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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '960.11100', 'ambient_counter': 11100, 'ambient_order': 960, 'ambient_tex': 'C_2\\times C_{20}.S_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 80, 'counter': 32, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.11100.6.i1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.i1.c1', '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': 6, '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': '160.171', 'subgroup_hash': 171, 'subgroup_order': 160, 'subgroup_tex': 'C_{20}.D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.11100', 'aut_centralizer_order': None, 'aut_label': '6.i1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '120.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.d1.c1', '3.a1.a1'], 'contains': ['12.b1.b1', '12.k1.c1', '12.m1.d1', '12.n1.c1', '12.o1.c1', '12.r1.a1', '12.t1.a1', '30.i1.c1'], 'core': '12.b1.b1', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [5527161, 619906, 31507001, 49744146, 21062184, 9826016], 'label': '960.11100.6.i1.c1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.d1.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '6.i1.c1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.i1.c1', 'subgroup_fusion': None, 'weyl_group': None}
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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': 2, 'aut_exponent': 20, 'aut_gen_orders': [2, 2, 2, 4, 20], 'aut_gens': [[1, 2, 16], [1, 14, 16], [9, 10, 16], [1, 2, 24], [1, 14, 112], [13, 130, 16]], 'aut_group': '640.21483', 'aut_hash': 21483, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 640, 'aut_permdeg': 13, 'aut_perms': [3669840, 41040, 90720, 479001613, 1041556420], 'aut_phi_ratio': 10.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 20, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 20, 1, 1], [5, 2, 2, 1], [8, 10, 2, 2], [10, 2, 2, 1], [10, 4, 2, 1], [10, 4, 4, 1], [20, 2, 4, 1], [20, 4, 2, 1], [20, 4, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{10}.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '80.50', 'autcentquo_hash': 50, 'autcentquo_nilpotent': False, 'autcentquo_order': 80, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [2, 20, 1], [4, 1, 2], [4, 2, 1], [4, 4, 1], [4, 20, 1], [5, 2, 2], [8, 10, 4], [10, 2, 2], [10, 4, 6], [20, 2, 4], [20, 4, 6]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '40.8', 'commutator_count': 1, 'commutator_label': '20.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 171, '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, 4, 1, 1], [2, 20, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 20, 1, 1], [5, 2, 2, 1], [8, 10, 2, 2], [10, 2, 2, 1], [10, 4, 2, 1], [10, 4, 4, 1], [20, 2, 4, 1], [20, 4, 2, 1], [20, 4, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2016, 'exponent': 40, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 0, 4]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '40.13', 'hash': 171, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [2, 4, 10], 'inner_gens': [[1, 14, 24], [5, 2, 144], [9, 34, 16]], 'inner_hash': 8, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': True, 'inner_tex': 'C_5:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 32, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 22], [4, 4]], 'label': '160.171', 'linC_count': 68, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C20.D4', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 18, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 34, 'number_divisions': 18, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 62, 'number_subgroup_classes': 62, 'number_subgroups': 184, 'old_label': None, 'order': 160, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 27], [4, 28], [5, 4], [8, 40], [10, 28], [20, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 10, 16], [1, 2, 24], [1, 2, 48]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [120, 5040, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 4], [8, 3], [16, 1]], 'representations': {'PC': {'code': 34880407789364652979740675, 'gens': [1, 2, 5], 'pres': [6, -2, -2, -2, -2, 2, -5, 169, 31, 218, 50, 724, 2170, 88, 2315]}, 'GLZN': {'d': 2, 'p': 85, 'gens': [5527161, 615571, 49744146, 31755201, 9826016, 22108559]}, 'Perm': {'d': 21, 'gens': [2461175261634186727, 5501776060824705360, 8073591402550544640, 10342032017634253440, 13044523156676127360, 37]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_{20}.D_4', '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': '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': [12, 20, 12, 12, 10, 10], 'aut_gens': [[21280226, 27058124, 31433901, 31502326, 31756646], [43679984, 14775629, 49744146, 51586569, 31826856], [31574814, 27171599, 31755201, 31831276, 31829746], [19327779, 27058124, 31755201, 31430926, 49748481], [33236986, 45916619, 12896706, 9826001, 31429566], [33235541, 27171599, 31828301, 31759876, 49745591], [21281671, 45481844, 31755201, 20990784, 31435346]], 'aut_group': None, 'aut_hash': 7817645015147102572, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 15360, 'aut_permdeg': 124, 'aut_perms': [297621466120979278579823546162500500906503419117395331540788691563773572987114184176225754690086838094953063820007659997385125032788847404271261498782807000731485094392853057088820774610480397448158349155959, 1240363878387224512663220684652727850358809471302878382345569774438665239969466529469347301163747097527580274118588647293233086614386643071555699198741045751583294949517925235916381012408937400025118815449745, 5679437113886244916933380403398434098979959043143930808981524963806226953070663137593653406575813157734111950245263497464575039436057107574326237136116641789695066754763061828320132028444482855571013034130, 1249640745940786836834407737528274163781067705462809074107178159065621929400187023464526829425231021917526711098020573483156631343939020799456458170692299188831629856228355764446225454336310175737196532767303, 7750577964101352543075760458931660971825114830174949078754846263424588514404333024123931715084584048923447541627436315214050750029818602710739590329927856466224542355593753728766888904255980319278907548043, 298844476173449061665341649333307534604244675974091108183124755445488214274614508288329693044540191542521362071439824233533783240976933240813066237572023299344436118972619996329287872245750989602387532466778], 'aut_phi_ratio': 60.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 6, 2, 1], [2, 60, 2, 1], [3, 8, 1, 1], [4, 1, 4, 1], [4, 6, 1, 2], [4, 60, 2, 1], [5, 2, 2, 1], [6, 8, 1, 1], [6, 8, 2, 1], [8, 30, 4, 2], [10, 2, 2, 1], [10, 2, 4, 1], [10, 12, 4, 1], [12, 8, 4, 1], [15, 8, 4, 1], [20, 2, 8, 1], [20, 12, 2, 2], [30, 8, 4, 1], [30, 8, 8, 1], [60, 8, 16, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_5\\times A_4).C_2^4.C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\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, 6, 2], [2, 60, 2], [3, 8, 1], [4, 1, 4], [4, 6, 2], [4, 60, 2], [5, 2, 2], [6, 8, 3], [8, 30, 8], [10, 2, 6], [10, 12, 4], [12, 8, 4], [15, 8, 4], [20, 2, 8], [20, 12, 4], [30, 8, 12], [60, 8, 16]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '120.38', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 11100, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['480.1031', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 6, 1, 2], [2, 60, 1, 2], [3, 8, 1, 1], [4, 1, 2, 2], [4, 6, 1, 2], [4, 60, 1, 2], [5, 2, 2, 1], [6, 8, 1, 3], [8, 30, 2, 4], [10, 2, 2, 3], [10, 12, 2, 2], [12, 8, 2, 2], [15, 8, 4, 1], [20, 2, 4, 2], [20, 12, 2, 2], [30, 8, 4, 3], [60, 8, 8, 2]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 15120, 'exponent': 120, 'exponents_of_order': [6, 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': '480.1199', 'hash': 11100, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 3, 2, 2, 10], 'inner_gens': [[21280226, 46030094, 31507001, 31831276, 31826856], [8887671, 27058124, 31755201, 31430926, 49745591], [21245801, 45916619, 31433901, 31502326, 31501221], [21245801, 45916619, 31433901, 31502326, 31501221], [21139041, 45595319, 31828301, 31759876, 31756646]], 'inner_hash': 38, 'inner_nilpotent': False, 'inner_order': 120, 'inner_split': True, 'inner_tex': 'C_5:S_4', 'inner_used': [1, 2, 3, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 36], [3, 8], [4, 28], [6, 8]], 'label': '960.11100', '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': 'C2*C20.S4', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 26, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 88, 'number_divisions': 40, 'number_normal_subgroups': 47, 'number_subgroup_autclasses': 174, 'number_subgroup_classes': 306, 'number_subgroups': 2710, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 135], [3, 8], [4, 136], [5, 4], [6, 24], [8, 240], [10, 60], [12, 32], [15, 32], [20, 64], [30, 96], [60, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4, 4], 'outer_gen_pows': [614126, 614126, 614126, 614126, 614126], 'outer_gens': [[21280226, 27606374, 31828301, 21319734, 31501221], [31720334, 45481844, 31755201, 31831276, 31435346], [21280226, 27606374, 31828301, 31759876, 31501221], [21280226, 27606374, 31828301, 31502326, 31502666], [21101726, 48552449, 31507001, 21391134, 31829746]], 'outer_group': '128.1031', 'outer_hash': 1031, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [1508416, 211977376, 40723320, 212340378, 261314640], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3.C_2^4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 23, '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, 8], [12, 4], [16, 2], [32, 2]], 'representations': {'PC': {'code': 7567729481006065028822347088807781756951266357035170429861092389422083, 'gens': [1, 2, 4, 5, 6], 'pres': [8, -2, -2, -3, -2, 2, 2, -2, -5, 161, 41, 194, 23043, 3851, 5875, 1307, 11044, 15132, 7460, 32837, 12973, 9381, 2141, 1093, 141, 48390, 166, 49159]}, 'GLZN': {'d': 2, 'p': 85, 'gens': [5527161, 615571, 9826016, 614141, 49744206, 31502326, 3721361, 42374694]}, 'Perm': {'d': 23, 'gens': [506520282718638846, 1131683576599619406721, 2455438497192474433920, 166864779438295680, 966, 3642223264519385871360, 4820439767847188123520, 1131683576599619406720]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{20}.S_4', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
