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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '350.7', 'ambient_counter': 7, 'ambient_order': 350, 'ambient_tex': 'C_5\\times D_{35}', 'central': False, 'central_factor': False, 'centralizer_order': 175, 'characteristic': False, 'core_order': 7, 'counter': 9, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '350.7.10.c1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '10.c1.b1', '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': 10, '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': '35.1', 'subgroup_hash': 1, 'subgroup_order': 35, 'subgroup_tex': 'C_{35}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '350.7', 'aut_centralizer_order': 35, 'aut_label': '10.c1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '24.9', 'aut_weyl_index': 140, 'centralizer': '2.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['50.a1.a1', '70.c1.b1'], 'core': '50.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [577, -1, 9613, -1, 625, -1, 535, -1], 'generators': [72, 50], 'label': '350.7.10.c1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '10.c1.b1', 'projective_image': '350.7', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '10.c1.b1', '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': '35.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1], [6], [3]], 'aut_group': '24.9', 'aut_hash': 9, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 24, 'aut_permdeg': 9, 'aut_perms': [42000, 867], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [7, 1, 6, 1], [35, 1, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{12}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.9', 'autcent_hash': 9, 'autcent_nilpotent': True, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{12}', '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], [5, 1, 4], [7, 1, 6], [35, 1, 24]], 'center_label': '35.1', 'center_order': 35, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['5.1', '7.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [7, 1, 6, 1], [35, 1, 24, 1]], 'element_repr_type': 'PC', 'elementary': 35, 'eulerian_function': 1, 'exponent': 35, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [5, 7], 'faithful_reps': [[1, 0, 24]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '35.1', 'hash': 1, 'hyperelementary': 35, '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': 24, 'irrQ_dim': 24, 'irrR_degree': 2, 'irrep_stats': [[1, 35]], 'label': '35.1', 'linC_count': 24, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 12, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C35', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 35, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 35, 'order_factorization_type': 11, 'order_stats': [[1, 1], [5, 4], [7, 6], [35, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[6], [3]], 'outer_group': '24.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [42000, 867], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{12}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [5, 7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [6, 1], [24, 1]], 'representations': {'PC': {'code': 203, 'gens': [1], 'pres': [2, -5, -7, 10]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [72212]}, 'Perm': {'d': 12, 'gens': [159667200, 4320]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [35], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{35}', 'transitive_degree': 35, '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': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [60, 30, 28, 12], 'aut_gens': [[1, 10], [327, 110], [311, 260], [151, 220], [223, 230]], 'aut_group': None, 'aut_hash': 6689779008057805882, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3360, 'aut_permdeg': 39, 'aut_perms': [12276634611259940266075176873975331271664283490, 19221019525894338726674953538768658820018229872, 19779225335434016187892073765821497684583046432, 1165022133730743137378744179309508987377555714], 'aut_phi_ratio': 28.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 35, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [7, 2, 3, 1], [10, 35, 4, 1], [35, 2, 12, 2], [35, 2, 48, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times F_5\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 420, 'autcentquo_group': '840.139', 'autcentquo_hash': 139, 'autcentquo_nilpotent': False, 'autcentquo_order': 840, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5\\times F_7', 'cc_stats': [[1, 1, 1], [2, 35, 1], [5, 1, 4], [5, 2, 10], [7, 2, 3], [10, 35, 4], [35, 2, 72]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '70.3', 'commutator_count': 1, 'commutator_label': '35.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '5.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['70.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 35, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [7, 2, 3, 1], [10, 35, 4, 1], [35, 2, 12, 2], [35, 2, 24, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 70, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': [[2, 0, 48]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '350.7', 'hash': 7, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [2, 35], 'inner_gens': [[1, 340], [21, 10]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 70, 'inner_split': True, 'inner_tex': 'D_{35}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 4, 'irrep_stats': [[1, 10], [2, 85]], 'label': '350.7', 'linC_count': 48, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 4, 'linQ_dim': 14, 'linQ_dim_count': 4, 'linR_count': 72, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*D35', 'ngens': 4, '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 95, 'number_divisions': 12, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 20, 'number_subgroups': 112, 'old_label': None, 'order': 350, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 35], [5, 24], [7, 6], [10, 140], [35, 144]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [4, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 80], [3, 160]], 'outer_group': '48.20', 'outer_hash': 20, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [864, 4032003], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times C_{12}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 3], [6, 1], [8, 2], [24, 2], [48, 2]], 'representations': {'PC': {'code': 144443913776339, 'gens': [1, 3], 'pres': [4, -2, -5, -5, -7, 8, 4082, 70, 4803]}, 'GLFp': {'d': 2, 'p': 71, 'gens': [19327195, 10379468, 5112]}, 'Perm': {'d': 17, 'gens': [21115948009087, 46032173550720, 66718256814720, 973]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_{35}', 'transitive_degree': 70, 'wreath_data': None, 'wreath_product': False}
