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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '714.2', 'ambient_counter': 2, 'ambient_order': 714, 'ambient_tex': 'C_{17}\\times F_7', 'central': False, 'central_factor': False, 'centralizer_order': 17, 'characteristic': True, 'core_order': 238, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 238, 'label': '714.2.3.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '3.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '3.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 3, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '238.1', 'subgroup_hash': 1, 'subgroup_order': 238, 'subgroup_tex': 'D_7\\times C_{17}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '714.2', 'aut_centralizer_order': 1, 'aut_label': '3.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '672.7', 'aut_weyl_index': 1, 'centralizer': '42.a1.a1', 'complements': ['238.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '21.a1.a1', '51.a1.a1'], 'core': '3.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3799, 3813, 5753, 3813, 4935, 1772, 4954, 1772], 'generators': [3, 42, 510], 'label': '714.2.3.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '3.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['6.a1.a1', '51.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '3.a1.a1', 'projective_image': '42.1', 'quotient_action_image': '3.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '3.a1.a1', 'subgroup_fusion': None, 'weyl_group': '42.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '34.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 336, 'aut_gen_orders': [6, 112], 'aut_gens': [[1, 2], [1, 104], [137, 184]], 'aut_group': '672.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 672, 'aut_permdeg': 23, 'aut_perms': [255820001847201792000, 2462103591737496140313], 'aut_phi_ratio': 7.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [7, 2, 3, 1], [17, 1, 16, 1], [34, 7, 16, 1], [119, 2, 48, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{16}\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 16, 'autcent_group': '16.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{16}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 7, 1], [7, 2, 3], [17, 1, 16], [34, 7, 16], [119, 2, 48]], 'center_label': '17.1', 'center_order': 17, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '7.1', '17.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['17.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [7, 2, 3, 1], [17, 1, 16, 1], [34, 7, 16, 1], [119, 2, 48, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 54, 'exponent': 238, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7, 17], 'faithful_reps': [[2, 0, 48]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '238.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 138], [103, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 96, 'irrQ_dim': 96, 'irrR_degree': 4, 'irrep_stats': [[1, 34], [2, 51]], 'label': '238.1', 'linC_count': 48, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 2, 'linQ_dim': 22, 'linQ_dim_count': 2, 'linR_count': 72, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D7*C17', 'ngens': 3, '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 85, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 20, 'old_label': None, 'order': 238, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 7], [7, 6], [17, 16], [34, 112], [119, 96]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 48, 'outer_gen_orders': [48], 'outer_gen_pows': [0], 'outer_gens': [[1, 150]], 'outer_group': '48.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 19, 'outer_perms': [6780385526348307], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{48}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 17], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [6, 1], [16, 2], [96, 1]], 'representations': {'PC': {'code': 2932498527, 'gens': [1, 2], 'pres': [3, -2, -7, -17, 829, 46]}, 'GLFp': {'d': 2, 'p': 239, 'gens': [57360, 136519214, 2907858840]}, 'Perm': {'d': 24, 'gens': [127, 27029669736328405574400, 973]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [34], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_7\\times C_{17}', 'transitive_degree': 119, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '102.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 336, 'aut_gen_orders': [6, 112], 'aut_gens': [[1, 6], [1, 312], [307, 342]], 'aut_group': '672.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 672, 'aut_permdeg': 23, 'aut_perms': [105328127519568246, 13098927475185897748118], 'aut_phi_ratio': 3.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [3, 7, 1, 2], [6, 7, 1, 2], [7, 6, 1, 1], [17, 1, 16, 1], [34, 7, 16, 1], [51, 7, 16, 2], [102, 7, 16, 2], [119, 6, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{16}\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 16, 'autcent_group': '16.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{16}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 7, 1], [3, 7, 2], [6, 7, 2], [7, 6, 1], [17, 1, 16], [34, 7, 16], [51, 7, 32], [102, 7, 32], [119, 6, 16]], 'center_label': '17.1', 'center_order': 17, 'central_product': True, 'central_quotient': '42.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '7.1', '17.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['42.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [3, 7, 2, 1], [6, 7, 2, 1], [7, 6, 1, 1], [17, 1, 16, 1], [34, 7, 16, 1], [51, 7, 32, 1], [102, 7, 32, 1], [119, 6, 16, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 432, 'exponent': 714, 'exponents_of_order': [1, 1, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7, 17], 'faithful_reps': [[6, 0, 16]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '714.2', 'hash': 2, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [6, 7], 'inner_gens': [[1, 618], [103, 6]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 42, 'inner_split': True, 'inner_tex': 'F_7', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 96, 'irrQ_dim': 96, 'irrR_degree': 12, 'irrep_stats': [[1, 102], [6, 17]], 'label': '714.2', 'linC_count': 16, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 2, 'linQ_dim': 22, 'linQ_dim_count': 2, 'linR_count': 48, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C17*F7', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 14, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 119, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 52, 'old_label': None, 'order': 714, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 7], [3, 14], [6, 14], [7, 6], [17, 16], [34, 112], [51, 224], [102, 224], [119, 96]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [16], 'outer_gen_pows': [0], 'outer_gens': [[1, 342]], 'outer_group': '16.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 16, 'outer_perms': [1401602636313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{16}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 17], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [6, 1], [16, 2], [32, 2], [96, 1]], 'representations': {'PC': {'code': 648724756603142495, 'gens': [1, 3], 'pres': [4, -2, -3, -7, -17, 8, 7418, 654, 94]}, 'GLZN': {'d': 2, 'p': 119, 'gens': [1685993, 116282041, 1687183, 30332863]}, 'Perm': {'d': 24, 'gens': [1126440032159490048000, 2355292434115805184000, 22324392524313, 28107316900464721920000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [102], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{17}\\times F_7', 'transitive_degree': 119, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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]], 'center_label': '3.1', 'center_order': 3, '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'], 'composition_length': 1, '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]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, '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': 'C3', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], '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': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
