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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '60.5', 'ambient_counter': 5, 'ambient_order': 60, 'ambient_tex': 'A_5', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 1, 'counter': 2, 'cyclic': False, 'direct': None, 'hall': 6, 'label': '60.5.5.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '5.a1.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': 5, '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': '12.3', 'subgroup_hash': 3, 'subgroup_order': 12, 'subgroup_tex': 'A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '60.5', 'aut_centralizer_order': 1, 'aut_label': '5.a1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '24.12', 'aut_weyl_index': 5, 'centralizer': '60.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['15.a1.a1', '20.a1.a1'], 'core': '60.a1.a1', 'coset_action_label': '5T4', 'count': 5, 'diagramx': [1255, -1, 1255, -1, 4856, -1, 4856, -1], 'generators': [15, 23, 7], 'label': '60.5.5.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '5.a1.a1', 'projective_image': '60.5', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '5.a1.a1', 'subgroup_fusion': None, 'weyl_group': '12.3'}
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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': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2], 'aut_gens': [[4, 16, 7], [11, 16, 23], [19, 23, 16], [8, 16, 7], [15, 16, 7]], 'aut_group': '24.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 4, 'aut_perms': [2, 4, 16, 7], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4', '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': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 4, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[4, 7, 23], [19, 16, 7], [15, 16, 7]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 3, 'irrQ_dim': 3, 'irrR_degree': 3, 'irrep_stats': [[1, 3], [3, 1]], 'label': '12.3', 'linC_count': 1, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 1, 'linQ_dim': 3, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'A4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 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': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 5, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 8]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3, 7, 16]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 4, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1]], 'representations': {'PC': {'code': 9572, 'gens': [1, 2, 3], 'pres': [3, -3, -2, 2, 55, 29]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [4828, 4708]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [31, 21], 'family': 'PSL'}, {'d': 2, 'q': 3, 'gens': [362, 766], 'family': 'PSU'}, {'d': 3, 'q': 3, 'gens': [10315, 1638], 'family': 'Omega'}, {'d': 3, 'q': 3, 'gens': [10315, 1638], 'family': 'POmega'}, {'d': 2, 'q': 3, 'gens': [31, 21], 'family': 'PSigmaL'}, {'d': 1, 'q': 4, 'gens': [3, 7], 'family': 'AGL'}], 'GLFp': {'d': 3, 'p': 2, 'gens': [458, 314, 124]}, 'Perm': {'d': 4, 'gens': [4, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_4', 'transitive_degree': 4, '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': '1.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 60, 'aut_gen_orders': [2, 5], 'aut_gens': [[33, 30], [42, 45], [52, 20]], 'aut_group': '120.34', 'aut_hash': 34, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 120, 'aut_permdeg': 5, 'aut_perms': [24, 33], 'aut_phi_ratio': 7.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 20, 1, 1], [5, 12, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_5', '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': 60, 'autcentquo_group': '120.34', 'autcentquo_hash': 34, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_5', 'cc_stats': [[1, 1, 1], [2, 15, 1], [3, 20, 1], [5, 12, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '60.5', 'commutator_count': 1, 'commutator_label': '60.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['60.5'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 20, 1, 1], [5, 12, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 19, 'exponent': 30, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 1, 2], [4, 1, 1], [5, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '60.5', 'hash': 5, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [5, 3], 'inner_gens': [[33, 8], [85, 30]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 60, 'inner_split': True, 'inner_tex': 'A_5', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 3, 'irrep_stats': [[1, 1], [3, 2], [4, 1], [5, 1]], 'label': '60.5', 'linC_count': 2, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'A5', 'ngens': 2, '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': 4, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 5, 'number_divisions': 4, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 9, 'number_subgroups': 59, 'old_label': None, 'order': 60, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 15], [3, 20], [5, 24]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [20], 'outer_gens': [[112, 56]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': None, 'perfect': True, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': True, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [5, 1], [6, 1]], 'representations': {'GLZ': {'b': 3, 'd': 4, 'gens': [40673960, 19414147]}, 'Lie': [{'d': 2, 'q': 4, 'gens': [194, 21], 'family': 'SL'}, {'d': 2, 'q': 4, 'gens': [194, 21], 'family': 'PSL'}, {'d': 2, 'q': 5, 'gens': [109, 377], 'family': 'PSL'}, {'d': 2, 'q': 4, 'gens': [66, 21], 'family': 'PGL'}, {'d': 3, 'q': 4, 'gens': [4373, 131331, 4368], 'family': 'SO'}, {'d': 2, 'q': 4, 'gens': [28678, 273], 'family': 'SU'}, {'d': 3, 'q': 4, 'gens': [4373, 131331, 4368], 'family': 'PSO'}, {'d': 2, 'q': 4, 'gens': [28678, 273], 'family': 'PSU'}, {'d': 2, 'q': 5, 'gens': [10579, 46877], 'family': 'PSU'}, {'d': 3, 'q': 4, 'gens': [4373, 131331, 4368], 'family': 'GO'}, {'d': 3, 'q': 4, 'gens': [196866, 4373], 'family': 'Omega'}, {'d': 3, 'q': 5, 'gens': [65611, 1563129], 'family': 'Omega'}, {'d': 4, 'q': 2, 'gens': [33377, 5931], 'family': 'OmegaMinus'}, {'d': 3, 'q': 4, 'gens': [4373, 131331, 4368], 'family': 'PGO'}, {'d': 2, 'q': 4, 'gens': [57346, 273], 'family': 'PGU'}, {'d': 3, 'q': 4, 'gens': [196866, 4373], 'family': 'POmega'}, {'d': 3, 'q': 5, 'gens': [65611, 1563129], 'family': 'POmega'}, {'d': 4, 'q': 2, 'gens': [33377, 5931], 'family': 'POmegaMinus'}, {'d': 4, 'q': 2, 'gens': [194, 21], 'family': 'SpinMinus'}, {'d': 2, 'q': 4, 'gens': [28678, 273], 'family': 'CSU'}, {'d': 2, 'q': 5, 'gens': [109, 377], 'family': 'PSigmaL'}], 'GLFp': {'d': 3, 'p': 5, 'gens': [1269282, 1294106]}, 'Perm': {'d': 5, 'gens': [33, 30]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
