-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1075.2', 'ambient_counter': 2, 'ambient_order': 1075, 'ambient_tex': 'C_5\\times C_{215}', 'central': True, 'central_factor': False, 'centralizer_order': 1075, 'characteristic': False, 'core_order': 5, 'counter': 13, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '1075.2.215.a1.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '215.a1.d1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '215.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 215, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{215}', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '5.1', 'subgroup_hash': 1, 'subgroup_order': 5, 'subgroup_tex': 'C_5', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1075.2', 'aut_centralizer_order': 840, 'aut_label': '215.a1', 'aut_quo_index': 1, 'aut_stab_index': 6, 'aut_weyl_group': '4.1', 'aut_weyl_index': 5040, 'centralizer': '1.a1.a1', 'complements': ['5.a1.a1', '5.a1.e1', '5.a1.b1', '5.a1.f1', '5.a1.c1'], 'conjugacy_class_count': 1, 'contained_in': ['5.a1.d1', '43.a1.a1'], 'contains': ['1075.a1.a1'], 'core': '215.a1.d1', 'coset_action_label': None, 'count': 1, 'diagramx': [5078, 5078, 7404, 7404, 2601, 2601, 6376, 6376], 'generators': [861], 'label': '1075.2.215.a1.d1', 'mobius_quo': -1, 'mobius_sub': 1, 'normal_closure': '215.a1.d1', 'normal_contained_in': ['5.a1.d1', '43.a1.a1'], 'normal_contains': ['1075.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '215.a1.d1', 'projective_image': '215.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '215.1', 'quotient_action_kernel_order': 215, 'quotient_fusion': None, 'short_label': '215.a1.d1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '5.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [2]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', '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': 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]], 'center_label': '5.1', 'center_order': 5, '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': ['5.1'], '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], [5, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '5.1', 'hash': 1, 'hyperelementary': 5, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 5]], 'label': '5.1', 'linC_count': 4, 'linC_degree': 1, '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': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5', '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': 5, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 5, 'order_factorization_type': 1, 'order_stats': [[1, 1], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 5, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -5]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26329372]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131]}, 'Perm': {'d': 5, 'gens': [96]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [5], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '1075.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 840, 'aut_gen_orders': [4, 3, 42], 'aut_gens': [[1, 5], [2, 5], [434, 654], [1, 230]], 'aut_group': None, 'aut_hash': 958266202237350053, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20160, 'aut_permdeg': 66, 'aut_perms': [8501426012765600467715580139178822124850960655680924251770854567275775348376076288000000000, 16882006246475932655673154591303259670730096312946391877527740929004871402651648000000000000, 34289376947494122614363304694584807557656420940313], 'aut_phi_ratio': 24.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [5, 1, 24, 1], [43, 1, 42, 1], [215, 1, 1008, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{42}\\times \\GL(2,5)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 840, 'autcent_group': None, 'autcent_hash': 958266202237350053, 'autcent_nilpotent': False, 'autcent_order': 20160, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_{42}\\times \\GL(2,5)', '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, 24], [43, 1, 42], [215, 1, 1008]], 'center_label': '1075.2', 'center_order': 1075, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['5.1', '5.1', '43.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['43.1', 1], ['5.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 6], [43, 1, 42, 1], [215, 1, 168, 6]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 44, 'exponent': 215, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [5, 43], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1075.2', 'hash': 2, 'hyperelementary': 5, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 5], [1, 5]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 1075]], 'label': '1075.2', 'linC_count': None, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*C215', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 4, 'number_conjugacy_classes': 1075, 'number_divisions': 14, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 1075, 'order_factorization_type': 22, 'order_stats': [[1, 1], [5, 24], [43, 42], [215, 1008]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 840, 'outer_gen_orders': [4, 3, 42], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[2, 5], [434, 654], [1, 230]], 'outer_group': None, 'outer_hash': 958266202237350053, 'outer_nilpotent': False, 'outer_order': 20160, 'outer_permdeg': 66, 'outer_perms': [8501426012765600467715580139178822124850960655680924251770854567275775348376076288000000000, 16882006246475932655673154591303259670730096312946391877527740929004871402651648000000000000, 34289376947494122614363304694584807557656420940313], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_{42}\\times \\GL(2,5)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 53, 'pgroup': 0, 'primary_abelian_invariants': [5, 5, 43], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 6], [42, 1], [168, 6]], 'representations': {'PC': {'code': 2524451, 'gens': [1, 2], 'pres': [3, -5, -5, -43, 34]}, 'GLFp': {'d': 2, 'p': 431, 'gens': [11048693155, 19615432890]}, 'Perm': {'d': 53, 'gens': [322632700683775514286642547425615067901158021763533111296000000000000, 1034492966044672722571857420614447919876790529556480000000000, 59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [5, 215], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times C_{215}', 'transitive_degree': 1075, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '215.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 84, 'aut_gen_orders': [2, 84], 'aut_gens': [[1], [171], [212]], 'aut_group': '168.39', 'aut_hash': 39, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 168, 'aut_permdeg': 16, 'aut_perms': [1321086412800, 6746343273], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [43, 1, 42, 1], [215, 1, 168, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{84}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.39', 'autcent_hash': 39, 'autcent_nilpotent': True, 'autcent_order': 168, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{84}', '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], [43, 1, 42], [215, 1, 168]], 'center_label': '215.1', 'center_order': 215, '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', '43.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['43.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [43, 1, 42, 1], [215, 1, 168, 1]], 'element_repr_type': 'PC', 'elementary': 215, 'eulerian_function': 1, 'exponent': 215, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [5, 43], 'faithful_reps': [[1, 0, 168]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '215.1', 'hash': 1, 'hyperelementary': 215, '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': 168, 'irrQ_dim': 168, 'irrR_degree': 2, 'irrep_stats': [[1, 215]], 'label': '215.1', 'linC_count': 168, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 46, 'linQ_degree_count': 1, 'linQ_dim': 46, 'linQ_dim_count': 1, 'linR_count': 84, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C215', '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': 215, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 215, 'order_factorization_type': 11, 'order_stats': [[1, 1], [5, 4], [43, 42], [215, 168]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [2, 84], 'outer_gen_pows': [0, 0], 'outer_gens': [[171], [212]], 'outer_group': '168.39', 'outer_hash': 39, 'outer_nilpotent': True, 'outer_order': 168, 'outer_permdeg': 16, 'outer_perms': [1321086412800, 6746343273], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{84}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 48, 'pgroup': 0, 'primary_abelian_invariants': [5, 43], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [42, 1], [168, 1]], 'representations': {'PC': {'code': 8987, 'gens': [1], 'pres': [2, -5, -43, 10]}, 'GLFp': {'d': 2, 'p': 431, 'gens': [10248063175]}, 'Perm': {'d': 48, 'gens': [1034492966044672722571857420614447919876790529556480000000000, 59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [215], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{215}', 'transitive_degree': 215, 'wreath_data': None, 'wreath_product': False}
