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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '960.10966', 'ambient_counter': 10966, 'ambient_order': 960, 'ambient_tex': '\\GL(2,3):D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 20, 'counter': 117, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.10966.24.j1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '24.j1.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': 24, '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': '40.5', 'subgroup_hash': 5, 'subgroup_order': 40, 'subgroup_tex': 'C_4\\times D_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.10966', 'aut_centralizer_order': 16, 'aut_label': '24.j1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '80.50', 'aut_weyl_index': 48, 'centralizer': '120.bc1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.b1.a1', '12.e1.a1', '12.m1.a1', '12.n1.a1', '12.o1.a1', '12.p1.a1', '12.q1.a1'], 'contains': ['48.b1.a1', '48.i1.a1', '48.j1.a1', '120.j1.a1'], 'core': '48.b1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [5462, -1, 4166, -1, 5199, -1, 5519, -1], 'generators': [12, 480, 192, 504], 'label': '960.10966.24.j1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '24.j1.a1', 'projective_image': '480.1193', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.j1.a1', 'subgroup_fusion': None, 'weyl_group': '40.13'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [2, 4, 10], 'aut_gens': [[1, 2], [21, 2], [1, 26], [17, 22]], 'aut_group': '80.50', 'aut_hash': 50, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 80, 'aut_permdeg': 9, 'aut_perms': [7, 15120, 97336], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 2, 1], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1], [20, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times F_5', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 2], [4, 1, 2], [4, 5, 2], [5, 2, 2], [10, 2, 2], [20, 2, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1], [20, 2, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 20, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '20.4', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 18], [25, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 8]], 'label': '40.5', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 10, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4*D5', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_conjugacy_classes': 16, 'number_divisions': 9, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 16, 'number_subgroups': 36, 'old_label': None, 'order': 40, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 11], [4, 12], [5, 4], [10, 4], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [1, 0, 0], 'outer_gens': [[1, 34], [1, 22], [21, 2]], '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': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 2], [8, 1]], 'representations': {'PC': {'code': 1040612736515, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -5, 145, 21, 434, 34, 515]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101743041058377, 58438765960015333]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 377, 501]}, 'Perm': {'d': 9, 'gens': [5167, 17, 7, 50520]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times D_5', 'transitive_degree': 20, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [12, 2, 2, 4, 2, 2, 2, 4, 10, 10], 'aut_gens': [[1, 2, 4, 24, 48], [9, 2, 4, 720, 168], [745, 2, 748, 24, 528], [481, 482, 244, 504, 528], [745, 2, 28, 504, 816], [481, 2, 436, 504, 912], [1, 482, 244, 504, 528], [265, 482, 748, 24, 528], [265, 2, 556, 24, 144], [17, 2, 788, 504, 72], [481, 2, 52, 504, 528]], 'aut_group': '3840.ho', 'aut_hash': 4584141657853489991, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3840, 'aut_permdeg': 24, 'aut_perms': [2253843814588285215111, 859805579988956793, 738228385321419513, 541662136986560154149328, 26007844155662101356959, 49698488061513, 859809503011335033, 131510761342195356592763, 129261184634863016844633, 129260205382678702959753], 'aut_phi_ratio': 15.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 10, 1, 1], [2, 12, 1, 2], [2, 30, 1, 1], [2, 60, 1, 1], [3, 8, 1, 1], [4, 6, 1, 2], [4, 10, 1, 1], [4, 30, 1, 1], [4, 60, 1, 1], [5, 2, 2, 1], [6, 8, 1, 1], [6, 16, 1, 1], [6, 80, 1, 1], [8, 6, 2, 1], [8, 12, 1, 1], [8, 60, 1, 2], [10, 2, 2, 1], [10, 2, 4, 1], [10, 24, 2, 2], [12, 80, 1, 1], [15, 16, 2, 1], [20, 12, 2, 2], [30, 16, 2, 1], [30, 16, 4, 1], [40, 12, 4, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3\\times F_5\\times S_4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '480.1189', 'autcentquo_hash': 1189, 'autcentquo_nilpotent': False, 'autcentquo_order': 480, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 10, 1], [2, 12, 2], [2, 30, 1], [2, 60, 1], [3, 8, 1], [4, 6, 2], [4, 10, 1], [4, 30, 1], [4, 60, 1], [5, 2, 2], [6, 8, 1], [6, 16, 1], [6, 80, 1], [8, 6, 2], [8, 12, 1], [8, 60, 2], [10, 2, 6], [10, 24, 4], [12, 80, 1], [15, 16, 2], [20, 12, 4], [30, 16, 6], [40, 12, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '480.1193', 'commutator_count': 1, 'commutator_label': '120.15', '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', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10966, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 10, 1, 1], [2, 12, 1, 2], [2, 30, 1, 1], [2, 60, 1, 1], [3, 8, 1, 1], [4, 6, 1, 2], [4, 10, 1, 1], [4, 30, 1, 1], [4, 60, 1, 1], [5, 2, 2, 1], [6, 8, 1, 1], [6, 16, 1, 1], [6, 80, 1, 1], [8, 6, 2, 1], [8, 12, 1, 1], [8, 60, 1, 2], [10, 2, 2, 1], [10, 2, 4, 1], [10, 24, 2, 2], [12, 80, 1, 1], [15, 16, 2, 1], [20, 12, 2, 2], [30, 16, 2, 1], [30, 16, 4, 1], [40, 12, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 60480, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 8], [8, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '480.1193', 'hash': 10966, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 2, 6, 2, 10], 'inner_gens': [[1, 2, 20, 240, 792], [1, 2, 484, 24, 48], [9, 482, 4, 720, 936], [265, 2, 748, 24, 528], [745, 2, 124, 504, 48]], 'inner_hash': 1193, 'inner_nilpotent': False, 'inner_order': 480, 'inner_split': True, 'inner_tex': 'D_{10}\\times S_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 4, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 12], [3, 8], [4, 14], [6, 8], [8, 5]], 'label': '960.10966', 'linC_count': 8, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 8, 'linQ_dim': 12, 'linQ_dim_count': 8, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'GL(2,3):D10', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 34, 'number_characteristic_subgroups': 37, 'number_conjugacy_classes': 55, 'number_divisions': 34, 'number_normal_subgroups': 37, 'number_subgroup_autclasses': 298, 'number_subgroup_classes': 304, 'number_subgroups': 2918, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 127], [3, 8], [4, 112], [5, 4], [6, 104], [8, 144], [10, 108], [12, 80], [15, 32], [20, 48], [30, 96], [40, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 2, 244, 504, 528], [1, 2, 4, 24, 816]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [3, 8], [4, 4], [8, 4], [12, 4], [32, 2]], 'representations': {'PC': {'code': 8593022825841698262663120202750109516673799501984128795148547, 'gens': [1, 2, 3, 5, 6], 'pres': [8, -2, -2, -2, -3, -2, 2, -2, -5, 482, 5818, 66, 515, 9604, 7220, 1348, 836, 38021, 11253, 1901, 1093, 141, 12118, 166, 12311]}, 'Perm': {'d': 21, 'gens': [2459999639040975367, 40949162930149200, 493359947036732880, 9270329331605040, 37, 5496238066015801440, 8074576096790193360, 10271542109627555280]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\GL(2,3):D_{10}', 'transitive_degree': 80, 'wreath_data': None, 'wreath_product': False}
