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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1728.34767', 'ambient_counter': 34767, 'ambient_order': 1728, 'ambient_tex': '(C_2\\times C_4).S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 48, 'counter': 135, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.34767.9.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '9.b1.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': 9, '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': '192.1195', 'subgroup_hash': 1195, 'subgroup_order': 192, 'subgroup_tex': 'C_2.(Q_8\\times D_6)', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.34767', 'aut_centralizer_order': None, 'aut_label': '9.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '432.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1.a1', '3.b1.a1'], 'contains': ['18.a1.a1', '18.b1.a1', '18.c1.a1', '18.d1.a1', '18.m1.b1', '18.n1.b1', '18.o1.b1', '18.p1.b1', '18.q1.a1', '18.r1.a1', '18.s1.a1', '18.t1.b1', '18.u1.b1', '18.v1.b1', '18.w1.b1', '27.a1.a1'], 'core': '36.a1.b1', 'coset_action_label': None, 'count': 9, 'diagramx': None, 'generators': [1, 864, 936, 6, 4, 36, 432], 'label': '1728.34767.9.b1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.b1.a1', 'old_label': '9.b1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '9.b1.a1', '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6, 2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 12, 48], [1, 10, 12, 48], [125, 2, 12, 48], [97, 2, 12, 48], [1, 122, 12, 48], [1, 98, 12, 48], [1, 2, 132, 48], [1, 2, 108, 48], [1, 2, 12, 168], [1, 2, 12, 144]], 'aut_group': '1536.408526597', 'aut_hash': 7468161104855904088, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 24, 'aut_perms': [666743587073014404531, 364079791595743667843483, 183014830560256148174836, 53397762472454400, 40948702010092800, 50502821760, 38088691920, 22572775080597208734990, 20327194077989707653366], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 6, 2, 1], [3, 2, 1, 1], [4, 2, 2, 1], [4, 4, 1, 5], [4, 6, 2, 3], [4, 12, 1, 4], [6, 2, 1, 3], [6, 4, 2, 1], [12, 4, 2, 2], [12, 8, 1, 4]], 'aut_supersolvable': True, 'aut_tex': 'C_2^8\\times S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', '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, 3], [2, 2, 2], [2, 6, 2], [3, 2, 1], [4, 2, 2], [4, 4, 5], [4, 6, 6], [4, 12, 4], [6, 2, 3], [6, 4, 2], [12, 4, 4], [12, 8, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '48.51', 'commutator_count': 1, 'commutator_label': '12.5', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1195, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 6, 1, 2], [3, 2, 1, 1], [4, 2, 2, 1], [4, 4, 1, 5], [4, 6, 1, 4], [4, 6, 2, 1], [4, 12, 1, 4], [6, 2, 1, 3], [6, 4, 1, 2], [12, 4, 2, 2], [12, 8, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 262080, 'exponent': 12, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '48.51', 'hash': 1195, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 2, 2], 'inner_gens': [[1, 130, 12, 48], [125, 2, 132, 168], [1, 122, 12, 144], [1, 122, 108, 48]], 'inner_hash': 51, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': True, 'inner_tex': 'C_2^2\\times D_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 16], [4, 7]], 'label': '192.1195', 'linC_count': 160, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 102, 'linQ_dim': 10, 'linQ_dim_count': 112, 'linR_count': 112, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2.(Q8*D6)', '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': 30, 'number_characteristic_subgroups': 91, 'number_conjugacy_classes': 39, 'number_divisions': 35, 'number_normal_subgroups': 105, 'number_subgroup_autclasses': 204, 'number_subgroup_classes': 228, 'number_subgroups': 528, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 19], [3, 2], [4, 108], [6, 14], [12, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[121, 2, 12, 48], [97, 2, 12, 48], [1, 122, 12, 48], [1, 98, 12, 48], [1, 2, 132, 48]], 'outer_group': '32.51', 'outer_hash': 51, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 19, '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, 5], [8, 2]], 'representations': {'PC': {'code': 169571095266449455242573072, 'gens': [1, 2, 4, 6], 'pres': [7, -2, -2, -3, -2, -2, 2, -2, 1821, 36, 170, 1858, 80, 3540, 530, 124]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [27019, 40966, 297011, 630313, 626113, 513019, 407655]}, 'Perm': {'d': 19, 'gens': [368056, 6446835113255398, 13897975637162842, 20715131677386973, 6446835113271709, 30109, 3991680]}}, 'schur_multiplier': [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.(Q_8\\times D_6)', 'transitive_degree': 96, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 4, 12, 6, 6, 4, 6, 2, 2, 6], 'aut_gens': [[1, 2, 12, 144], [5, 58, 636, 1584], [54, 53, 1192, 1656], [918, 993, 332, 144], [77, 122, 348, 1008], [945, 26, 1236, 216], [6, 53, 40, 792], [102, 97, 904, 1080], [945, 874, 1236, 792], [945, 946, 1164, 1656], [945, 26, 84, 1584]], 'aut_group': None, 'aut_hash': 7214043090240141899, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 110592, 'aut_permdeg': 40, 'aut_perms': [709979369774842847576604511590086135563423899222, 761601611040396596764831259211403316787327720283, 41410108889281508068580353050681315835909274323, 424167849300852069823472102349708414626437284912, 121476680672375753227938030096996211022622188990, 223793418295878241191928730688343871969558402632, 659894018634537844879589211446491437586343091582, 733344698777967944419604615237865187888101110768, 6695292109535124329982622180950296452040573825, 750848933792582221923010973771372605293620331980], 'aut_phi_ratio': 192.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 6, 4, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [4, 4, 1, 1], [4, 6, 2, 1], [4, 12, 1, 1], [4, 12, 2, 1], [4, 18, 2, 2], [4, 36, 2, 2], [4, 54, 2, 1], [4, 108, 1, 1], [6, 2, 1, 3], [6, 2, 2, 3], [6, 4, 1, 3], [6, 4, 2, 3], [6, 8, 1, 3], [6, 12, 4, 2], [6, 24, 4, 1], [12, 4, 2, 1], [12, 8, 2, 2], [12, 8, 4, 2], [12, 12, 4, 2], [12, 18, 4, 1], [12, 24, 2, 4], [12, 24, 4, 1], [12, 36, 2, 1], [12, 36, 4, 1], [12, 72, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times C_3^3.C_2^6.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '432.741', 'autcentquo_hash': 741, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^3:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 6, 4], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 4, 1], [4, 6, 2], [4, 12, 3], [4, 18, 4], [4, 36, 4], [4, 54, 2], [4, 108, 1], [6, 2, 9], [6, 4, 9], [6, 8, 3], [6, 12, 8], [6, 24, 4], [12, 4, 2], [12, 8, 12], [12, 12, 8], [12, 18, 4], [12, 24, 12], [12, 36, 6], [12, 72, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '432.759', 'commutator_count': 1, 'commutator_label': '108.45', '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', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 34767, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 6, 1, 4], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 4, 1, 1], [4, 6, 2, 1], [4, 12, 1, 3], [4, 18, 1, 2], [4, 18, 2, 1], [4, 36, 1, 4], [4, 54, 1, 2], [4, 108, 1, 1], [6, 2, 1, 9], [6, 4, 1, 9], [6, 8, 1, 3], [6, 12, 1, 8], [6, 24, 1, 4], [12, 4, 2, 1], [12, 8, 1, 4], [12, 8, 2, 4], [12, 12, 2, 4], [12, 18, 4, 1], [12, 24, 1, 4], [12, 24, 2, 4], [12, 36, 1, 2], [12, 36, 2, 2], [12, 72, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 22145760, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '432.759', 'hash': 34767, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 6, 6], 'inner_gens': [[1, 946, 12, 1080], [941, 2, 60, 1080], [1, 98, 12, 1584], [937, 938, 300, 144]], 'inner_hash': 759, 'inner_nilpotent': False, 'inner_order': 432, 'inner_split': True, 'inner_tex': 'C_2\\times S_3^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 40], [4, 41], [8, 14]], 'label': '1728.34767', '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*C4).S3^3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 54, 'number_characteristic_subgroups': 58, 'number_conjugacy_classes': 111, 'number_divisions': 91, 'number_normal_subgroups': 194, 'number_subgroup_autclasses': 634, 'number_subgroup_classes': 1168, 'number_subgroups': 8356, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 27], [3, 26], [4, 484], [6, 270], [12, 920]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 4], 'outer_gen_pows': [96, 584, 624, 4, 0, 676, 0], 'outer_gens': [[1, 898, 12, 720], [9, 2, 588, 1008], [1, 50, 1524, 1008], [941, 866, 1500, 1656], [869, 874, 948, 792], [77, 26, 588, 144], [102, 965, 976, 1584]], 'outer_group': '256.55643', 'outer_hash': 55643, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 64, 'outer_perms': [12257309289760343774605855458287405213698076116231869265937468697619720813973254821366121, 73831087184710904455939441741154406504506842877844678601083159089852749803360527546906393, 118465307525375087883419688714462959740646022452006035163261107839277225071770426936632564, 28453496941954504967169382965599332177989788091405576281442614978188062893975419221223127, 52866728992654941797193116356984683845200993163334619838458962509130021815571356789552991, 40161574247444892958827969444148661024489788135526272700405701746597403786759565019663472, 86109635487182336369671667448550370881728094034728774797973941769446814441087000109645145], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5:D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 28], [4, 23], [8, 22], [16, 2]], 'representations': {'PC': {'code': 19085572857874713721845307296823019314483703932644263148893257, 'gens': [1, 2, 4, 7], 'pres': [9, 2, 2, 3, 2, 2, 3, 2, 2, 3, 17029, 46, 218, 1092, 102, 2713, 130, 2606, 68046, 34035, 8349, 186, 8674, 214, 7811]}, 'Perm': {'d': 25, 'gens': [622798591475511783129121, 622798591475512259630064, 1319781461471106746319624, 1969307671116938609260344, 622798591475511779328000, 12176904, 6706022400, 3, 22230464256000]}}, 'schur_multiplier': [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\\times C_4).S_3^3', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
