-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '240.124', 'ambient_counter': 124, 'ambient_order': 240, 'ambient_tex': 'D_{10}.D_6', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 10, 'counter': 31, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '240.124.12.e1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.e1.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': 12, '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': '20.3', 'subgroup_hash': 3, 'subgroup_order': 20, 'subgroup_tex': 'F_5', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '240.124', 'aut_centralizer_order': 4, 'aut_label': '12.e1', 'aut_quo_index': None, 'aut_stab_index': 12, 'aut_weyl_group': '20.3', 'aut_weyl_index': 48, 'centralizer': '120.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.e1.b1', '6.b1.b1'], 'contains': ['24.b1.b1', '60.e1.b1'], 'core': '24.b1.b1', 'coset_action_label': None, 'count': 6, 'diagramx': [1927, -1, 7472, -1, 9336, -1, 8114, -1], 'generators': [57, 48, 54], 'label': '240.124.12.e1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.b1.b1', 'old_label': '12.e1.b1', 'projective_image': '240.124', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.e1.b1', 'subgroup_fusion': None, 'weyl_group': '20.3'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 5], 'aut_gens': [[1, 4], [1, 12], [5, 4]], 'aut_group': '20.3', 'aut_hash': 3, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20, 'aut_permdeg': 5, 'aut_perms': [18, 34], 'aut_phi_ratio': 2.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 5, 1, 2], [5, 4, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_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': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [4, 5, 2], [5, 4, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '20.3', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1'], '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, 5, 1, 1], [4, 5, 2, 1], [5, 4, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '20.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 5], 'inner_gens': [[1, 12], [13, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'F_5', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [4, 1]], 'label': '20.3', 'linC_count': 1, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'F5', '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': 5, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 14, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 5], [4, 10], [5, 4]], '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': 5, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1]], 'representations': {'PC': {'code': 2551363, 'gens': [1, 3], 'pres': [3, -2, -2, -5, 6, 110, 77]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [23561683, 28172809]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33, 10], 'family': 'AGL'}, {'d': 1, 'q': 5, 'gens': [33, 10], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 127]}, 'Perm': {'d': 5, 'gens': [10, 23, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'F_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [2, 2, 4, 60], 'aut_gens': [[1, 4, 8], [1, 4, 88], [1, 4, 12], [1, 4, 188], [57, 4, 12]], 'aut_group': '960.11038', 'aut_hash': 11038, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 960, 'aut_permdeg': 12, 'aut_perms': [720, 87091200, 127371618, 87464231], 'aut_phi_ratio': 15.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 5, 2, 1], [2, 10, 1, 1], [3, 2, 1, 1], [4, 30, 2, 2], [5, 4, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 10, 2, 2], [10, 4, 1, 1], [10, 4, 2, 1], [15, 4, 2, 1], [30, 4, 2, 1], [30, 4, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times D_4\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '240.195', 'autcentquo_hash': 195, 'autcentquo_nilpotent': False, 'autcentquo_order': 240, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 5, 2], [2, 10, 1], [3, 2, 1], [4, 30, 4], [5, 4, 1], [6, 2, 3], [6, 10, 4], [10, 4, 3], [15, 4, 2], [30, 4, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '120.41', 'commutator_count': 1, 'commutator_label': '30.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 124, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 5, 1, 2], [2, 10, 1, 1], [3, 2, 1, 1], [4, 30, 2, 2], [5, 4, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 10, 1, 2], [6, 10, 2, 1], [10, 4, 1, 1], [10, 4, 2, 1], [15, 4, 2, 1], [30, 4, 2, 1], [30, 4, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 60, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '120.41', 'hash': 124, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [4, 1, 30], 'inner_gens': [[1, 4, 140], [1, 4, 8], [117, 4, 8]], 'inner_hash': 41, 'inner_nilpotent': False, 'inner_order': 120, 'inner_split': False, 'inner_tex': 'C_{30}:C_4', 'inner_used': [1, 3], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 10], [4, 12]], 'label': '240.124', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 20, 'linQ_dim': 8, 'linQ_dim_count': 12, 'linR_count': 4, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D10.D6', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 30, 'number_divisions': 20, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 52, 'number_subgroup_classes': 68, 'number_subgroups': 320, 'old_label': None, 'order': 240, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 23], [3, 2], [4, 120], [5, 4], [6, 46], [10, 12], [15, 8], [30, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[121, 4, 8], [1, 4, 12], [1, 4, 88]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 4], [8, 3], [16, 1]], 'representations': {'PC': {'code': 142030415629314272430516703239, 'gens': [1, 3, 4], 'pres': [6, -2, -2, -2, 2, -3, -5, 12, 3363, 1833, 69, 964, 970, 118, 3461, 3467]}, 'GLZN': {'d': 2, 'p': 20, 'gens': [12211, 24181, 92201, 8081, 72001, 130311]}, 'Perm': {'d': 12, 'gens': [40280409, 7620480, 16, 87091200, 5760, 46]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{10}.D_6', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
