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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1512.780', 'ambient_counter': 780, 'ambient_order': 1512, 'ambient_tex': 'C_3\\times \\SL(2,8)', 'central': False, 'central_factor': False, 'centralizer_order': 27, 'characteristic': False, 'core_order': 1, 'counter': 16, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1512.780.168.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '168.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': 168, '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': '9.1', 'subgroup_hash': 1, 'subgroup_order': 9, 'subgroup_tex': 'C_9', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1512.780', 'aut_centralizer_order': 9, 'aut_label': '168.c1', 'aut_quo_index': None, 'aut_stab_index': 56, 'aut_weyl_group': '6.2', 'aut_weyl_index': 504, 'centralizer': '56.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['56.a1.a1'], 'contains': ['504.b1.a1'], 'core': '1512.a1.a1', 'coset_action_label': None, 'count': 56, 'diagramx': [8679, -1, 9318, -1, 9126, -1, 9126, -1], 'generators': [8696646, 336192], 'label': '1512.780.168.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '56.a1.a1', 'old_label': '168.c1.a1', 'projective_image': '1512.780', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '168.c1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '9.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [6], 'aut_gens': [[1], [2]], 'aut_group': '6.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 6, 'aut_permdeg': 5, 'aut_perms': [27], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', '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], [3, 1, 2], [9, 1, 6]], 'center_label': '9.1', 'center_order': 9, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 9, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 6]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 9]], 'label': '9.1', 'linC_count': 6, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C9', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 9, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 9, 'order_factorization_type': 2, 'order_stats': [[1, 1], [3, 2], [9, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [9], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [6, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -3, -3, 6]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41907234971425459]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [78572]}, 'Perm': {'d': 9, 'gens': [357120, 80884]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [9], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9', 'transitive_degree': 9, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 126, 'aut_gen_orders': [6, 6], 'aut_gens': [[3445490, 3670030, 2751618], [5314875, 3932175, 13144781], [10109117, 3670030, 9761484]], 'aut_group': '3024.r', 'aut_hash': 5679810977767407995, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3024, 'aut_permdeg': 58, 'aut_perms': [40587677830871890315952890912460436003477704980204715423312414011009144090527, 187656311126021048722961117309089771409918398446222299971241631962507660173], 'aut_phi_ratio': 7.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 63, 1, 1], [3, 1, 2, 1], [3, 56, 1, 1], [3, 56, 2, 1], [6, 63, 2, 1], [7, 72, 3, 1], [9, 56, 3, 1], [9, 56, 6, 1], [21, 72, 6, 1]], 'aut_supersolvable': False, 'aut_tex': '\\SL(2,8):C_6', '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': 126, 'autcentquo_group': '1512.779', 'autcentquo_hash': 779, 'autcentquo_nilpotent': False, 'autcentquo_order': 1512, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '{}^2G(2,3)', 'cc_stats': [[1, 1, 1], [2, 63, 1], [3, 1, 2], [3, 56, 3], [6, 63, 2], [7, 72, 3], [9, 56, 9], [21, 72, 6]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '504.156', 'commutator_count': 1, 'commutator_label': '504.156', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '504.156'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 780, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['504.156', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 63, 1, 1], [3, 1, 2, 1], [3, 56, 1, 1], [3, 56, 2, 1], [6, 63, 2, 1], [7, 72, 3, 1], [9, 56, 3, 1], [9, 56, 6, 1], [21, 72, 6, 1]], 'element_repr_type': 'GLFq', 'elementary': 1, 'eulerian_function': 568, 'exponent': 126, 'exponents_of_order': [3, 3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[7, 0, 8], [8, 0, 2], [9, 0, 6]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1512.780', 'hash': 780, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 126, 'inner_gen_orders': [7, 1, 7], 'inner_gens': [[3445490, 3670030, 7918806], [3445490, 3670030, 2751618], [15741891, 3670030, 2751618]], 'inner_hash': 156, 'inner_nilpotent': False, 'inner_order': 504, 'inner_split': True, 'inner_tex': '\\SL(2,8)', 'inner_used': [1, 3], 'irrC_degree': 7, 'irrQ_degree': 14, 'irrQ_dim': 14, 'irrR_degree': 14, 'irrep_stats': [[1, 3], [7, 12], [8, 3], [9, 9]], 'label': '1512.780', 'linC_count': 8, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 1, 'linQ_dim': 9, 'linQ_dim_count': 1, 'linR_count': 4, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C3*SL(2,8)', 'ngens': 3, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 27, 'number_divisions': 10, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 26, 'number_subgroups': 884, 'old_label': None, 'order': 1512, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 63], [3, 170], [6, 126], [7, 216], [9, 504], [21, 432]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [262145], 'outer_gens': [[4749018, 3932175, 14393581]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': None, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [7, 1], [8, 1], [14, 1], [16, 1], [21, 1], [27, 1], [42, 1], [54, 1]], 'representations': {'GLFq': {'d': 2, 'q': 64, 'gens': [3445490, 3670030, 2751618]}, 'Perm': {'d': 12, 'gens': [14516644, 80015044]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times \\SL(2,8)', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
