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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '480.1199', 'ambient_counter': 1199, 'ambient_order': 480, 'ambient_tex': 'C_2^3:D_{30}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 5, 'counter': 26, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '480.1199.12.h1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.h1', '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': 12, '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': '40.8', 'subgroup_hash': 8, 'subgroup_order': 40, 'subgroup_tex': 'C_5:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '480.1199', 'aut_centralizer_order': 4, 'aut_label': '12.h1', 'aut_quo_index': None, 'aut_stab_index': 36, 'aut_weyl_group': '80.50', 'aut_weyl_index': 144, 'centralizer': '60.f1', 'complements': None, 'conjugacy_class_count': 12, 'contained_in': ['6.b1', '6.c1'], 'contains': ['24.f1', '24.g1', '24.h1', '60.h1'], 'core': '96.a1', 'coset_action_label': None, 'count': 36, 'diagramx': [3251, -1, 9516, -1], 'generators': [8079, 88211, 8081, 156019], 'label': '480.1199.12.h1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '12.h1', 'projective_image': '480.1199', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.h1', 'subgroup_fusion': None, 'weyl_group': '20.4'}
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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': 2, 'aut_exponent': 20, 'aut_gen_orders': [2, 4, 10], 'aut_gens': [[7988, 132, 13311], [7988, 1320, 13311], [9325, 132, 13311], [7988, 517, 1341]], 'aut_group': '80.50', 'aut_hash': 50, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 80, 'aut_permdeg': 9, 'aut_perms': [7, 16560, 50536], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 10, 1, 1], [4, 10, 1, 1], [5, 2, 2, 1], [10, 2, 2, 1], [10, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 10, 1], [4, 10, 1], [5, 2, 2], [10, 2, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '20.4', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 10, 1, 1], [4, 10, 1, 1], [5, 2, 2, 1], [10, 2, 2, 1], [10, 2, 4, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 6, 'exponent': 20, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '20.4', 'hash': 8, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [5, 2, 2], 'inner_gens': [[7988, 517, 13311], [2668, 132, 1341], [7988, 1320, 13311]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'D_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 9]], 'label': '40.8', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 2, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 6, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5:D4', 'ngens': 3, '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': 8, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 13, 'number_divisions': 8, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 40, 'old_label': None, 'order': 40, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 13], [4, 10], [5, 4], [10, 12]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [1332], 'outer_gens': [[9325, 132, 13311]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 2], [8, 1]], 'representations': {'PC': {'code': 702690307, 'gens': [1, 2, 3], 'pres': [4, -2, -2, 2, -5, 458, 34, 515]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [25038648174890735, 91793129359348471]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [7988, 132, 13311]}, 'Perm': {'d': 9, 'gens': [5167, 5, 16, 50520]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5:D_4', 'transitive_degree': 20, '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': '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}
