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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '3072.cc', 'ambient_counter': 55, 'ambient_order': 3072, 'ambient_tex': '(C_4\\times C_8).\\GL(2,\\mathbb{Z}/4)', 'central': False, 'central_factor': False, 'centralizer_order': 128, 'characteristic': True, 'core_order': 768, 'counter': 7, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '3072.cc.4.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '768.1084932', 'subgroup_hash': 1084932, 'subgroup_order': 768, 'subgroup_tex': 'C_2^4:C_{48}', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3072.cc', 'aut_centralizer_order': None, 'aut_label': '4.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '24.d1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.c1.a1', '8.d1.a1', '8.d1.b1', '8.m1.a1', '8.p1.a1', '12.b1.a1', '16.p1.a1'], 'core': '4.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [294921, 573457, 172813, 836105, 383041, 632172, 49665, 557073, 1015839], 'label': '3072.cc.4.b1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.b1.a1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['8.c1.a1', '8.d1.b1', '8.d1.a1', '12.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.b1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.b1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '192.935', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [12, 4, 4, 4, 4], 'aut_gens': [[648556, 1032719, 573457, 49665, 453063], [974004, 49681, 49665, 557585, 907345], [140653, 573953, 49665, 573457, 236109], [957620, 1032719, 573457, 49665, 175055], [664957, 1032719, 557585, 49665, 907345], [664957, 1032719, 557585, 49665, 43979]], 'aut_group': None, 'aut_hash': 7474513708491177873, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 18432, 'aut_permdeg': 40, 'aut_perms': [809423928496687167271487085757255527994810877675, 295345899726570506854301515652037996798773058441, 180323696592935524080812416574114644051812565223, 75645435193711474216406918532443387415632573939, 67180695819489304252814337793113993042085955482], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 3, 1, 2], [2, 3, 6, 1], [3, 4, 2, 1], [4, 1, 2, 1], [4, 1, 6, 1], [4, 3, 2, 1], [4, 3, 6, 1], [6, 4, 2, 1], [6, 4, 12, 1], [8, 1, 4, 1], [8, 1, 12, 1], [8, 3, 4, 1], [8, 3, 12, 1], [12, 4, 4, 1], [12, 4, 12, 1], [16, 1, 32, 1], [16, 3, 32, 1], [24, 4, 8, 1], [24, 4, 24, 1], [48, 4, 64, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2.C_2^6.D_6^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '768.1090215', 'autcent_hash': 1090215, 'autcent_nilpotent': False, 'autcent_order': 768, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '(C_2^3\\times C_4):S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 3, 8], [3, 4, 2], [4, 1, 8], [4, 3, 8], [6, 4, 14], [8, 1, 16], [8, 3, 16], [12, 4, 16], [16, 1, 32], [16, 3, 32], [24, 4, 32], [48, 4, 64]], 'center_label': '64.183', 'center_order': 64, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', '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'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1084932, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['16.1', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 3, 1, 8], [3, 4, 2, 1], [4, 1, 2, 4], [4, 3, 2, 4], [6, 4, 2, 7], [8, 1, 4, 4], [8, 3, 4, 4], [12, 4, 4, 4], [16, 1, 8, 4], [16, 3, 8, 4], [24, 4, 8, 4], [48, 4, 16, 4]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 7280, 'exponent': 48, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '96.228', 'hash': 1084932, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 1, 2, 2, 1], 'inner_gens': [[648556, 1032719, 557585, 573457, 453063], [648556, 1032719, 573457, 49665, 453063], [107900, 1032719, 573457, 49665, 453063], [632684, 1032719, 573457, 49665, 453063], [648556, 1032719, 573457, 49665, 453063]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 192], [3, 64]], 'label': '768.1084932', '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': 'C2^4:C48', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 256, 'number_divisions': 60, 'number_normal_subgroups': 147, 'number_subgroup_autclasses': 176, 'number_subgroup_classes': 595, 'number_subgroups': 1540, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 31], [3, 8], [4, 32], [6, 56], [8, 64], [12, 64], [16, 128], [24, 128], [48, 256]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2, 4, 2, 2, 2, 2, 2], 'outer_gen_pows': [32769, 32769, 32769, 32769, 32769, 32769, 32769, 32769, 32769, 32769], 'outer_gens': [[157053, 1032719, 49665, 573457, 453063], [433844, 1015839, 573457, 49665, 437207], [648556, 1032719, 573457, 49665, 321987], [416931, 1032719, 49665, 573457, 453063], [416931, 1032719, 49665, 573457, 190943], [648556, 1032719, 573457, 49665, 437207], [648556, 1032719, 573457, 49665, 645193], [108396, 508447, 573457, 49665, 629337], [108396, 1032719, 573457, 49665, 120921], [648556, 1032719, 573457, 49665, 977367]], 'outer_group': '1536.408641001', 'outer_hash': 5942603984477037536, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 20, 'outer_perms': [122007708013328040, 11212469452800, 23, 122002101778601656, 377744146144662983, 513696337505971200, 1541100224635776000, 1149404675007012480, 1663101012472709160, 122002101778601640], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '(C_2^4\\times C_4):S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 16, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 8], [6, 4], [8, 8], [12, 4], [16, 4], [24, 4]], 'representations': {'PC': {'code': '45511043742870652668070063131177108096', 'gens': [1, 3, 4, 5, 6], 'pres': [9, -2, -3, -2, -2, 2, -2, -2, -2, -2, 18, 1299, 444, 544, 823, 158, 186, 214]}, 'GLZN': {'d': 2, 'p': 28, 'gens': [340649, 214622, 375039, 22345, 26725, 124067, 33321, 33335, 285389]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [32785, 294921, 124284, 33281, 245767, 557073, 573953, 713485, 453063]}, 'Perm': {'d': 24, 'gens': [181, 28260974647749700644480, 82, 56212811007499316626109, 56212811007499316625920, 56212811007499316625840, 83490668053409500934400, 110527028898728529830400, 56212811007499316620800]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 48], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4:C_{48}', 'transitive_degree': 192, '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': '32.3', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 48, 'aut_gen_orders': [8, 12, 12, 4, 4, 12, 16, 12], 'aut_gens': [[591870, 236718, 49665, 1015839, 573457], [212778, 93245, 49665, 1015839, 557585], [1043921, 574578, 573457, 1015839, 557585], [122317, 126014, 573457, 1015839, 557585], [48271, 614073, 573457, 1015839, 49665], [536032, 72873, 49665, 491535, 557585], [249801, 524222, 573457, 1015839, 557585], [9392, 575074, 49665, 1015839, 573457], [834039, 253630, 557585, 491535, 49665]], 'aut_group': None, 'aut_hash': 5314746377555349766, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 98304, 'aut_permdeg': 768, 'aut_perms': 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841303265433437328376376945895559522122940655844490035610447737820804434065698864701375073079495589870948954549655527453096937735752733137620235044404864239011822877913196015709078072866314267306838397211634893870342986371353020573066074970021319661852531093128296326488186083215609954651015044133532162580497588790301107135010149317318263331341596668228945003736897973385217973649395433413594450761042651672205510295923061557768349855131929164385281362357908051858782920150329798887139178923115839712114170], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4.C_2^5', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 40, 'pgroup': 0, 'primary_abelian_invariants': [4, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 14], [3, 4], [4, 20], [6, 10], [8, 8], [12, 10], [16, 6], [24, 2], [32, 4], [48, 4]], 'representations': {'PC': {'code': '4374104001971812637210868657296110215910690691632974045662714270831884873597020723817377879952630871178771851970344968', 'gens': [1, 4, 9, 10, 11], 'pres': [11, -2, -2, -2, -2, -2, -2, -2, -3, -2, 2, 2, 22, 56, 1608, 134819, 42430, 124, 41364, 42695, 158, 149957, 64432, 192, 54214, 61617, 226, 191495, 90130, 19049, 11932, 4815, 3044, 1273, 232330, 29083, 2958, 7325, 802, 1902]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [32785, 98307, 67566, 444639, 294921, 124284, 33281, 557073, 573953, 713485, 828161]}, 'Perm': {'d': 40, 'gens': [106725278220396658682310360437065777294176856320, 85946506876778786839613506412246644828980518400, 43980516372435047537341729429260515801035532285, 21583860631528492200752571982553339407830281647, 127885534293755012324800517846834471011426579200, 10080, 106890825587930069984075720886007484472064190760, 106890825587930069984075720886007484472064185600, 193075178158260629138208243878660206787155200, 86167098932475055551957252829593389915556369960, 179311424803903420894810092865203991933591696]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 8], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_4\\times C_8).\\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
