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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1760.326', 'ambient_counter': 326, 'ambient_order': 1760, 'ambient_tex': '\\SD_{16}:F_{11}', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': False, 'core_order': 8, 'counter': 57, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1760.326.22.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '22.g1.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': 22, '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': '80.27', 'subgroup_hash': 27, 'subgroup_order': 80, 'subgroup_tex': 'C_5\\times Q_{16}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.326', 'aut_centralizer_order': 20, 'aut_label': '22.g1', 'aut_quo_index': None, 'aut_stab_index': 11, 'aut_weyl_group': '16.11', 'aut_weyl_index': 220, 'centralizer': '88.f1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.g1.a1', '11.a1.a1'], 'contains': ['44.h1.a1', '44.i1.a1', '44.k1.a1', '110.g1.a1'], 'core': '220.b1.a1', 'coset_action_label': None, 'count': 11, 'diagramx': [9536, -1, 9797, -1, 9541, -1, 8230, -1], 'generators': [225, 2, 440, 110, 880], 'label': '1760.326.22.g1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.g1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '11.a1.a1', 'old_label': '22.g1.a1', 'projective_image': '880.118', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '22.g1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 4, 8], 'aut_gens': [[1, 2], [41, 42], [1, 78], [1, 34], [71, 2]], 'aut_group': '128.1676', 'aut_hash': 1676, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 128, 'aut_permdeg': 12, 'aut_perms': [83589127, 94404360, 90842417, 280869120], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 4, 2, 1], [5, 1, 4, 1], [8, 2, 2, 1], [10, 1, 4, 1], [20, 2, 4, 1], [20, 4, 8, 1], [40, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4^2.D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 4, 2], [5, 1, 4], [8, 2, 2], [10, 1, 4], [20, 2, 4], [20, 4, 8], [40, 2, 8]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '8.3', 'commutator_count': 1, 'commutator_label': '4.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 27, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.9', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 4, 1, 2], [5, 1, 4, 1], [8, 2, 2, 1], [10, 1, 4, 1], [20, 2, 4, 1], [20, 4, 4, 2], [40, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 18, 'exponent': 40, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 0, 8]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '20.5', 'hash': 27, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4], 'inner_gens': [[1, 62], [21, 2]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 20], [2, 15]], 'label': '80.27', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*Q16', 'ngens': 5, 'nilpotency_class': 3, '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 35, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 18, 'number_subgroups': 22, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [4, 10], [5, 4], [8, 4], [10, 4], [20, 40], [40, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [2, 0, 0], 'outer_gens': [[11, 2], [1, 42], [1, 34]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 5], [8, 1], [16, 1]], 'representations': {'PC': {'code': 397207715878861827, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -2, -5, 200, 621, 26, 662, 42, 58]}, 'GLFp': {'d': 2, 'p': 31, 'gens': [743914, 867130]}, 'Perm': {'d': 21, 'gens': [22929945340561459200, 12684525064368161760, 96, 7542040774495983840, 2439325392333218640]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times Q_{16}', 'transitive_degree': 80, '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': 220, 'aut_gen_orders': [10, 10, 10, 20, 44, 10], 'aut_gens': [[1, 10, 220], [981, 70, 1540], [881, 170, 660], [101, 70, 1540], [1021, 1450, 1100], [181, 1330, 220], [161, 150, 1540]], 'aut_group': None, 'aut_hash': 4036028994375102156, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3520, 'aut_permdeg': 41, 'aut_perms': [19751248832416541930352156256337636278321138494809, 32115968499453893700894342514176327371227170061861, 12530921681244009197912957590253296268365653083272, 11775903384477820067506690875113728293753308970643, 25412920679509947884607496988862466165592845074154, 14665844938348699503779162520385512620318108933813], 'aut_phi_ratio': 5.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [2, 22, 1, 1], [2, 44, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 11, 2, 1], [4, 44, 1, 1], [5, 11, 1, 4], [8, 2, 2, 1], [8, 22, 2, 1], [10, 11, 1, 4], [10, 22, 1, 4], [10, 44, 1, 8], [11, 10, 1, 1], [20, 11, 2, 4], [20, 22, 1, 4], [20, 44, 1, 8], [22, 10, 1, 1], [22, 40, 1, 1], [40, 22, 2, 8], [44, 20, 1, 1], [44, 40, 1, 1], [88, 20, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{22}.(C_2^4\\times C_{10})', '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': 110, 'autcentquo_group': '440.42', 'autcentquo_hash': 42, 'autcentquo_nilpotent': False, 'autcentquo_order': 440, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 1], [2, 22, 1], [2, 44, 1], [4, 2, 1], [4, 4, 1], [4, 11, 2], [4, 44, 1], [5, 11, 4], [8, 2, 2], [8, 22, 2], [10, 11, 4], [10, 22, 4], [10, 44, 8], [11, 10, 1], [20, 11, 8], [20, 22, 4], [20, 44, 8], [22, 10, 1], [22, 40, 1], [40, 22, 16], [44, 20, 1], [44, 40, 1], [88, 20, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '880.118', 'commutator_count': 1, 'commutator_label': '44.2', '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': 326, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [2, 22, 1, 1], [2, 44, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 11, 2, 1], [4, 44, 1, 1], [5, 11, 4, 1], [8, 2, 2, 1], [8, 22, 2, 1], [10, 11, 4, 1], [10, 22, 4, 1], [10, 44, 4, 2], [11, 10, 1, 1], [20, 11, 8, 1], [20, 22, 4, 1], [20, 44, 4, 2], [22, 10, 1, 1], [22, 40, 1, 1], [40, 22, 8, 2], [44, 20, 1, 1], [44, 40, 1, 1], [88, 20, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 499968, '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': '4.1', 'frattini_quotient': '440.42', 'hash': 326, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 220, 'inner_gen_orders': [10, 22, 4], 'inner_gens': [[1, 1010, 220], [101, 10, 660], [1, 1330, 220]], 'inner_hash': 118, 'inner_nilpotent': False, 'inner_order': 880, 'inner_split': True, 'inner_tex': 'D_4\\times F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': 20, 'irrQ_degree': 40, 'irrQ_dim': 80, 'irrR_degree': 40, 'irrep_stats': [[1, 40], [2, 30], [10, 4], [20, 3]], 'label': '1760.326', 'linC_count': 80, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 4, 'linQ_dim': 18, 'linQ_dim_count': 4, 'linR_count': 40, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'SD16:F11', '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': 61, 'number_characteristic_subgroups': 47, 'number_conjugacy_classes': 77, 'number_divisions': 28, 'number_normal_subgroups': 47, 'number_subgroup_autclasses': 124, 'number_subgroup_classes': 124, 'number_subgroups': 954, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 71], [4, 72], [5, 44], [8, 48], [10, 484], [11, 10], [20, 528], [22, 50], [40, 352], [44, 60], [88, 40]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 210, 1100], [881, 210, 220]], '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': 3, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 8], [8, 3], [10, 4], [20, 1], [32, 1], [40, 1]], 'representations': {'PC': {'code': 383206884651950194117485708333060653183330309000, 'gens': [1, 3, 5], 'pres': [7, -2, -5, -2, -11, -2, -2, -2, 14, 21212, 10824, 58, 25763, 13450, 1256, 2328, 102, 5563, 124]}, 'Perm': {'d': 27, 'gens': [15537113407107378012851520, 1355224626139, 3074465924804, 32317378649298032209920000, 4497733002751, 5621565927546, 449828623964764313837568000]}}, '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': '\\SD_{16}:F_{11}', 'transitive_degree': 176, 'wreath_data': None, 'wreath_product': False}
