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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '79860.e', 'ambient_counter': 5, 'ambient_order': 79860, 'ambient_tex': 'C_{10}\\times C_{11}\\wr S_3', 'central': False, 'central_factor': False, 'centralizer_order': 1210, 'characteristic': False, 'core_order': 605, 'counter': 36, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '79860.e.66.k1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '66.k1', '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': 66, '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': '1210.22', 'subgroup_hash': 22, 'subgroup_order': 1210, 'subgroup_tex': 'D_{11}\\times C_{55}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '79860.e', 'aut_centralizer_order': None, 'aut_label': '66.k1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '66.b1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['6.b1', '22.b1', '33.d1'], 'contains': ['132.a1', '330.k1', '726.j1', '726.k1'], 'core': '132.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [4809, -1, 6046, -1], 'generators': [1771681, 99207483, 2580810, 143496522], 'label': '79860.e.66.k1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '11.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '66.k1', 'projective_image': '15972.c', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '66.k1', 'subgroup_fusion': None, 'weyl_group': '22.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '110.6', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [10, 4, 10, 11], 'aut_gens': [[1, 22], [19, 462], [21, 704], [1, 1122], [111, 22]], 'aut_group': None, 'aut_hash': 2648413785721325108, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4400, 'aut_permdeg': 25, 'aut_perms': [10816639139114723033792129, 15159796166042493648653033, 369701743670580087296936, 163345427532720414996842], 'aut_phi_ratio': 10.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 11, 1, 1], [5, 1, 4, 1], [10, 11, 4, 1], [11, 1, 10, 1], [11, 2, 5, 1], [11, 2, 50, 1], [22, 11, 10, 1], [55, 1, 40, 1], [55, 2, 20, 1], [55, 2, 200, 1], [110, 11, 40, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{11}:(C_2\\times C_{10}\\times C_{20})', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '40.9', 'autcent_hash': 9, 'autcent_nilpotent': True, 'autcent_order': 40, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{20}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 11, 1], [5, 1, 4], [10, 11, 4], [11, 1, 10], [11, 2, 55], [22, 11, 10], [55, 1, 40], [55, 2, 220], [110, 11, 40]], 'center_label': '55.2', 'center_order': 55, 'central_product': True, 'central_quotient': '22.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '11.1', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 22, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['22.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 11, 1, 1], [5, 1, 4, 1], [10, 11, 4, 1], [11, 1, 10, 1], [11, 2, 5, 1], [11, 2, 10, 5], [22, 11, 10, 1], [55, 1, 40, 1], [55, 2, 20, 1], [55, 2, 40, 5], [110, 11, 40, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 216, 'exponent': 110, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[2, 0, 200]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1210.22', 'hash': 22, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [2, 11], 'inner_gens': [[1, 462], [771, 22]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 22, 'inner_split': True, 'inner_tex': 'D_{11}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 4, 'irrep_stats': [[1, 110], [2, 275]], 'label': '1210.22', 'linC_count': 200, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 14, 'linQ_dim': 24, 'linQ_dim_count': 14, 'linR_count': 300, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D11*C55', 'ngens': 4, '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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 385, 'number_divisions': 20, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 26, 'number_subgroups': 76, 'old_label': None, 'order': 1210, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 11], [5, 4], [10, 44], [11, 120], [22, 110], [55, 480], [110, 440]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [10, 20], 'outer_gen_pows': [0, 0], 'outer_gens': [[13, 462], [21, 814]], 'outer_group': '200.37', 'outer_hash': 37, 'outer_nilpotent': True, 'outer_order': 200, 'outer_permdeg': 16, 'outer_perms': [1313941305664, 13452370560], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{10}\\times C_{20}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 5, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [10, 3], [20, 5], [40, 3], [80, 5]], 'representations': {'PC': {'code': 81663522125330939, 'gens': [1, 3], 'pres': [4, -2, -11, -5, -11, 8, 5546, 70, 17603]}, 'GLFp': {'d': 2, 'p': 331, 'gens': [10625554733, 3807792770, 109892]}, 'Perm': {'d': 27, 'gens': [368047, 1401558681600, 419450149241383865020416000, 4364893]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [110], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{11}\\times C_{55}', 'transitive_degree': 110, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '220.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [60, 30, 60, 20, 20, 10], 'aut_gens': [[1771681, 120998121, 80529571, 99207483], [106810028, 97016897, 196643283, 197299466], [109074058, 208318546, 99542895, 21915026], [107718012, 96853228, 41234480, 139158757], [108210481, 170704113, 159287459, 120145433], [2911017, 212972242, 21915026, 79720323], [107887049, 150354167, 79720323, 80529571]], 'aut_group': None, 'aut_hash': 5829061629890111487, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 580800, 'aut_permdeg': 80, 'aut_perms': [33188016233193426781273674270817718954018114055911915530220068770420851156121747611000230842941700845443922637531113527, 5722488144838667397840341547382775980114388581849480560114296128431542105215180037304172865296674224333136322097979054, 63997518964409171148513625392989359423982284457948155414118618123996999267000160319640175749691135564316049445502323756, 49808939110855322790496924176709397520566745137564127973989412996800228385740863357766103673854673545711370608461196036, 37387514122649573901173268091882738672759794929048551641098033092993471867403827592500508113190780867475358588871287242, 16296840414111789950058044751351532476463049837744979642404461868727518743041694904675032237650419920708306851972416863], 'aut_phi_ratio': 30.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 33, 2, 1], [3, 242, 1, 1], [5, 1, 4, 1], [6, 242, 1, 1], [10, 1, 4, 1], [10, 33, 8, 1], [11, 1, 10, 1], [11, 3, 10, 1], [11, 3, 100, 1], [11, 6, 5, 1], [11, 6, 10, 1], [11, 6, 50, 1], [11, 6, 100, 1], [15, 242, 4, 1], [22, 1, 10, 1], [22, 3, 10, 1], [22, 3, 100, 1], [22, 6, 5, 1], [22, 6, 10, 1], [22, 6, 50, 1], [22, 6, 100, 1], [22, 33, 20, 2], [22, 33, 200, 1], [30, 242, 4, 1], [33, 242, 10, 1], [55, 1, 40, 1], [55, 3, 40, 1], [55, 3, 400, 1], [55, 6, 20, 1], [55, 6, 40, 1], [55, 6, 200, 1], [55, 6, 400, 1], [66, 242, 10, 1], [110, 1, 40, 1], [110, 3, 40, 1], [110, 3, 400, 1], [110, 6, 20, 1], [110, 6, 40, 1], [110, 6, 200, 1], [110, 6, 400, 1], [110, 33, 80, 2], [110, 33, 800, 1], [165, 242, 40, 1], [330, 242, 40, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{11}^2.C_{15}.C_{20}.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '80.45', 'autcent_hash': 45, 'autcent_nilpotent': True, 'autcent_order': 80, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{20}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '7260.bg', 'autcentquo_hash': 7237787727408842954, 'autcentquo_nilpotent': False, 'autcentquo_order': 7260, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{11}^2:(S_3\\times C_{10})', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 33, 2], [3, 242, 1], [5, 1, 4], [6, 242, 1], [10, 1, 4], [10, 33, 8], [11, 1, 10], [11, 3, 110], [11, 6, 165], [15, 242, 4], [22, 1, 10], [22, 3, 110], [22, 6, 165], [22, 33, 240], [30, 242, 4], [33, 242, 10], [55, 1, 40], [55, 3, 440], [55, 6, 660], [66, 242, 10], [110, 1, 40], [110, 3, 440], [110, 6, 660], [110, 33, 960], [165, 242, 40], [330, 242, 40]], 'center_label': '110.6', 'center_order': 110, 'central_product': True, 'central_quotient': '726.5', 'commutator_count': 1, 'commutator_label': '363.2', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '3.1', '5.1', '11.1', '11.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['2.1', 1], ['5.1', 1], ['726.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 33, 1, 2], [3, 242, 1, 1], [5, 1, 4, 1], [6, 242, 1, 1], [10, 1, 4, 1], [10, 33, 4, 2], [11, 1, 10, 1], [11, 3, 10, 11], [11, 6, 5, 1], [11, 6, 10, 16], [15, 242, 4, 1], [22, 1, 10, 1], [22, 3, 10, 11], [22, 6, 5, 1], [22, 6, 10, 16], [22, 33, 10, 24], [30, 242, 4, 1], [33, 242, 10, 1], [55, 1, 40, 1], [55, 3, 40, 11], [55, 6, 20, 1], [55, 6, 40, 16], [66, 242, 10, 1], [110, 1, 40, 1], [110, 3, 40, 11], [110, 6, 20, 1], [110, 6, 40, 16], [110, 33, 40, 24], [165, 242, 40, 1], [330, 242, 40, 1]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 2592, 'exponent': 330, 'exponents_of_order': [3, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[3, 0, 800], [6, 0, 600]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '79860.e', 'hash': 5510129331216617240, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [2, 3, 11, 11], 'inner_gens': [[1771681, 121720491, 80697277, 99207483], [107394579, 120998121, 80697277, 60400869], [3388846, 42728710, 80529571, 99207483], [1771681, 121486598, 80529571, 99207483]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 726, 'inner_split': True, 'inner_tex': 'C_{11}^2:S_3', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': None, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 220], [2, 110], [3, 2200], [6, 1650]], 'label': '79860.e', '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': True, 'name': 'C10*C11wrS3', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 48, 'number_characteristic_subgroups': 28, 'number_conjugacy_classes': 4180, 'number_divisions': 180, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 136, 'number_subgroup_classes': 448, 'number_subgroups': 6536, 'old_label': None, 'order': 79860, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 67], [3, 242], [5, 4], [6, 242], [10, 268], [11, 1330], [15, 968], [22, 9250], [30, 968], [33, 2420], [55, 5320], [66, 2420], [110, 37000], [165, 9680], [330, 9680]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 20, 'outer_gen_orders': [2, 2, 10, 20], 'outer_gen_pows': [201142769, 1771562, 201016445, 201142769], 'outer_gens': [[107133461, 23241550, 157668963, 157836669], [1771681, 23721953, 139158757, 118694643], [109011743, 101991485, 118694643, 120145433], [109011743, 96212951, 60233163, 61851659]], 'outer_group': '800.1185', 'outer_hash': 1185, 'outer_nilpotent': True, 'outer_order': 800, 'outer_permdeg': 20, 'outer_perms': [355693695033600, 1307674368000, 121645100408832064, 122002115150770560], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{10}\\times C_{20}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 40, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': None, 'representations': {'PC': {'code': '5497376773593720357490282277375688418018166352082055180013999', 'gens': [1, 2, 6, 7], 'pres': [7, -2, -2, -3, -5, -11, -11, 11, 98589, 36, 1058066, 79, 164, 2106725, 1053372, 381169, 145543, 905540]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 28745646345613891, 42771567939938742, 33418192856010432, 19063078487671420, 4764541194236758, 17298483428108842]}, 'GLZq': {'d': 2, 'q': 121, 'gens': [106969989, 47832174, 1771617, 41081349, 196643382, 212587440, 1771681]}, 'Perm': {'d': 40, 'gens': [43496542703628312127224630586176285245408353920, 2520, 244913462535524065563175693811150487349325397120, 131839829285316931676463608873698560, 22546539549097035499542455338957416078694542720, 1, 82965253667880196418319025646639736218803201]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 110], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}\\times C_{11}\\wr S_3', 'transitive_degree': 330, 'wreath_data': None, 'wreath_product': False}
