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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1056.237', 'ambient_counter': 237, 'ambient_order': 1056, 'ambient_tex': 'C_{24}:D_{22}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 3, 'counter': 120, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1056.237.176.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '176.e1.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': 176, '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': '6.1', 'subgroup_hash': 1, 'subgroup_order': 6, 'subgroup_tex': 'S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1056.237', 'aut_centralizer_order': 80, 'aut_label': '176.e1', 'aut_quo_index': None, 'aut_stab_index': 44, 'aut_weyl_group': '6.1', 'aut_weyl_index': 3520, 'centralizer': '264.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['16.e1.a1', '88.e1.a1', '88.h1.a1', '88.h1.a2'], 'contains': ['352.a1.a1', '528.e1.a1'], 'core': '352.a1.a1', 'coset_action_label': None, 'count': 44, 'diagramx': [9622, -1, 8491, -1, 1859, -1, 8206, -1], 'generators': [3, 704], 'label': '1056.237.176.e1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '44.d1.a1', 'old_label': '176.e1.a1', 'projective_image': '1056.237', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '176.e1.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 4], [1, 3], [2, 4]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 2, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 3], [2, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 1]], 'label': '6.1', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'S3', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 3, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 6, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 3], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 3, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 25, 'gens': [1, 2], 'pres': [2, -2, -3, 17]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [28, 45]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'GL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'SL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGL'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'SO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'SU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PSO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'PSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'GO'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'Omega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PGO'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'POmega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CSO'}, {'d': 2, 'q': 4, 'gens': [20, 194], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'CSOMinus'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'CSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CO'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'COMinus'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGammaL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSigmaL'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGammaU'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGL'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11, 6]}, 'Perm': {'d': 3, 'gens': [1, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3', 'transitive_degree': 3, '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': 2, 'aut_exponent': 660, 'aut_gen_orders': [10, 30, 10, 22, 10, 20, 22], 'aut_gens': [[1, 2, 4], [793, 434, 1004], [969, 2, 652], [793, 626, 284], [705, 146, 356], [1, 866, 692], [89, 914, 1028], [353, 962, 884]], 'aut_group': None, 'aut_hash': 5956886757194812668, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21120, 'aut_permdeg': 38, 'aut_perms': [179318835710342783238405196764992015968399248, 182496588282230297793417176716876278688962975, 419678733201357678420843324760103043707996732, 9506470598686600946487509362604506573542223, 240417652427883643727047067879485055475485493, 2592225788443939701570949459952975758137623, 246217146693830508807115602378671168615704567], 'aut_phi_ratio': 66.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 12, 1, 2], [2, 22, 1, 1], [2, 132, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 22, 1, 1], [4, 132, 1, 1], [6, 2, 1, 1], [6, 44, 1, 1], [8, 4, 1, 1], [8, 44, 1, 1], [11, 2, 5, 1], [12, 2, 2, 1], [12, 44, 1, 1], [22, 2, 5, 1], [22, 24, 5, 2], [24, 4, 2, 1], [24, 44, 2, 1], [33, 4, 5, 1], [44, 4, 5, 1], [66, 4, 5, 1], [88, 4, 10, 1], [132, 4, 10, 1], [264, 4, 20, 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, 12, 2], [2, 22, 1], [2, 132, 1], [3, 2, 1], [4, 2, 1], [4, 22, 1], [4, 132, 1], [6, 2, 1], [6, 44, 1], [8, 4, 1], [8, 44, 1], [11, 2, 5], [12, 2, 2], [12, 44, 1], [22, 2, 5], [22, 24, 10], [24, 4, 2], [24, 44, 2], [33, 4, 5], [44, 4, 5], [66, 4, 5], [88, 4, 10], [132, 4, 10], [264, 4, 20]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '528.107', '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', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 237, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 12, 1, 2], [2, 22, 1, 1], [2, 132, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 22, 1, 1], [4, 132, 1, 1], [6, 2, 1, 1], [6, 44, 1, 1], [8, 4, 1, 1], [8, 44, 1, 1], [11, 2, 5, 1], [12, 2, 2, 1], [12, 44, 1, 1], [22, 2, 5, 1], [22, 24, 5, 2], [24, 4, 2, 1], [24, 44, 2, 1], [33, 4, 5, 1], [44, 4, 5, 1], [66, 4, 5, 1], [88, 4, 10, 1], [132, 4, 10, 1], [264, 4, 20, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 16128, 'exponent': 264, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[4, 0, 20]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '264.34', 'hash': 237, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 132, 'inner_gen_orders': [2, 2, 132], 'inner_gens': [[1, 2, 92], [1, 2, 436], [969, 626, 4]], 'inner_hash': 107, 'inner_nilpotent': False, 'inner_order': 528, 'inner_split': True, 'inner_tex': 'D_{11}\\times D_{12}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 30], [4, 58]], 'label': '1056.237', 'linC_count': 20, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 16, 'linQ_dim': 16, 'linQ_dim_count': 16, 'linR_count': 40, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C24:D22', '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': 28, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 96, 'number_divisions': 28, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 132, 'number_subgroup_classes': 136, 'number_subgroups': 1840, 'old_label': None, 'order': 1056, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 179], [3, 2], [4, 156], [6, 46], [8, 48], [11, 10], [12, 48], [22, 250], [24, 96], [33, 20], [44, 20], [66, 20], [88, 40], [132, 40], [264, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 10], 'outer_gen_pows': [176, 0, 352], 'outer_gens': [[705, 2, 964], [1, 2, 260], [705, 530, 292]], 'outer_group': '40.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 40, 'outer_permdeg': 11, 'outer_perms': [720, 3628800, 40353], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{10}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 3], [8, 1], [10, 4], [20, 3], [40, 2], [80, 1]], 'representations': {'PC': {'code': 23235754551153712560360511363992838482754252120000019, 'gens': [1, 2, 3], 'pres': [7, -2, -2, -2, -2, -2, -3, -11, 1934, 4587, 58, 5155, 12218, 80, 12884, 12051, 102, 30917, 6732, 166, 23533]}, 'Perm': {'d': 22, 'gens': [7426796651898992047, 10232060244516633600, 63761596730447616000, 112177098605924352000, 161058154786873344000, 6706022400, 4364893]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{24}:D_{22}', 'transitive_degree': 264, 'wreath_data': None, 'wreath_product': False}
