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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '10368.rp', 'ambient_counter': 458, 'ambient_order': 10368, 'ambient_tex': 'A_4^2:\\SOPlus(4,2)', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 12, 'counter': 1336, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '10368.rp.432.ct1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '432.ct1', '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': 432, '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': '24.13', 'subgroup_hash': 13, 'subgroup_order': 24, 'subgroup_tex': 'C_2\\times A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '10368.rp', 'aut_centralizer_order': None, 'aut_label': '432.ct1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '216.fi1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['144.cc1', '144.ce1', '144.ch1', '216.cy1', '216.ed1', '216.eg1', '216.ej1', '216.ek1'], 'contains': ['864.a1', '1296.bk1', '1728.n1'], 'core': '864.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [8057, -1, 3875, -1], 'generators': [86, 18720979200, 19678982400, 13412044800], 'label': '10368.rp.432.ct1', 'mobius_quo': None, 'mobius_sub': -24, 'normal_closure': '48.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1', 'old_label': '432.ct1', 'projective_image': '10368.rp', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '432.ct1', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2], 'aut_gens': [[127, 576, 7, 126], [127, 304, 121, 126], [127, 576, 121, 7], [127, 702, 7, 126], [127, 583, 7, 126]], 'aut_group': '24.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 4, 'aut_perms': [2, 4, 16, 7], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 4, 2, 1], [6, 4, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4', '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': 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, 4, 2], [6, 4, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 4, 2, 1], [6, 4, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '24.13', 'hash': 13, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 3, 2, 2], 'inner_gens': [[127, 576, 7, 126], [127, 576, 126, 121], [127, 702, 7, 126], [127, 697, 7, 126]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [2, 3], 'irrC_degree': 3, 'irrQ_degree': 3, 'irrQ_dim': 3, 'irrR_degree': 3, 'irrep_stats': [[1, 6], [3, 2]], 'label': '24.13', 'linC_count': 1, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 1, 'linQ_dim': 3, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*A4', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 8, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 26, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [3, 8], [6, 8]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[127, 304, 126, 7]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [3, 2]], 'representations': {'PC': {'code': 5789945620, 'gens': [1, 3, 4], 'pres': [4, -2, -3, -2, 2, 8, 218, 78, 99, 151]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [14854, 7404]}, 'Lie': [{'d': 1, 'q': 8, 'gens': [19941, 186, 5381, 12289], 'family': 'ASigmaL'}], 'GLFp': {'d': 3, 'p': 3, 'gens': [16120, 18391, 13286, 6922]}, 'Perm': {'d': 6, 'gens': [127, 576, 7, 126]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times A_4', 'transitive_degree': 6, 'wreath_data': ['12.b1.a1', '8.a1.a1', '24.a1.a1', '3T1'], 'wreath_product': True}
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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': 12, 'aut_gen_orders': [6, 12, 2, 3, 6, 4, 12, 2, 2, 2, 2, 2, 3], 'aut_gens': [[19160870858, 771120, 1174320, 13412044800, 367920, 18720979200, 19678982400, 19161193395, 520087944, 13453131573, 6267340886], [958371468, 1093680, 806400, 19678982400, 1174320, 7224940800, 13412044800, 958452085, 12973403784, 19639509692, 6267341244], [6707197178, 771120, 367920, 19678982400, 806400, 19160064000, 13412044800, 6707151795, 18722068230, 13453050383, 6268031619], [19161238778, 1128960, 806400, 13412044800, 367920, 18720979200, 19678982400, 19160835555, 519730104, 13452773733, 6267386246], [19160432378, 771120, 806400, 13412044800, 1174320, 18720979200, 19678982400, 19161193395, 519362184, 13452405813, 6267018326], [7225309045, 403200, 806400, 13412044800, 1174320, 12972960000, 19678982400, 7225666908, 12454772790, 19639796938, 6268031391], [6706390778, 725760, 1174320, 6266937600, 806400, 518918400, 19678982400, 6706425948, 12973368320, 18681471333, 13412493625], [6706829258, 80640, 1174320, 19678982400, 367920, 19160064000, 13412044800, 6706068195, 18722148870, 13453131573, 6268031545], [19160870858, 403200, 1174320, 13412044800, 367920, 18720979200, 19678982400, 19160790195, 519730104, 13452773733, 6267708806], [12454848458, 771120, 1174320, 13412044800, 367920, 958003200, 19678982400, 12455170995, 7226110344, 19640235573, 6267340886], [6706829258, 771120, 1174320, 13412044800, 367920, 7224940800, 6266937600, 6707151795, 959172744, 12495128373, 19679385686], [19160870858, 1093680, 1174320, 13412044800, 367920, 18720979200, 19678982400, 19160512995, 519730104, 13452773733, 6267018326], [519725258, 771120, 1174320, 13412044800, 367920, 6706022400, 19678982400, 520047795, 19161233544, 13453131573, 6267340886], [7225747658, 771120, 1174320, 19678982400, 367920, 6706022400, 6266937600, 7226070195, 520087944, 18682232373, 13412448086]], 'aut_group': '41472.kk', 'aut_hash': 6814181777269209381, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 41472, 'aut_permdeg': 144, 'aut_perms': [4929158283788984380440344713904346691750168945248275595148954493708392571580929230249158705028363512126864289807847559930253063973194553665346660132988313159438705969444239394923826470344627051301669881361528204150446800455471372215089395880559773971, 4043229555553745919208681476512357917593369288641391964027817733732241065889992970397597716829190768727168415732092372224167891759170770693941583171734014274880047438177418264421867679794278025936613302689596524886592442204250530052500038738395728091, 142051349352990037371253503370276332089146445362374064665996137333491248478922714284692260684368149224502935889488113365796281121722077956258581523502091870411259996184416956186094548274166156388066715839160282765046267616320423258581491695502870, 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5162973831646263232893952597279789609328023572718735572954760745747673563289633283035489971595445411422271670652146131358826670020924025725650655311471793887623198770525238171199025089799760456679533475205325803975072651322079674820494842295233262600, 402765772213643946072460087847705401171970016951911649086517393294598221393445668729093360344225996930884320556448179120660529975874695411766739326097476338670779255433798936641828217162885804208771230400948053187906615584136364160000, 3363112081656652871693706807619997094138395553067876984570061321688522808792412520060897057180742432856557062077077808263627473670973012211302042024719347283221525755665727006001705558076469817869345829445653397254486537354035360488315252493, 1319654107473043670899237949088435635620591873114974390002160197665122762049608745186861615572203727267801396365370948193451213898804833140701670340914274603155051733657419769724868191567291483200927566379016026843274659633126646527856673583058710920], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 2], [2, 9, 1, 2], [2, 27, 1, 2], [2, 36, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [2, 216, 1, 1], [3, 4, 1, 2], [3, 8, 1, 2], [3, 16, 2, 1], [3, 32, 1, 3], [3, 64, 1, 1], [3, 128, 2, 2], [4, 36, 1, 1], [4, 108, 1, 3], [4, 216, 1, 3], [4, 324, 1, 2], [6, 12, 1, 4], [6, 24, 1, 2], [6, 36, 1, 2], [6, 48, 2, 1], [6, 72, 1, 3], [6, 96, 1, 3], [6, 144, 2, 1], [6, 216, 1, 3], [6, 288, 2, 1], [6, 432, 1, 1], [6, 576, 1, 1], [12, 72, 1, 1], [12, 144, 2, 1], [12, 216, 1, 1], [12, 288, 2, 1], [12, 432, 1, 3], [12, 432, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'S_4^2.\\SOPlus(4,2)', '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': 12, 'autcentquo_group': '41472.kk', 'autcentquo_hash': 6814181777269209381, 'autcentquo_nilpotent': False, 'autcentquo_order': 41472, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4^2.\\SOPlus(4,2)', 'cc_stats': [[1, 1, 1], [2, 3, 2], [2, 9, 2], [2, 27, 2], [2, 36, 1], [2, 81, 1], [2, 108, 1], [2, 216, 1], [3, 4, 2], [3, 8, 2], [3, 16, 2], [3, 32, 3], [3, 64, 1], [3, 128, 4], [4, 36, 1], [4, 108, 3], [4, 216, 3], [4, 324, 2], [6, 12, 4], [6, 24, 2], [6, 36, 2], [6, 48, 2], [6, 72, 3], [6, 96, 3], [6, 144, 2], [6, 216, 3], [6, 288, 2], [6, 432, 1], [6, 576, 1], [12, 72, 1], [12, 144, 2], [12, 216, 1], [12, 288, 2], [12, 432, 7]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '10368.rp', 'commutator_count': 1, 'commutator_label': '2592.oh', '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', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 458, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 2], [2, 9, 1, 2], [2, 27, 1, 2], [2, 36, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [2, 216, 1, 1], [3, 4, 1, 2], [3, 8, 1, 2], [3, 16, 2, 1], [3, 32, 1, 3], [3, 64, 1, 1], [3, 128, 1, 2], [3, 128, 2, 1], [4, 36, 1, 1], [4, 108, 1, 3], [4, 216, 1, 3], [4, 324, 1, 2], [6, 12, 1, 4], [6, 24, 1, 2], [6, 36, 1, 2], [6, 48, 2, 1], [6, 72, 1, 3], [6, 96, 1, 3], [6, 144, 2, 1], [6, 216, 1, 3], [6, 288, 2, 1], [6, 432, 1, 1], [6, 576, 1, 1], [12, 72, 1, 1], [12, 144, 2, 1], [12, 216, 1, 1], [12, 288, 2, 1], [12, 432, 1, 3], [12, 432, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 216, 'exponent': 12, 'exponents_of_order': [7, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[36, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '10368.rp', 'hash': 1838968437832093872, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 3, 2, 2, 2, 3, 2, 3, 12, 4, 6], 'inner_gens': [[19160870858, 403200, 1174320, 19678982400, 367920, 958003200, 6266937600, 19160790195, 12454853430, 6746750783, 13412816259], [19160432378, 771120, 806400, 13412044800, 1174320, 18720979200, 19678982400, 19161193395, 519362184, 13452405813, 6267018326], [19160870858, 1093680, 1174320, 13412044800, 367920, 18720979200, 19678982400, 19160512995, 519730104, 13452773733, 6267018326], [12454848458, 771120, 1174320, 13412044800, 367920, 958003200, 19678982400, 12455170995, 7226110344, 19640235573, 6267340886], [19160870858, 80640, 1174320, 13412044800, 367920, 18720979200, 19678982400, 19160109795, 520087944, 13453131573, 6268031366], [519725258, 771120, 1174320, 6266937600, 367920, 18720979200, 13412044800, 520047795, 12455211144, 7186193973, 19679385686], [7225747658, 771120, 1174320, 13412044800, 367920, 12972960000, 19678982400, 7226070195, 12455211144, 19640235573, 6267340886], [19161238778, 771120, 367920, 19678982400, 806400, 958003200, 6266937600, 19161193395, 12455130630, 6747028533, 13413138745], [7226115468, 448560, 806400, 19678982400, 367920, 6706022400, 6266937600, 7226034805, 520087944, 18682232252, 13413174204], [6707197045, 448560, 806400, 19678982400, 367920, 19160064000, 13412044800, 6707116515, 18722148481, 13453131573, 6268066646], [519286645, 1093680, 806400, 13412044800, 1174320, 6706022400, 19678982400, 519367308, 19160507910, 13452405813, 6267340886]], 'inner_hash': 1838968437832093872, 'inner_nilpotent': False, 'inner_order': 10368, 'inner_split': True, 'inner_tex': 'A_4^2:\\SOPlus(4,2)', 'inner_used': [1, 2, 4, 6, 8, 9, 10], 'irrC_degree': 36, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 36, 'irrep_stats': [[1, 4], [2, 9], [3, 8], [4, 10], [6, 10], [8, 7], [9, 4], [12, 12], [18, 1], [24, 3], [36, 4]], 'label': '10368.rp', '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': 'A4^2:SO+(4,2)', 'ngens': 11, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 62, 'number_characteristic_subgroups': 35, 'number_conjugacy_classes': 72, 'number_divisions': 63, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 1986, 'number_subgroup_classes': 2154, 'number_subgroups': 120036, 'old_label': None, 'order': 10368, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 519], [3, 728], [4, 1656], [6, 3288], [12, 4176]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[12973766858, 771120, 1174320, 19678982400, 367920, 12454041600, 13412044800, 12974089395, 959172744, 19640235573, 6267340886], [19161238778, 1128960, 806400, 13412044800, 367920, 18720979200, 19678982400, 19160835555, 519730104, 13452773733, 6267386246]], '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': 6, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [3, 8], [4, 8], [6, 6], [8, 7], [9, 4], [12, 10], [16, 1], [18, 1], [24, 5], [36, 4]], 'representations': {'PC': {'code': '134355869908109786860574259462214790236194036503626508284525621278176580088639627111531285867215912311185173007842152505245462230258525071498410113193', 'gens': [1, 2, 5, 7, 9, 10], 'pres': [11, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 1320, 136181, 56, 3038, 90, 231971, 36975, 71636, 36337, 158, 188512, 781710, 174653, 115066, 2118, 21302, 1678, 226, 633607, 316818, 12701, 641528, 26782, 11646, 958329, 495020, 97051, 24792, 328, 836362, 453045, 209120]}, 'Perm': {'d': 14, 'gens': [19160870858, 771120, 1174320, 13412044800, 367920, 18720979200, 19678982400, 19161193395, 520087944, 13453131573, 6267340886]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_4^2:\\SOPlus(4,2)', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
