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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '11520.ec', 'ambient_counter': 107, 'ambient_order': 11520, 'ambient_tex': 'C_2^5:C_6\\times A_5', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 8, 'counter': 906, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '11520.ec.240.bi1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '240.bi1', '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': 240, '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': '48.44', 'subgroup_hash': 44, 'subgroup_order': 48, 'subgroup_tex': 'C_2^2\\times C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '11520.ec', 'aut_centralizer_order': None, 'aut_label': '240.bi1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '120.bz1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['48.i1', '60.x1', '80.e1', '120.t1', '120.by1', '120.bz1'], 'contains': ['480.h1', '480.n1', '480.o1', '480.cu1', '480.dc1', '720.be1'], 'core': '1440.b1', 'coset_action_label': None, 'count': 60, 'diagramx': None, 'generators': [367926, 593, 19679350919, 834774, 121], 'label': '11520.ec.240.bi1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '60.n1', 'old_label': '240.bi1', 'projective_image': '2880.gn', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '240.bi1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '48.44', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [3, 2, 4], [1, 2, 20], [2, 3, 4], [1, 2, 6], [1, 2, 7], [1, 26, 29], [25, 2, 30], [1, 2, 28]], 'aut_group': '384.20089', 'aut_hash': 20089, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 10, 'aut_perms': [85945, 848929, 166680, 1961646, 1270440, 1538646, 2682390, 848928], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [3, 1, 2, 1], [4, 1, 8, 1], [6, 1, 2, 1], [6, 1, 12, 1], [12, 1, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '384.20089', 'autcent_hash': 20089, 'autcent_nilpotent': False, 'autcent_order': 384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4:S_4', '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, 7], [3, 1, 2], [4, 1, 8], [6, 1, 14], [12, 1, 16]], 'center_label': '48.44', 'center_order': 48, 'central_product': True, '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', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 44, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 1], [4, 1, 2, 4], [6, 1, 2, 7], [12, 1, 4, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 91, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '24.15', 'hash': 44, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48]], 'label': '48.44', 'linC_count': 5824, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 96, 'linQ_dim': 5, 'linQ_dim_count': 96, 'linR_count': 96, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C12', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 48, 'number_divisions': 24, 'number_normal_subgroups': 54, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 54, 'number_subgroups': 54, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 8], [6, 14], [12, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[3, 2, 4], [1, 2, 20], [2, 3, 4], [1, 2, 6], [1, 2, 7], [1, 26, 29], [25, 2, 30], [1, 2, 28]], 'outer_group': '384.20089', 'outer_hash': 20089, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 10, 'outer_perms': [85945, 848929, 166680, 1961646, 1270440, 1538646, 2682390, 848928], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 4]], 'representations': {'PC': {'code': 26739073, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -3, 42, 58]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [705946296092, 141602720900, 706461794101]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [10319436418, 9789111396, 6526074264, 4548416880, 6762482606]}, 'Perm': {'d': 11, 'gens': [2400, 3628800, 40320, 4, 744]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 12], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{12}', 'transitive_degree': 48, '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [20, 4, 6, 30, 12, 20], 'aut_gens': [[362886, 363599, 2424528, 5167, 5047, 13413242449, 5040, 349234870728, 1], [367920, 368512, 1197654, 5167, 362881, 281176358695, 6, 354981295128, 367927], [362886, 362887, 2424528, 5167, 5638, 268721095536, 5040, 93408099817, 592], [367920, 367921, 1192729, 5167, 5759, 281176364143, 127, 519748416, 713], [5046, 5638, 2782368, 368041, 7, 88177045969, 362880, 355421611848, 367927], [5046, 5047, 2424649, 368041, 598, 180624717942, 363001, 349234865977, 368518], [362886, 363599, 2424649, 5167, 362881, 88177040935, 5161, 355421606814, 367927]], 'aut_group': None, 'aut_hash': 6293584833498415370, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 184320, 'aut_permdeg': 31, 'aut_perms': [4478945397383180209950366877038068, 198719575044961869040821523165454, 5180759045212816889729792409698710, 8142179124918957848175911260254388, 2671505500455493811344944802997601, 175432144929647036742395551854506], 'aut_phi_ratio': 60.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 6, 4, 1], [2, 15, 1, 2], [2, 15, 2, 1], [2, 30, 4, 1], [2, 45, 1, 2], [2, 45, 2, 1], [2, 90, 4, 1], [3, 4, 2, 1], [3, 20, 1, 1], [3, 80, 2, 1], [4, 2, 2, 1], [4, 6, 2, 1], [4, 30, 2, 1], [4, 90, 2, 1], [5, 12, 2, 1], [6, 4, 2, 1], [6, 4, 4, 1], [6, 8, 8, 1], [6, 20, 1, 1], [6, 20, 2, 1], [6, 40, 4, 1], [6, 60, 1, 2], [6, 60, 2, 3], [6, 60, 4, 1], [6, 80, 2, 1], [6, 80, 4, 1], [6, 120, 4, 1], [6, 120, 8, 1], [6, 160, 8, 1], [10, 12, 2, 1], [10, 12, 4, 1], [10, 24, 8, 1], [10, 36, 2, 2], [10, 36, 4, 1], [10, 72, 8, 1], [12, 8, 4, 1], [12, 40, 2, 1], [12, 120, 2, 1], [12, 120, 4, 1], [12, 160, 4, 1], [15, 48, 4, 1], [20, 24, 4, 1], [20, 72, 4, 1], [30, 48, 4, 1], [30, 48, 8, 1], [30, 96, 16, 1], [60, 96, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_5\\times C_2\\wr C_2^2\\times S_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': '5760.co', 'autcentquo_hash': 6537139331229567448, 'autcentquo_nilpotent': False, 'autcentquo_order': 5760, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_4\\times S_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [2, 3, 4], [2, 6, 4], [2, 15, 4], [2, 30, 4], [2, 45, 4], [2, 90, 4], [3, 4, 2], [3, 20, 1], [3, 80, 2], [4, 2, 2], [4, 6, 2], [4, 30, 2], [4, 90, 2], [5, 12, 2], [6, 4, 6], [6, 8, 8], [6, 20, 3], [6, 40, 4], [6, 60, 12], [6, 80, 6], [6, 120, 12], [6, 160, 8], [10, 12, 6], [10, 24, 8], [10, 36, 8], [10, 72, 8], [12, 8, 4], [12, 40, 2], [12, 120, 6], [12, 160, 4], [15, 48, 4], [20, 24, 4], [20, 72, 4], [30, 48, 12], [30, 96, 16], [60, 96, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '2880.gn', 'commutator_count': 1, 'commutator_label': '480.1187', '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', '60.5'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 107, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['2.1', 1], ['60.5', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [2, 3, 1, 4], [2, 6, 1, 4], [2, 15, 1, 4], [2, 30, 1, 4], [2, 45, 1, 4], [2, 90, 1, 4], [3, 4, 2, 1], [3, 20, 1, 1], [3, 80, 2, 1], [4, 2, 1, 2], [4, 6, 1, 2], [4, 30, 1, 2], [4, 90, 1, 2], [5, 12, 2, 1], [6, 4, 2, 3], [6, 8, 2, 4], [6, 20, 1, 3], [6, 40, 1, 4], [6, 60, 1, 4], [6, 60, 2, 4], [6, 80, 2, 3], [6, 120, 1, 4], [6, 120, 2, 4], [6, 160, 2, 4], [10, 12, 2, 3], [10, 24, 2, 4], [10, 36, 2, 4], [10, 72, 2, 4], [12, 8, 2, 2], [12, 40, 1, 2], [12, 120, 1, 2], [12, 120, 2, 2], [12, 160, 2, 2], [15, 48, 4, 1], [20, 24, 2, 2], [20, 72, 2, 2], [30, 48, 4, 3], [30, 96, 4, 4], [60, 96, 4, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2276820, 'exponent': 60, 'exponents_of_order': [8, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '5760.gx', 'hash': 3725610209488732842, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 2, 3, 2, 2, 6, 2, 6, 2], 'inner_gens': [[362886, 363599, 2419494, 5167, 5047, 13412874529, 5040, 349234865694, 1], [362886, 363599, 2419494, 5167, 5166, 13412874648, 5040, 349234865575, 120], [5046, 5759, 2424528, 368041, 367921, 13412874529, 362880, 349234507854, 1], [362886, 363599, 2782368, 5167, 5047, 13412879575, 5040, 349234502808, 1], [362886, 363478, 2782368, 5167, 5047, 13412879575, 5040, 349234502808, 1], [367920, 368512, 2787414, 363007, 362887, 13413242449, 6, 181065387294, 1], [362886, 363599, 2419494, 5167, 5047, 13412874529, 5040, 349234865694, 1], [5046, 5638, 2787414, 368041, 367921, 274468751575, 362880, 349234870728, 1], [362886, 363478, 2424528, 5167, 5047, 13413242449, 5040, 349234870728, 1]], 'inner_hash': 5094570882729245637, 'inner_nilpotent': False, 'inner_order': 2880, 'inner_split': False, 'inner_tex': 'C_2^4:\\GL(2,4)', 'inner_used': [1, 2, 3, 5, 6, 8], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 6], [3, 56], [4, 24], [5, 24], [6, 14], [8, 6], [9, 16], [10, 6], [12, 8], [15, 8], [18, 4], [24, 2], [30, 2]], 'label': '11520.ec', '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': 'C2^5:C6*A5', '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, 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': 60, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 200, 'number_divisions': 120, 'number_normal_subgroups': 114, 'number_subgroup_autclasses': 2815, 'number_subgroup_classes': 12547, 'number_subgroups': 564134, 'old_label': None, 'order': 11520, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 767], [3, 188], [4, 256], [5, 24], [6, 4228], [10, 1128], [12, 1472], [15, 192], [20, 384], [30, 2112], [60, 768]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 712, 0, 0], 'outer_gens': [[367920, 368633, 1192608, 5167, 5047, 13414469335, 6, 349232913048, 1], [367920, 719, 1192608, 5167, 362881, 19681764769, 6, 354981300174, 367927], [362886, 5753, 2424649, 5167, 362881, 19679812135, 5161, 354982522014, 367927], [367920, 126, 1192608, 5167, 363472, 19681765056, 6, 354981300463, 368518], [362886, 5160, 2424528, 5167, 5638, 13413242736, 5040, 349234871017, 592]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [806400, 806520, 368040, 368056, 368062], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [3, 8], [4, 10], [5, 8], [6, 10], [8, 10], [10, 10], [12, 18], [15, 8], [16, 2], [18, 8], [20, 2], [24, 4], [30, 2], [36, 2]], 'representations': {'Perm': {'d': 15, 'gens': [362886, 363599, 2424528, 5167, 5047, 13413242449, 5040, 349234870728, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^5:C_6\\times A_5', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
