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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '3840.bf', 'ambient_counter': 32, 'ambient_order': 3840, 'ambient_tex': 'C_2\\times F_5\\times \\GL(2,\\mathbb{Z}/4)', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 40, 'counter': 114, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '3840.bf.12.Q', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.q1', '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': '320.1159', 'subgroup_hash': 1159, 'subgroup_order': 320, 'subgroup_tex': 'C_2^3:D_{20}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3840.bf', 'aut_centralizer_order': 8, 'aut_label': '12.Q', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '2560.ec', 'aut_weyl_index': 24, 'centralizer': '480.BX', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.F'], 'contains': ['24.BU', '24.BV', '24.BW', '24.DB', '24.DC', '24.DE', '24.DG', '24.EO', '24.EP', '60.M'], 'core': '96.C', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [12211, 88211, 88111, 12209, 72009, 8161, 88011], 'label': '3840.bf.12.Q', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.A', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '12.q1', 'projective_image': '960.11361', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.Q', 'subgroup_fusion': None, 'weyl_group': '160.236'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 40, 'aut_gen_orders': [20, 8, 8, 4, 8, 4], 'aut_gens': [[72009, 8259, 92211, 88211, 8021], [152219, 12379, 92211, 88211, 92221], [72209, 76171, 76009, 88211, 92181], [72209, 12179, 76209, 88211, 92061], [72209, 152281, 76009, 88211, 88151], [72209, 156321, 156219, 88211, 156229], [152019, 156361, 92011, 88211, 92261]], 'aut_group': None, 'aut_hash': 7222038130993120333, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 81920, 'aut_permdeg': 56, 'aut_perms': [437757844498877182553273250253568994854101304106662506892864837735969334248, 426312557473182517193116669079775916360969060892924536576624822005371360227, 429846651227703843233008918937098638098043197439862521582417492775969454915, 710727148900220320715257389449114798131146006554731857468798462673090832488, 150524780653297793107113503460883533579359988584724009546293458005078080890, 278116024594539173731200261962209444134229545648123400644554991460867246851], 'aut_phi_ratio': 640.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 1, 4, 1], [2, 2, 4, 1], [2, 10, 8, 1], [2, 20, 2, 1], [4, 4, 4, 1], [4, 20, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1], [10, 2, 4, 1], [10, 2, 8, 1], [10, 4, 8, 1], [20, 4, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_5:(C_4\\times C_2^6.D_4^2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 8092891664598236742, 'autcent_nilpotent': True, 'autcent_order': 2048, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7.C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 4], [2, 10, 8], [2, 20, 2], [4, 4, 4], [4, 20, 2], [5, 2, 2], [10, 2, 14], [10, 4, 8], [20, 4, 16]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '40.13', 'commutator_count': 1, 'commutator_label': '20.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1159, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['160.103', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 4], [2, 10, 1, 8], [2, 20, 1, 2], [4, 4, 1, 4], [4, 20, 1, 2], [5, 2, 2, 1], [10, 2, 2, 7], [10, 4, 2, 4], [20, 4, 4, 4]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 39060, 'exponent': 20, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '80.51', 'hash': 1159, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [1, 2, 2, 1, 10], 'inner_gens': [[72009, 8259, 92211, 88211, 8021], [72009, 8259, 92211, 88211, 8381], [72009, 8259, 92211, 88211, 88031], [72009, 8259, 92211, 88211, 8021], [72009, 8219, 12001, 88211, 8021]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': False, 'inner_tex': 'C_2\\times D_{10}', 'inner_used': [2, 3, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 44], [4, 8]], 'label': '320.1159', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3:D20', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 15, 'number_characteristic_subgroups': 17, 'number_conjugacy_classes': 68, 'number_divisions': 44, 'number_normal_subgroups': 143, 'number_subgroup_autclasses': 128, 'number_subgroup_classes': 662, 'number_subgroups': 3038, 'old_label': None, 'order': 320, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 135], [4, 56], [5, 4], [10, 60], [20, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 8, 'outer_gen_orders': [4, 8, 4, 4, 2, 4], 'outer_gen_pows': [8001, 8001, 8001, 8001, 8001, 8001], 'outer_gens': [[72009, 76171, 156219, 88211, 12351], [72209, 92049, 76009, 88211, 76399], [72209, 76171, 92211, 88211, 92221], [152219, 152361, 92011, 88211, 8221], [72209, 88049, 92011, 88211, 8221], [152219, 152361, 76009, 88211, 152199]], 'outer_group': None, 'outer_hash': 5198450426854685914, 'outer_nilpotent': True, 'outer_order': 2048, 'outer_permdeg': 24, 'outer_perms': [162009742576174721930250, 82315184648548985052741, 187869071175300687729859, 105663380280519232453104, 25956631882254348503165, 105658501668793165613312], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5.C_2^5.C_2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 8], [8, 8]], 'representations': {'PC': {'code': 147643418137663496507883523, 'gens': [1, 2, 3, 4, 5], 'pres': [7, -2, -2, -2, -2, 2, -2, -5, 5331, 228, 102, 6060, 124, 6285]}, 'GLZN': {'d': 2, 'p': 20, 'gens': [8101, 8201, 92049, 72009, 8081, 152019, 12201]}, 'Perm': {'d': 15, 'gens': [997960446, 88216168321, 1967535361, 193637162881, 88216168320, 288369849600, 966]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3:D_{20}', 'transitive_degree': 80, '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': [4, 12, 12, 2, 10, 12, 12], 'aut_gens': [[12211, 72009, 88111, 90312, 88211, 12281], [12211, 72009, 72309, 34143, 88211, 92251], [156209, 72009, 94011, 14062, 92011, 88051], [92201, 72009, 50316, 14062, 12201, 92251], [76219, 72009, 152319, 14142, 88211, 12121], [12211, 72009, 90011, 130237, 92011, 12281], [92201, 152019, 14001, 10262, 92011, 88051], [92201, 72009, 114304, 14142, 12201, 92251]], 'aut_group': None, 'aut_hash': 1483974879331033022, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 61440, 'aut_permdeg': 48, 'aut_perms': [7270097062406548665379863880022850115030821524940660411832391, 174894643565875825335786362160824267687670753713596420668292, 147381237708647174285383740880237405794003485426611046730414, 40079281769968161433988783371233597627783243094825022242073, 1711710200357906132947618862651571380897195208005912864115299, 232975830393164902411894669673216302036052213885257993453012, 5718962853188742813148899057515018623512941228647728101580527], '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, 5, 4, 2], [4, 10, 2, 2], [4, 12, 2, 3], [4, 15, 4, 2], [4, 30, 2, 2], [4, 60, 2, 11], [5, 4, 1, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 8, 4, 1], [6, 40, 1, 2], [6, 40, 2, 1], [6, 40, 4, 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, 40, 4, 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, 5, 8], [4, 10, 4], [4, 12, 6], [4, 15, 8], [4, 30, 4], [4, 60, 22], [5, 4, 1], [6, 8, 7], [6, 40, 8], [10, 4, 3], [10, 8, 2], [10, 12, 4], [10, 24, 2], [10, 48, 2], [12, 40, 16], [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': 32, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['20.3', 1], ['96.195', 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, 5, 2, 4], [4, 10, 2, 2], [4, 12, 1, 6], [4, 15, 2, 4], [4, 30, 2, 2], [4, 60, 1, 6], [4, 60, 2, 8], [5, 4, 1, 1], [6, 8, 1, 3], [6, 8, 2, 2], [6, 40, 1, 4], [6, 40, 2, 2], [10, 4, 1, 3], [10, 8, 1, 2], [10, 12, 1, 4], [10, 24, 1, 2], [10, 48, 1, 2], [12, 40, 2, 4], [12, 40, 4, 2], [15, 32, 1, 1], [20, 48, 1, 6], [30, 32, 1, 3], [30, 32, 2, 2]], 'element_repr_type': 'GLZN', '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': 5245989847885864621, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 1, 4, 12, 2, 10], 'inner_gens': [[12211, 72009, 8301, 90312, 88211, 12281], [12211, 72009, 88111, 90312, 88211, 12281], [92201, 72009, 88111, 50117, 88211, 92091], [12211, 72009, 134106, 90312, 12201, 92251], [12211, 72009, 88111, 10102, 88211, 12281], [12211, 72009, 8101, 94192, 88211, 12281]], '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, 3, 4, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 40], [3, 32], [4, 8], [6, 8], [8, 10], [12, 8], [24, 2]], 'label': '3840.bf', '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*F5*GL(2,Z/4)', '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': 140, 'number_divisions': 104, 'number_normal_subgroups': 264, 'number_subgroup_autclasses': 2228, 'number_subgroup_classes': 6074, 'number_subgroups': 80772, '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': [8001, 8001, 8001, 8001, 8001], 'outer_gens': [[12211, 72009, 8101, 118313, 88211, 92091], [76219, 72009, 72109, 118313, 88211, 92091], [12211, 72009, 72109, 54317, 88211, 92091], [12211, 72009, 88111, 14112, 88211, 12281], [76219, 152019, 88111, 154308, 88211, 12281]], 'outer_group': '64.202', 'outer_hash': 202, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [362880, 16, 368047, 367920, 1174342], '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, 8], [12, 10], [16, 2], [24, 2]], 'representations': {'PC': {'code': '23916915778682086780133533227577233948904804313086516374677291868213750050162200939212046403', 'gens': [1, 2, 3, 5, 8, 9], 'pres': [10, -2, -2, -2, -2, -2, -2, -3, -2, 2, -5, 6122, 82, 3424, 13334, 144, 31225, 14675, 175, 5626, 19076, 9647, 5337, 2467, 12988, 7608, 4918, 878, 268, 14449, 9659]}, 'GLZN': {'d': 2, 'p': 20, 'gens': [8009, 76219, 8101, 12201, 8081, 72009, 130311, 156209, 8003, 8201]}, 'Perm': {'d': 15, 'gens': [362886, 268719897600, 354940548607, 5167, 5047, 5040, 362125600566, 1, 175314585726, 3260759]}}, '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 F_5\\times \\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
