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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '5280.r', 'ambient_counter': 18, 'ambient_order': 5280, 'ambient_tex': '\\SL(2,11):C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 39, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5280.r.264.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '264.c1', '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': 264, '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': '20.1', 'subgroup_hash': 1, 'subgroup_order': 20, 'subgroup_tex': 'C_5:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5280.r', 'aut_centralizer_order': None, 'aut_label': '264.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '660.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['44.a1', '132.a1'], 'contains': ['528.a1', '1320.c1'], 'core': '2640.a1', 'coset_action_label': None, 'count': 66, 'diagramx': [2171, -1, 3172, -1], 'generators': [44735138646218703, 17575135970880675, 41772741070013040], 'label': '5280.r.264.c1', 'mobius_quo': None, 'mobius_sub': 2, 'normal_closure': '4.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '66.a1', 'old_label': '264.c1', 'projective_image': '2640.bt', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '264.c1', 'subgroup_fusion': None, 'weyl_group': '10.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 10], 'aut_gens': [[1, 4], [1, 12], [11, 4]], 'aut_group': '40.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 40, 'aut_permdeg': 7, 'aut_perms': [361, 1855], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 5, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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], [4, 5, 2], [5, 2, 2], [10, 2, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 5, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, -1, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '10.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 16], [9, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 4]], 'label': '20.1', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5:C4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 8, 'number_divisions': 5, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 10, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 10], [5, 4], [10, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [1, 0], 'outer_gens': [[1, 8], [3, 4]], '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': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2]], 'representations': {'PC': {'code': 133699, 'gens': [1, 3], 'pres': [3, -2, -2, -5, 6, 146]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [91655872471905395, 125101739906136769]}, 'Lie': [{'d': 2, 'q': 11, 'gens': [440, 6664, 7988, 13320], 'family': 'CSOPlus'}, {'d': 2, 'q': 9, 'gens': [1460, 2020, 6241], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 377]}, 'Perm': {'d': 9, 'gens': [5169, 16, 50520]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5:C_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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [20, 10, 12], 'aut_gens': [[44734985446128011, 24133814979398195, 12415902285472309, 6422340100995270, 41772741070013040], [44734985446128011, 13501442332363978, 45822493231479854, 10842363402478390, 41772741070013040], [44734985446128011, 42568814764959284, 36128724638041774, 6422340100995270, 41772741070013040], [6422340100995270, 31837183975835682, 28254458061896337, 5809717384314194, 41772741070013040]], 'aut_group': None, 'aut_hash': 5402696758532746038, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 31680, 'aut_permdeg': 116, 'aut_perms': [1472766104441299305816507186695686492830069762895372869336913247901266981356453061551294454998691458154290289763672656148073600991446578767262053069610686375189605303008560317282282965915399, 6676314203635320993064048394365558297980858493865981886081934449340653024170891281763142445638916548662376851280976856060643371863045719640963122044330905333285598629477133127046, 882615227518360134927206637111469474861849069064443026638903397230903003929000513536810485256798917187394457562936808798515343343179190296269253523215965941636332397365080899455898488697569], 'aut_phi_ratio': 24.75, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 110, 3, 1], [3, 110, 1, 1], [4, 2, 3, 1], [4, 110, 1, 1], [5, 132, 1, 2], [6, 110, 1, 1], [6, 220, 3, 1], [10, 132, 1, 2], [11, 60, 2, 1], [12, 110, 1, 2], [12, 220, 3, 1], [20, 264, 3, 2], [22, 60, 2, 1], [44, 120, 6, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,11).C_2\\times S_4', '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': 660, 'autcentquo_group': '7920.q', 'autcentquo_hash': 1157788818429169362, 'autcentquo_nilpotent': False, 'autcentquo_order': 7920, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times \\PGL(2,11)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 110, 3], [3, 110, 1], [4, 2, 3], [4, 110, 1], [5, 132, 2], [6, 110, 1], [6, 220, 3], [10, 132, 2], [11, 60, 2], [12, 110, 2], [12, 220, 3], [20, 264, 6], [22, 60, 2], [44, 120, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '2640.bt', 'commutator_count': 1, 'commutator_label': '1320.13', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '660.13'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 18, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 110, 1, 3], [3, 110, 1, 1], [4, 2, 1, 3], [4, 110, 1, 1], [5, 132, 2, 1], [6, 110, 1, 1], [6, 220, 1, 3], [10, 132, 2, 1], [11, 60, 2, 1], [12, 110, 2, 1], [12, 220, 1, 3], [20, 264, 2, 3], [22, 60, 2, 1], [44, 120, 2, 3]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 254, 'exponent': 660, 'exponents_of_order': [5, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[12, 0, 2], [20, 1, 3], [24, 1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '2640.bt', 'hash': 4597536282602998348, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [2, 2, 6, 2, 1], 'inner_gens': [[44734985446128011, 24133814979398195, 38125662136079015, 44087814079863762, 41772741070013040], [44734985446128011, 24133814979398195, 41147329439887960, 44087814079863762, 41772741070013040], [5809717384314194, 35215812647737389, 12415902285472309, 6422340100995270, 41772741070013040], [5809717384314194, 26376338987100625, 12415902285472309, 6422340100995270, 41772741070013040], [44734985446128011, 24133814979398195, 12415902285472309, 6422340100995270, 41772741070013040]], 'inner_hash': 6613815745579852445, 'inner_nilpotent': False, 'inner_order': 2640, 'inner_split': True, 'inner_tex': 'C_2^2\\times \\PSL(2,11)', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 20, 'irrep_stats': [[1, 4], [5, 8], [10, 8], [11, 4], [12, 10], [20, 3], [24, 2]], 'label': '5280.r', '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': False, 'name': 'SL(2,11):C2^2', 'ngens': 5, '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, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 39, 'number_divisions': 28, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 69, 'number_subgroup_classes': 170, 'number_subgroups': 11958, 'old_label': None, 'order': 5280, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 331], [3, 110], [4, 116], [5, 264], [6, 770], [10, 264], [11, 120], [12, 880], [20, 1584], [22, 120], [44, 720]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [4177274107001304, 23054351874780899], 'outer_gens': [[6422340100995270, 15713267665316284, 42277907939197929, 44734985446128011, 41772741070013040], [6422340100995270, 29435959871950416, 20594434871319655, 10842363402478390, 41772741070013040]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': None, 'perfect': False, 'permutation_degree': 96, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [10, 12], [11, 4], [20, 1], [24, 5], [40, 1], [48, 1]], 'representations': {'GLFp': {'d': 4, 'p': 11, 'gens': [44734985446128011, 24133814979398195, 12415902285472309, 6422340100995270, 41772741070013040]}, 'Perm': {'d': 96, 'gens': [10339370311181988351046690860219824167752582690464438371460610516989893884451851079408286962005810173694871061406171202140365261579228320182738708506, 21210681968368207302762313911197426671188073837622274806268655254593373239725883492901391202774887809582339316799335757108565342935881437091438997656, 31652816422273714129797368568885609885067582559555568952837277543871407377480362475417030613135682776180642755980871517012221945289905527629481195348, 42092787808822108442840047016235917814623343543992921414296501299834491005069225052426343032016452110422133733185913897629184977717247116571757657628, 52532673456466909767657060937022399580741151165179415121841732413318500496090552229468187461067455319852105342042901017591011927690755172335468241839]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SL(2,11):C_2^2', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
