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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '3840.ft', 'ambient_counter': 150, 'ambient_order': 3840, 'ambient_tex': 'C_2\\times \\GL(2,\\mathbb{Z}/4):F_5', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 480, 'counter': 56, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '3840.ft.8.F', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.f1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '8.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '480.1199', 'subgroup_hash': 1199, 'subgroup_order': 480, 'subgroup_tex': 'C_2^3:D_{30}', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3840.ft', 'aut_centralizer_order': 16, 'aut_label': '8.F', 'aut_quo_index': 8, 'aut_stab_index': 1, 'aut_weyl_group': '3840.fq', 'aut_weyl_index': 16, 'centralizer': '960.A', 'complements': ['480.FN', '480.JD', '480.JA', '480.FM'], 'conjugacy_class_count': 1, 'contained_in': ['4.C', '4.D', '4.K'], 'contains': ['16.C', '16.I', '16.Q', '24.DE', '32.R', '40.R'], 'core': '8.F', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [5041, 88177852086, 7, 363007, 187768650551, 180625507337, 187768627200], 'label': '3840.ft.8.F', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.F', 'normal_contained_in': ['4.C', '4.D', '4.K'], 'normal_contains': ['16.C', '16.I'], 'normalizer': '1.a1', 'old_label': '8.f1', 'projective_image': '960.11361', 'quotient_action_image': '8.2', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '8.F', 'subgroup_fusion': None, 'weyl_group': '960.11361'}
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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': 60, 'aut_gen_orders': [2, 2, 2, 3, 4, 2, 2, 3, 2, 6, 60], 'aut_gens': [[8079, 54311, 12041, 72009, 88211], [8079, 134101, 12041, 72009, 88211], [8079, 90106, 92251, 72009, 88211], [88069, 130301, 12041, 72009, 88211], [50274, 54311, 92251, 72009, 12201], [34116, 154304, 88131, 88011, 12201], [8079, 114109, 12041, 88011, 88211], [72151, 54311, 12041, 72009, 88211], [8079, 118319, 12041, 88011, 88211], [88269, 54311, 12041, 72009, 88211], [88229, 114119, 12361, 88011, 88211], [88309, 158314, 92251, 88011, 88211]], 'aut_group': '11520.fw', 'aut_hash': 8752884374094103541, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11520, 'aut_permdeg': 120, 'aut_perms': [1798743022754102287877016945737512201580627173951354329305692021264015257593217809686467671426066966132620428899802581874379896466606661626896260143313745581186243857268493408754261005605561940, 5972371765102814854572437758062027192763072507416461342812531661326299695645192575600550551708963470283305585726919458240816971224362920920546312646209598979932758834607108287458731194616641053362052, 5972369627394628640189641038405543459632813927750489950787138970702872312384348590150422732542218342957726585409634112683815778800169440347147310748297183756331411318293433988238585990860968009349148, 1113975265626300021135750746019474533972520056408549874461609555726075982761432549754571382418160326159179947184687466768106682901661125216552507598155223821348468806600330462474936767947154838208003, 534533259720302177238428438657556723101696497834850870695052896870026333868970391922659731430606164541437447200108513745818553462595157672330649851334274503128826068590222515192462509269180352625552, 1004310801502070703462019714391420660433532442556390980327579036388980287101098758027366001493511413975388006823460097442832875383898966253477939335769985680246594276657028728494877842365448200770410, 1518160526760622284849630214794811131174681460329279785351335252344170308295826921752722112814981955749760746510901701452854979813135271377079801946109152167799319707411180639434734807335017896071968, 1506023551624702602815781271741532973386830144903306501492600335461253441045038498543167437894737424794769157292849611143912989028868708702845241715349679823464043642715077154947778773157518130950092, 6248737109867419147553920989414324263279921190177517250585418483511157617962446082189158016087728039497205804767587894521461522308674037987260038285046650268164525302090628923461562600232896022677420, 765454958747035168181619460709486586570677987953543054815831596510218709022360662598474130789653005026910930086120092531391330632386857451003218227527315543389483730668753643752127944116418900429687, 2910074084254156399735450956149824835967410618941624558292866457470581415728566390609829107786685028750749452062780383589764309784639340220380054463291538098422226061162143026737982059828745404157101], 'aut_phi_ratio': 90.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 1, 1], [2, 3, 3, 1], [2, 30, 4, 1], [3, 8, 1, 1], [4, 30, 4, 1], [5, 2, 2, 1], [6, 8, 3, 1], [10, 2, 6, 1], [10, 6, 2, 1], [10, 6, 6, 1], [15, 8, 4, 1], [30, 8, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\times S_4^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '480.1189', 'autcentquo_hash': 1189, 'autcentquo_nilpotent': False, 'autcentquo_order': 480, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [2, 30, 4], [3, 8, 1], [4, 30, 4], [5, 2, 2], [6, 8, 3], [10, 2, 6], [10, 6, 8], [15, 8, 4], [30, 8, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '120.38', 'commutator_count': 1, 'commutator_label': '60.9', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1199, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['120.38', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [2, 30, 1, 4], [3, 8, 1, 1], [4, 30, 1, 4], [5, 2, 2, 1], [6, 8, 1, 3], [10, 2, 2, 3], [10, 6, 2, 4], [15, 8, 4, 1], [30, 8, 4, 3]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 2520, 'exponent': 60, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '480.1199', 'hash': 1199, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 3, 10, 1, 2], 'inner_gens': [[8079, 90106, 92171, 72009, 88211], [14179, 54311, 88051, 72009, 92011], [88389, 50111, 12041, 72009, 88211], [8079, 54311, 12041, 72009, 88211], [8079, 134101, 12041, 72009, 88211]], 'inner_hash': 38, 'inner_nilpotent': False, 'inner_order': 120, 'inner_split': True, 'inner_tex': 'C_5:S_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 28], [3, 8], [6, 8]], 'label': '480.1199', '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': 'C2^3:D30', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 52, 'number_divisions': 32, 'number_normal_subgroups': 41, 'number_subgroup_autclasses': 84, 'number_subgroup_classes': 262, 'number_subgroups': 2152, 'old_label': None, 'order': 480, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 135], [3, 8], [4, 120], [5, 4], [6, 24], [10, 60], [15, 32], [30, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 4, 4, 2, 6], 'outer_gen_pows': [8001, 8001, 8001, 8001, 8001, 8001], 'outer_gens': [[8079, 90106, 92251, 72009, 88211], [72151, 90106, 92251, 72009, 88211], [152341, 54311, 12281, 72009, 88211], [8079, 54311, 12121, 72009, 88211], [8079, 78314, 92251, 88011, 88211], [8079, 78314, 92251, 152019, 88211]], 'outer_group': '96.186', 'outer_hash': 186, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [16, 16696, 11529, 18, 10816, 5776], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [3, 8], [4, 4], [8, 4], [12, 4]], 'representations': {'PC': {'code': 86264473983076818201781500747730551088, 'gens': [1, 2, 4, 6, 7], 'pres': [7, 2, 2, 3, 2, 5, 2, 2, 141, 36, 170, 9747, 4378, 1781, 80, 3364, 7363, 755]}, 'GLZN': {'d': 2, 'p': 20, 'gens': [8201, 92011, 72009, 8081, 130311, 50274, 12201]}, 'Perm': {'d': 13, 'gens': [40284842, 5160, 11520, 3, 559198080, 7, 16]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3:D_{30}', 'transitive_degree': 60, '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': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [12, 4, 20, 4, 4, 4, 4], 'aut_gens': [[362886, 7187011321, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1], [5160, 262055416326, 19679017368, 368392147200, 367927, 348713532720, 368047, 363001, 99632695817, 363006], [126, 7186648327, 274467945305, 19678982400, 367927, 187768990086, 368047, 363001, 355898228704, 368040], [367921, 193558187527, 368392533430, 88176211200, 120, 187768632361, 368047, 363001, 99632730671, 368040], [367926, 355899818161, 274468308071, 180623520000, 120, 348713169966, 368047, 368040, 193556949535, 363006], [126, 262055784366, 274467951768, 368392147200, 367927, 93924593286, 368047, 368040, 7185426904, 363001], [362881, 262055421361, 88176239705, 368392147200, 5167, 280215936121, 5047, 5040, 355898580169, 6], [367921, 7186648327, 88176602471, 19678982400, 120, 348713169961, 368047, 363001, 262053826865, 368040]], 'aut_group': None, 'aut_hash': 1483974879331033022, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 61440, 'aut_permdeg': 48, 'aut_perms': [10055918268190300748547729577975912796417288146240168627905847, 9571827215033413321987878642561080868727378845530293091411501, 9571928231412665242382007266748958550722707603140080096325928, 10470344567759487031756284357298966805834717009831816552502962, 9573664062381657176684232901619813429037581969677903382944978, 4712618572699495250271753400423613303665718036802127174568110, 5622561253974010938964782955054597445118451544131763180202539], 'aut_phi_ratio': 60.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 5, 1, 2], [2, 5, 2, 1], [2, 6, 2, 1], [2, 10, 2, 1], [2, 12, 2, 1], [2, 15, 1, 2], [2, 15, 2, 1], [2, 30, 2, 1], [2, 60, 2, 1], [3, 8, 1, 1], [4, 10, 2, 4], [4, 12, 2, 3], [4, 30, 2, 4], [4, 30, 4, 4], [4, 60, 2, 7], [5, 4, 1, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 2, 1], [6, 40, 1, 2], [6, 40, 2, 1], [6, 80, 2, 1], [10, 4, 1, 1], [10, 4, 2, 1], [10, 8, 2, 1], [10, 12, 1, 2], [10, 12, 2, 1], [10, 24, 2, 1], [10, 48, 2, 1], [12, 80, 2, 4], [15, 32, 1, 1], [20, 48, 2, 3], [30, 32, 1, 1], [30, 32, 2, 1], [30, 32, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_5\\times A_4).C_2^4.C_2^6', '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': 60, 'autcentquo_group': '480.1189', 'autcentquo_hash': 1189, 'autcentquo_nilpotent': False, 'autcentquo_order': 480, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 3, 4], [2, 5, 4], [2, 6, 2], [2, 10, 2], [2, 12, 2], [2, 15, 4], [2, 30, 2], [2, 60, 2], [3, 8, 1], [4, 10, 8], [4, 12, 6], [4, 30, 24], [4, 60, 14], [5, 4, 1], [6, 8, 3], [6, 16, 2], [6, 40, 4], [6, 80, 2], [10, 4, 3], [10, 8, 2], [10, 12, 4], [10, 24, 2], [10, 48, 2], [12, 80, 8], [15, 32, 1], [20, 48, 6], [30, 32, 7]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '960.11361', 'commutator_count': 1, 'commutator_label': '120.43', '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', '5.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 150, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['1920.240035', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 3, 1, 4], [2, 5, 1, 4], [2, 6, 1, 2], [2, 10, 1, 2], [2, 12, 1, 2], [2, 15, 1, 4], [2, 30, 1, 2], [2, 60, 1, 2], [3, 8, 1, 1], [4, 10, 2, 4], [4, 12, 1, 6], [4, 30, 2, 12], [4, 60, 1, 6], [4, 60, 2, 4], [5, 4, 1, 1], [6, 8, 1, 3], [6, 16, 1, 2], [6, 40, 1, 4], [6, 80, 1, 2], [10, 4, 1, 3], [10, 8, 1, 2], [10, 12, 1, 4], [10, 24, 1, 2], [10, 48, 1, 2], [12, 80, 2, 4], [15, 32, 1, 1], [20, 48, 1, 6], [30, 32, 1, 3], [30, 32, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1023684480, 'exponent': 60, 'exponents_of_order': [8, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '1920.240396', 'hash': 7250606812080537101, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 4, 6, 2, 2, 10, 1, 2, 4, 2], 'inner_gens': [[362886, 7186648321, 274467951768, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1], [126, 7187011321, 88176613974, 19678982400, 367927, 348713527686, 5047, 5040, 262054189865, 1], [367920, 193558187526, 274467951775, 88176211200, 5167, 280215941160, 5047, 1, 262054155022, 6], [362886, 7187011321, 180623554975, 19678982400, 5167, 280215936126, 5047, 5040, 99632735704, 1], [362886, 7186648321, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1], [362886, 262055416321, 180623554968, 368392147200, 5167, 187768627326, 5047, 5040, 193556966104, 1], [362886, 7187011321, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1], [362886, 7187011321, 274467956809, 19678982400, 5167, 187768627326, 5047, 5040, 355898586665, 1], [362886, 99634325160, 368392175705, 274467916800, 5167, 348713164926, 5047, 1, 355898591704, 5040], [362886, 7187011321, 274467956814, 19678982400, 5167, 187768627326, 5047, 5040, 355898586665, 1]], 'inner_hash': 11361, 'inner_nilpotent': False, 'inner_order': 960, 'inner_split': False, 'inner_tex': 'C_2\\times F_5\\times S_4', 'inner_used': [1, 2, 3, 9], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [3, 32], [4, 12], [6, 8], [8, 10], [12, 8], [24, 2]], 'label': '3840.ft', '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*GL(2,Z/4):F5', 'ngens': 10, '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': 68, 'number_characteristic_subgroups': 76, 'number_conjugacy_classes': 128, 'number_divisions': 102, 'number_normal_subgroups': 248, 'number_subgroup_autclasses': 2198, 'number_subgroup_classes': 5338, 'number_subgroups': 77316, 'old_label': None, 'order': 3840, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 335], [3, 8], [4, 1712], [5, 4], [6, 376], [10, 220], [12, 640], [15, 32], [20, 288], [30, 224]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 19678982400, 0], 'outer_gens': [[367921, 7187016366, 274467956808, 19678982400, 120, 187768632361, 5047, 5040, 355898591704, 1], [362886, 7187016366, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898586671, 1], [362886, 7187016366, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1], [362886, 7187011321, 368392182175, 19678982400, 5167, 93924230526, 5047, 5040, 193556966104, 1], [126, 7186653366, 274467951775, 19678982400, 367927, 187768990086, 368047, 368040, 355898586671, 363001]], 'outer_group': '64.202', 'outer_hash': 202, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [1174447, 368063, 368040, 367920, 363022], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3:D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [3, 16], [4, 18], [6, 12], [8, 6], [12, 10], [16, 2], [24, 2]], 'representations': {'PC': {'code': '339095252087333614047069559593955708549354442679144545081731099001340733320034667623523907108046481807596665612617646192', 'gens': [1, 2, 4, 6, 9, 10], 'pres': [10, -2, -2, -2, -2, -2, -2, -3, -5, -2, 2, 42681, 51, 50093, 7783, 113, 19614, 5024, 59525, 64335, 9145, 11315, 5925, 175, 206086, 49296, 4506, 17956, 13766, 276, 15377, 15387, 3877, 3887, 32438, 5458, 4118, 9059, 1569]}, 'Perm': {'d': 15, 'gens': [362886, 7187011321, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times \\GL(2,\\mathbb{Z}/4):F_5', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
