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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '4800.bk', 'ambient_counter': 37, 'ambient_order': 4800, 'ambient_tex': 'C_5\\times \\GL(2,3):D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 200, 'characteristic': False, 'core_order': 2, 'counter': 670, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4800.bk.600.z1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '600.z1.a1', 'outer_equivalence': False, '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': 600, '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': '8.4', 'subgroup_hash': 4, 'subgroup_order': 8, 'subgroup_tex': 'Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4800.bk', 'aut_centralizer_order': 640, 'aut_label': '600.z1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '8.3', 'aut_weyl_index': 1920, 'centralizer': '24.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['120.cd1.a1', '120.ce1.a1', '120.cf1.a1', '120.cf1.b1', '300.a1.a1', '300.m1.a1', '300.t1.a1', '300.y1.a1', '300.bb1.a1', '300.bh1.a1', '300.bk1.a1'], 'contains': ['1200.g1.a1', '1200.h1.a1'], 'core': '2400.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [5456561382101072, 18688216819942578, 41772741070013040], 'label': '4800.bk.600.z1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '300.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '600.z1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '600.z1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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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': [2, 3, 2, 2], 'aut_gens': [[1, 2], [2, 1], [3, 5], [5, 2], [1, 6]], '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': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 3, 1]], 'aut_supersolvable': False, 'aut_tex': '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': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', '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': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [[2, -1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.4', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'Q8', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [2, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 2343, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 12, 37, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [25793872, 22766861]}, 'Lie': [{'d': 2, 'q': 5, 'gens': [90, 377, 504, 253], 'family': 'CSOPlus'}, {'d': 2, 'q': 3, 'gens': [56, 67, 15], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [67, 21]}, 'Perm': {'d': 8, 'gens': [28776, 16577, 5167]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_8', 'transitive_degree': 8, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [4, 2, 4, 4, 4, 4, 4], 'aut_gens': [[12531822321003912, 23748917383739229, 10416975381833665, 2062427169602618, 13512873342060371, 13258070444499114, 33418192856010432, 5224234401115719, 40126814940569095], [12531822321003912, 24218583233012708, 4682200994275411, 9915186924868891, 15159679308277411, 6423481269729558, 33418192856010432, 25439335536803650, 4681554916613431], [16709096428005216, 2817913196904748, 44806504720391178, 19743145391825526, 38052650225453868, 23305270394581862, 29240918749009128, 1760573775587444, 25761959957325597], [20886370535006520, 2136651745894348, 24509150887497213, 38567088971765453, 36189597143216864, 6423481269729558, 25063644642007824, 3103425093954039, 13513050616864378], [20886370535006520, 30383912584721251, 18269735138871499, 35480334712170867, 19514336391991876, 5455705447089206, 25063644642007824, 12934962647909777, 13503243904194602], [37595466963011736, 38045468877784248, 41218247313730868, 30714329216484550, 12464964508161683, 44086672911129474, 8354548214002608, 26310319439537720, 11021399039939440], [20886370535006520, 35699124086834751, 24509150887497213, 38567088971765453, 35699449720210030, 5455705447089206, 25063644642007824, 34671877733064810, 32302565329882962], [37595466963011736, 26114547575168155, 30212540757762856, 44894018275452650, 19550038542624636, 45054448733769826, 8354548214002608, 1012392581584673, 15620444181639910]], 'aut_group': None, 'aut_hash': 5442753492121977074, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 15360, 'aut_permdeg': 28, 'aut_perms': [44378203321031426768380028607, 142387824686052061458737046375, 43982975902250613066315583846, 164164002899523615312677776845, 229497228173423687057403475207, 65761193495287407662707518102, 141982671879724188231860970486], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 10, 1, 1], [2, 12, 1, 2], [2, 30, 1, 1], [2, 60, 1, 1], [3, 8, 1, 1], [4, 6, 1, 2], [4, 10, 1, 1], [4, 30, 1, 1], [4, 60, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [6, 8, 1, 1], [6, 16, 1, 1], [6, 80, 1, 1], [8, 6, 2, 1], [8, 12, 1, 1], [8, 60, 1, 2], [10, 1, 4, 1], [10, 2, 2, 1], [10, 2, 4, 2], [10, 2, 8, 1], [10, 2, 16, 1], [10, 10, 4, 1], [10, 12, 4, 2], [10, 24, 2, 2], [10, 24, 8, 2], [10, 30, 4, 1], [10, 60, 4, 1], [12, 80, 1, 1], [15, 8, 4, 1], [15, 16, 2, 1], [15, 16, 8, 1], [20, 6, 4, 2], [20, 10, 4, 1], [20, 12, 2, 2], [20, 12, 8, 2], [20, 30, 4, 1], [20, 60, 4, 1], [30, 8, 4, 1], [30, 16, 2, 1], [30, 16, 4, 2], [30, 16, 8, 1], [30, 16, 16, 1], [30, 80, 4, 1], [40, 6, 8, 1], [40, 12, 4, 3], [40, 12, 16, 2], [40, 60, 4, 2], [60, 80, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times A_4\\times F_5).C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.45', 'autcent_hash': 45, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_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, 1], [2, 2, 1], [2, 10, 1], [2, 12, 2], [2, 30, 1], [2, 60, 1], [3, 8, 1], [4, 6, 2], [4, 10, 1], [4, 30, 1], [4, 60, 1], [5, 1, 4], [5, 2, 10], [6, 8, 1], [6, 16, 1], [6, 80, 1], [8, 6, 2], [8, 12, 1], [8, 60, 2], [10, 1, 4], [10, 2, 34], [10, 10, 4], [10, 12, 8], [10, 24, 20], [10, 30, 4], [10, 60, 4], [12, 80, 1], [15, 8, 4], [15, 16, 10], [20, 6, 8], [20, 10, 4], [20, 12, 20], [20, 30, 4], [20, 60, 4], [30, 8, 4], [30, 16, 34], [30, 80, 4], [40, 6, 8], [40, 12, 44], [40, 60, 8], [60, 80, 4]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '480.1193', 'commutator_count': 1, 'commutator_label': '120.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1', '5.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 37, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['960.10966', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 10, 1, 1], [2, 12, 1, 2], [2, 30, 1, 1], [2, 60, 1, 1], [3, 8, 1, 1], [4, 6, 1, 2], [4, 10, 1, 1], [4, 30, 1, 1], [4, 60, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [6, 8, 1, 1], [6, 16, 1, 1], [6, 80, 1, 1], [8, 6, 2, 1], [8, 12, 1, 1], [8, 60, 1, 2], [10, 1, 4, 1], [10, 2, 2, 1], [10, 2, 4, 8], [10, 10, 4, 1], [10, 12, 4, 2], [10, 24, 2, 2], [10, 24, 4, 4], [10, 30, 4, 1], [10, 60, 4, 1], [12, 80, 1, 1], [15, 8, 4, 1], [15, 16, 2, 1], [15, 16, 4, 2], [20, 6, 4, 2], [20, 10, 4, 1], [20, 12, 2, 2], [20, 12, 4, 4], [20, 30, 4, 1], [20, 60, 4, 1], [30, 8, 4, 1], [30, 16, 2, 1], [30, 16, 4, 8], [30, 80, 4, 1], [40, 6, 8, 1], [40, 12, 4, 3], [40, 12, 8, 4], [40, 60, 4, 2], [60, 80, 4, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 1874880, 'exponent': 120, 'exponents_of_order': [6, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 32], [8, 0, 16]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': None, 'hash': 7518590142126048533, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [1, 10, 10, 5, 10, 2, 1, 30, 2], 'inner_gens': [[12531822321003912, 23748917383739229, 10416975381833665, 2062427169602618, 13512873342060371, 13258070444499114, 33418192856010432, 5224234401115719, 40126814940569095], [12531822321003912, 23748917383739229, 10416975381833665, 2062427169602618, 37031829507869005, 23305270394581862, 33418192856010432, 8772335526487693, 6455414532407917], [12531822321003912, 23748917383739229, 10416975381833665, 2062427169602618, 13512873342060371, 5455705447089206, 33418192856010432, 5224234401115719, 12444495279276036], [12531822321003912, 23748917383739229, 10416975381833665, 2062427169602618, 13512873342060371, 21268714169688133, 33418192856010432, 5224234401115719, 32363042457659126], [12531822321003912, 26795785466188805, 10416975381833665, 2062427169602618, 13512873342060371, 27204883786277170, 33418192856010432, 42915620762010941, 4681554916613431], [12531822321003912, 3174366939167738, 31773179254093103, 23418916351761106, 30152431942652599, 13258070444499114, 33418192856010432, 23271641402866208, 32363042457659126], [12531822321003912, 23748917383739229, 10416975381833665, 2062427169602618, 13512873342060371, 13258070444499114, 33418192856010432, 5224234401115719, 40126814940569095], [12531822321003912, 16430578810115752, 10416975381833665, 2062427169602618, 13997265710498781, 6423481269729558, 33418192856010432, 5224234401115719, 39970791001296850], [12531822321003912, 35079398428398021, 31773179254093103, 23418916351761106, 20392270907276777, 23305270394581862, 33418192856010432, 28543047641810673, 40126814940569095]], 'inner_hash': 1193, 'inner_nilpotent': False, 'inner_order': 480, 'inner_split': True, 'inner_tex': 'D_{10}\\times S_4', 'inner_used': [2, 3, 5, 6, 8], 'irrC_degree': 4, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 8, 'irrep_stats': [[1, 40], [2, 60], [3, 40], [4, 70], [6, 40], [8, 25]], 'label': '4800.bk', 'linC_count': 32, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 80, 'linQ_dim': 16, 'linQ_dim_count': 80, 'linR_count': 16, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C5*GL(2,3):D10', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 68, 'number_characteristic_subgroups': 74, 'number_conjugacy_classes': 275, 'number_divisions': 84, 'number_normal_subgroups': 74, 'number_subgroup_autclasses': 639, 'number_subgroup_classes': 710, 'number_subgroups': 6608, 'old_label': None, 'order': 4800, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 127], [3, 8], [4, 112], [5, 24], [6, 104], [8, 144], [10, 1048], [12, 80], [15, 192], [20, 688], [30, 896], [40, 1056], [60, 320]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4], 'outer_gen_pows': [4177274107001304, 4177274107001304, 4177274107001304], 'outer_gens': [[12531822321003912, 26795785466188805, 40127727466324150, 2062427169602618, 37031829507869005, 37252083736359918, 33418192856010432, 45320468234454765, 10417602595918109], [12531822321003912, 27152239208453137, 45862501855653965, 9915186924868891, 15159679308277411, 21268714169688133, 33418192856010432, 26012899538804739, 38100207570653340], [37595466963011736, 20670667591746986, 29745560322841063, 504926887274107, 40120583959999258, 13258070444499114, 8354548214002608, 24940446178354432, 20800656327731650]], 'outer_group': '32.21', 'outer_hash': 21, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 11640, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [3, 8], [4, 12], [8, 16], [12, 12], [16, 4], [24, 8], [32, 12]], 'representations': {'PC': {'code': '12311680077605695455695248057865064001048852265247678724404778506285967284629717251', 'gens': [1, 3, 4, 6, 7], 'pres': [9, -2, -5, -2, -2, -3, -2, 2, -2, -5, 18, 52563, 52212, 9021, 102, 382, 13631, 3596, 4901, 1130, 2292, 10995, 4389, 1437, 186, 15586, 214, 15587]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 23748917383739229, 10416975381833665, 2062427169602618, 13512873342060371, 13258070444499114, 33418192856010432, 5224234401115719, 40126814940569095]}, 'Perm': {'d': 26, 'gens': [466439, 16265577903217243040589146, 1275023, 49791463565716111830021890, 31643127362686418162077756, 34750999093082280039828623, 31648917067060989889696346, 64716702945695770878779492, 77773413233082088478901422]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times \\GL(2,3):D_{10}', 'transitive_degree': 80, 'wreath_data': None, 'wreath_product': False}
