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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '5280.l', 'ambient_counter': 12, 'ambient_order': 5280, 'ambient_tex': 'C_{24}:C_2\\times F_{11}', 'central': False, 'central_factor': False, 'centralizer_order': 240, 'characteristic': False, 'core_order': 4, 'counter': 242, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5280.l.660.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '660.c1.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': 660, '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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5280.l', 'aut_centralizer_order': None, 'aut_label': '660.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '22.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['60.a1.a1', '132.b1.a1', '220.e1.a1', '330.a1.a1', '330.b1.a1', '330.c1.a1'], 'contains': ['1320.a1.a1', '1320.b1.a1', '1320.f1.a1'], 'core': '1320.a1.a1', 'coset_action_label': None, 'count': 11, 'diagramx': [6454, -1, 3554, -1, 5541, -1, 3964, -1], 'generators': [5, 1320], 'label': '5280.l.660.c1.a1', 'mobius_quo': None, 'mobius_sub': -6, 'normal_closure': '60.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '11.a1.a1', 'old_label': '660.c1.a1', 'projective_image': '2640.x', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '660.c1.a1', '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_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, 3], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', 'transitive_degree': 8, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [10, 10, 10, 10, 20, 60, 30], 'aut_gens': [[1, 10, 20], [3121, 4250, 4300], [3121, 810, 4660], [1681, 5050, 1700], [1681, 4010, 4060], [4801, 3570, 2020], [3601, 4850, 1220], [3841, 1410, 2540]], 'aut_group': None, 'aut_hash': 5956886757194812668, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21120, 'aut_permdeg': 38, 'aut_perms': [334784766771750714765567746750204820611589375, 10212800402108678825526549856042727626254411, 358529843608810321782796320526048808124924089, 482472868688125166741347152976637663836434582, 368890856753190538917988343212023462760358644, 362864636238509311977018059360820296313007280, 336128383470905682413776866880492065784532633], 'aut_phi_ratio': 16.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 2, 1], [2, 12, 1, 1], [2, 132, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 12, 1, 1], [4, 22, 1, 1], [4, 132, 1, 1], [5, 11, 1, 4], [6, 2, 1, 1], [6, 22, 2, 1], [8, 2, 2, 1], [8, 22, 2, 1], [10, 11, 1, 4], [10, 11, 2, 4], [10, 132, 1, 8], [11, 10, 1, 1], [12, 2, 2, 1], [12, 22, 2, 1], [15, 22, 1, 4], [20, 22, 1, 8], [20, 132, 1, 8], [22, 10, 1, 1], [22, 120, 1, 1], [24, 2, 4, 1], [24, 22, 4, 1], [30, 22, 1, 4], [30, 22, 2, 4], [33, 20, 1, 1], [40, 22, 2, 8], [44, 20, 1, 1], [44, 120, 1, 1], [60, 22, 2, 8], [66, 20, 1, 1], [88, 20, 2, 1], [120, 22, 4, 8], [132, 20, 2, 1], [264, 20, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{66}.C_{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': 330, 'autcentquo_group': None, 'autcentquo_hash': 4845733366138083661, 'autcentquo_nilpotent': False, 'autcentquo_order': 2640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{11}:(C_2^2\\times C_{10}\\times S_3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 11, 2], [2, 12, 1], [2, 132, 1], [3, 2, 1], [4, 2, 1], [4, 12, 1], [4, 22, 1], [4, 132, 1], [5, 11, 4], [6, 2, 1], [6, 22, 2], [8, 2, 2], [8, 22, 2], [10, 11, 12], [10, 132, 8], [11, 10, 1], [12, 2, 2], [12, 22, 2], [15, 22, 4], [20, 22, 8], [20, 132, 8], [22, 10, 1], [22, 120, 1], [24, 2, 4], [24, 22, 4], [30, 22, 12], [33, 20, 1], [40, 22, 16], [44, 20, 1], [44, 120, 1], [60, 22, 16], [66, 20, 1], [88, 20, 2], [120, 22, 32], [132, 20, 2], [264, 20, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '2640.x', 'commutator_count': 1, 'commutator_label': '132.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1', '11.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['110.1', 1], ['48.6', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 1, 2], [2, 12, 1, 1], [2, 132, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 12, 1, 1], [4, 22, 1, 1], [4, 132, 1, 1], [5, 11, 4, 1], [6, 2, 1, 1], [6, 22, 1, 2], [8, 2, 2, 1], [8, 22, 2, 1], [10, 11, 4, 3], [10, 132, 4, 2], [11, 10, 1, 1], [12, 2, 2, 1], [12, 22, 2, 1], [15, 22, 4, 1], [20, 22, 4, 2], [20, 132, 4, 2], [22, 10, 1, 1], [22, 120, 1, 1], [24, 2, 4, 1], [24, 22, 4, 1], [30, 22, 4, 3], [33, 20, 1, 1], [40, 22, 8, 2], [44, 20, 1, 1], [44, 120, 1, 1], [60, 22, 8, 2], [66, 20, 1, 1], [88, 20, 2, 1], [120, 22, 16, 2], [132, 20, 2, 1], [264, 20, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1999872, 'exponent': 1320, 'exponents_of_order': [5, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[20, 0, 4]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '1320.142', 'hash': 3340380322487147873, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 660, 'inner_gen_orders': [10, 2, 132], 'inner_gens': [[1, 1450, 3860], [3841, 10, 2620], [1441, 2690, 20]], 'inner_hash': 1114821233248874711, 'inner_nilpotent': False, 'inner_order': 2640, 'inner_split': True, 'inner_tex': 'D_{12}\\times F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': 20, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 40, 'irrep_stats': [[1, 40], [2, 110], [10, 4], [20, 11]], 'label': '5280.l', 'linC_count': 160, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 32, 'linQ_dim': 16, 'linQ_dim_count': 32, 'linR_count': 80, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C24:C2*F11', '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, 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': 100, 'number_characteristic_subgroups': 69, 'number_conjugacy_classes': 165, 'number_divisions': 50, 'number_normal_subgroups': 77, 'number_subgroup_autclasses': 256, 'number_subgroup_classes': 272, 'number_subgroups': 4136, 'old_label': None, 'order': 5280, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 167], [3, 2], [4, 168], [5, 44], [6, 46], [8, 48], [10, 1188], [11, 10], [12, 48], [15, 88], [20, 1232], [22, 130], [24, 96], [30, 264], [33, 20], [40, 352], [44, 140], [60, 352], [66, 20], [88, 40], [120, 704], [132, 40], [264, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 490, 2180], [2641, 490, 4820], [1, 490, 1300]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 12], [8, 8], [10, 4], [16, 4], [20, 3], [32, 2], [40, 2], [80, 1]], 'representations': {'PC': {'code': '46322998854108250826595521280476493151650528079774433168774427595788748881483859469375884103000019', 'gens': [1, 3, 4], 'pres': [8, -2, -5, -2, -2, -2, -2, -3, -11, 16, 34802, 57730, 123523, 8011, 8403, 91, 97604, 20012, 20980, 116, 234245, 48013, 24981, 141, 250886, 112014, 28694, 222, 184327, 46095, 30743]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [31471887339564514, 44892999667169041, 15858444949379930, 26934596962789267, 29311048591218838, 9794019781044287, 32390444358457867, 41772741070013040]}, 'Perm': {'d': 22, 'gens': [10463720175840647544, 61328800643894784000, 112805267711768755200, 158145787614360268803, 216657398669661348000, 270067105113437238506, 104885904124511604024, 3719544]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{24}:C_2\\times F_{11}', 'transitive_degree': 264, 'wreath_data': None, 'wreath_product': False}
