 83629560, 94525320]], 'inner_hash': 138, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': False, 'inner_tex': 'C_2\\wr C_2^2', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [4, 24]], 'label': '512.7530050', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'D4^2:C2^3', 'ngens': 9, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [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': 17, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 80, 'number_divisions': 80, 'number_normal_subgroups': 578, 'number_subgroup_autclasses': 466, 'number_subgroup_classes': 7242, 'number_subgroups': 27282, 'old_label': None, 'order': 512, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 175], [4, 272], [8, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [4, 4, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[207290296, 167183, 18558007, 7, 94525336, 177960, 139725360, 14928480, 94525320], [41141520, 167167, 90882856, 94525336, 94525327, 177960, 139725360, 14928480, 94525320], [207290280, 90882840, 167160, 16, 94525343, 139725360, 177960, 83629560, 94525320]], 'outer_group': None, 'outer_hash': 7369186840958827619, 'outer_nilpotent': False, 'outer_order': 12288, 'outer_permdeg': 20, 'outer_perms': [1270063923519864381, 1916621141797368736, 1539500538872303417], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6.C_2^4.D_6', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 24], [4, 24]], 'representations': {'PC': {'code': '21948546922097849121396062733480935605416', 'gens': [1, 2, 3, 4, 6, 8], 'pres': [9, 2, 2, 2, 2, 2, 2, 2, 2, 2, 867, 102, 2408, 365, 158, 29959, 2050, 1771, 214, 25928, 3275]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [24992906222183252, 125101724435340314, 25038659807093174, 125101740294620861, 125101750006154014]}, 'Perm': {'d': 12, 'gens': [41141527, 90882847, 94399200, 94525336, 94525327, 139725360, 177960, 83629560, 94525320]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4^2:C_2^3', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
