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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1760.54', 'ambient_counter': 54, 'ambient_order': 1760, 'ambient_tex': '(C_2\\times C_{44}).C_{20}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 110, 'counter': 17, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1760.54.8.c1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '8.c1.b1', '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': 8, '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': '220.2', 'subgroup_hash': 2, 'subgroup_order': 220, 'subgroup_tex': 'C_{11}:C_{20}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.54', 'aut_centralizer_order': 16, 'aut_label': '8.c1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '220.7', 'aut_weyl_index': 64, 'centralizer': '220.b1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.b1', '4.c1.a1', '4.c1.b1'], 'contains': ['16.a1.a1', '40.c1.b1', '88.c1.b1'], 'core': '16.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [2940, -1, 6852, -1, 1378, -1, 2774, -1], 'generators': [441, 880, 160, 4], 'label': '1760.54.8.c1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '8.c1.b1', 'projective_image': '880.24', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.c1.b1', 'subgroup_fusion': None, 'weyl_group': '110.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '20.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 110, 'aut_gen_orders': [10, 22], 'aut_gens': [[1, 5], [1, 65], [201, 115]], 'aut_group': '220.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 220, 'aut_permdeg': 13, 'aut_perms': [143842590, 982342849], 'aut_phi_ratio': 2.75, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 11, 1, 4], [10, 11, 1, 4], [11, 5, 2, 1], [20, 11, 2, 4], [22, 5, 2, 1], [44, 5, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_{11}', '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': 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, 1, 1], [4, 1, 2], [5, 11, 4], [10, 11, 4], [11, 5, 2], [20, 11, 8], [22, 5, 2], [44, 5, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '55.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 11, 4, 1], [10, 11, 4, 1], [11, 5, 2, 1], [20, 11, 8, 1], [22, 5, 2, 1], [44, 5, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 220, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[5, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '110.2', 'hash': 2, 'hyperelementary': 5, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 55, 'inner_gen_orders': [5, 11], 'inner_gens': [[1, 185], [41, 5]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 55, 'inner_split': True, 'inner_tex': 'C_{11}:C_5', 'inner_used': [1, 2], 'irrC_degree': 5, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 10, 'irrep_stats': [[1, 20], [5, 8]], 'label': '220.2', 'linC_count': 4, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 2, 'linQ_dim': 12, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C11:C20', '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': 18, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 28, 'number_divisions': 9, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 42, 'old_label': None, 'order': 220, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 44], [10, 44], [11, 10], [20, 88], [22, 10], [44, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [2, 0], 'outer_gens': [[1, 145], [1, 215]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1], [10, 2], [20, 1]], 'representations': {'PC': {'code': 36365577375343009, 'gens': [1, 2], 'pres': [4, -5, -2, -2, -11, 1481, 21, 1802, 34, 1283]}, 'GLFq': {'d': 2, 'q': 121, 'gens': [11317501, 83263414, 165132033, 17715620]}, 'Perm': {'d': 15, 'gens': [9, 7277200680, 16, 100725742080]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [20], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}:C_{20}', 'transitive_degree': 44, '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': '40.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 440, 'aut_gen_orders': [20, 2, 20, 20, 10], 'aut_gens': [[1, 2, 20], [11, 1042, 500], [1331, 1122, 1740], [451, 82, 1470], [451, 722, 781], [891, 162, 1580]], 'aut_group': None, 'aut_hash': 7302058934348597158, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14080, 'aut_permdeg': 27, 'aut_perms': [9929899746724593700042478235, 7951320114563568411817064585, 1143623997170616454647884638, 2012508894981026206688935052, 1882428796649744935755804709], 'aut_phi_ratio': 22.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 2, 2, 1], [4, 4, 2, 1], [5, 11, 1, 4], [8, 4, 4, 1], [10, 11, 1, 4], [10, 22, 1, 4], [11, 5, 2, 1], [20, 22, 2, 4], [20, 44, 2, 4], [22, 5, 2, 1], [22, 10, 2, 1], [40, 44, 4, 4], [44, 10, 4, 1], [44, 20, 4, 1], [88, 20, 8, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_{22}\\times D_4).C_5.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 220, 'autcentquo_group': '3520.b', 'autcentquo_hash': 1968696860554166433, 'autcentquo_nilpotent': False, 'autcentquo_order': 3520, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\wr C_2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [4, 2, 2], [4, 4, 2], [5, 11, 4], [8, 4, 4], [10, 11, 4], [10, 22, 4], [11, 5, 2], [20, 22, 8], [20, 44, 8], [22, 5, 2], [22, 10, 2], [40, 44, 16], [44, 10, 4], [44, 20, 4], [88, 20, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '880.24', 'commutator_count': 1, 'commutator_label': '44.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 54, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['32.8', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 2], [5, 11, 4, 1], [8, 4, 2, 2], [10, 11, 4, 1], [10, 22, 4, 1], [11, 5, 2, 1], [20, 22, 4, 2], [20, 44, 4, 2], [22, 5, 2, 1], [22, 10, 2, 1], [40, 44, 8, 2], [44, 10, 2, 2], [44, 20, 2, 2], [88, 20, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 440, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[20, 0, 2]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '220.8', 'hash': 54, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 220, 'inner_gen_orders': [2, 10, 44], 'inner_gens': [[1, 2, 910], [1, 2, 1060], [11, 722, 20]], 'inner_hash': 24, 'inner_nilpotent': False, 'inner_order': 880, 'inner_split': True, 'inner_tex': '(C_2\\times C_{22}):C_{20}', 'inner_used': [1, 2, 3], 'irrC_degree': 20, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 40, 'irrep_stats': [[1, 40], [2, 10], [4, 5], [5, 16], [10, 4], [20, 2]], 'label': '1760.54', 'linC_count': 80, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 4, 'linQ_dim': 18, 'linQ_dim_count': 4, 'linR_count': 24, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*C44).C20', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 36, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 77, 'number_divisions': 27, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 52, 'number_subgroup_classes': 76, 'number_subgroups': 364, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [4, 12], [5, 44], [8, 16], [10, 132], [11, 10], [20, 528], [22, 30], [40, 704], [44, 120], [88, 160]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [890, 0, 0, 0], 'outer_gens': [[1, 2, 1311], [441, 882, 860], [1, 882, 860], [881, 882, 1340]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [288, 121, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5], [8, 4], [10, 4], [16, 1], [20, 4], [40, 1]], 'representations': {'PC': {'code': 11032763158563456826840427535768207622519739974051356750030009, 'gens': [1, 2, 4], 'pres': [7, 2, 2, 5, 2, 2, 2, 11, 6160, 36, 25483, 14850, 11357, 80, 32204, 6311, 12968, 102, 15132, 12619, 124, 35293, 7860]}, 'Perm': {'d': 27, 'gens': [1610606972932, 2979643062695, 16833093532491537100800000, 4404290803109, 50350415795, 5807914735709, 436283137921489848557568000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 20], '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_{44}).C_{20}', 'transitive_degree': 176, 'wreath_data': None, 'wreath_product': False}
