-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '47520.a', 'ambient_counter': 1, 'ambient_order': 47520, 'ambient_tex': 'S_3\\times M_{11}', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 3, 'counter': 225, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '47520.a.7920.h1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '7920.h1.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': 7920, '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': '6.1', 'subgroup_hash': 1, 'subgroup_order': 6, 'subgroup_tex': 'S_3', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '47520.a', 'aut_centralizer_order': None, 'aut_label': '7920.h1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '990.l1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1584.c1.a1', '2640.h1.a1', '2640.k1.a1', '2640.k2.a1', '3960.c1.a1', '3960.d1.a1'], 'contains': ['15840.a1.a1', '23760.c1.a1'], 'core': '15840.a1.a1', 'coset_action_label': None, 'count': 165, 'diagramx': [8215, -1, 8404, -1, 8493, -1, 7032, -1], 'generators': [2921718482, 3], 'label': '47520.a.7920.h1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '165.b1.a1', 'old_label': '7920.h1.a1', 'projective_image': '47520.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '7920.h1.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 4], [1, 3], [2, 4]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', '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': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 2, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 3], [2, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 1]], 'label': '6.1', 'linC_count': 1, 'linC_degree': 2, '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': 'S3', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 3, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 6, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 3], [3, 2]], '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': 3, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 25, 'gens': [1, 2], 'pres': [2, -2, -3, 17]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [28, 45]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'GL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'SL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGL'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'SO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'SU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PSO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'PSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'GO'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'Omega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PGO'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'POmega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CSO'}, {'d': 2, 'q': 4, 'gens': [20, 194], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'CSOMinus'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'CSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CO'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'COMinus'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGammaL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSigmaL'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGammaU'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGL'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11, 6]}, 'Perm': {'d': 3, 'gens': [1, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1320, 'aut_gen_orders': [4, 6], 'aut_gens': [[7320831841, 14061657723], [9788457145, 32195217628], [63555172805, 27636960987]], 'aut_group': '47520.a', 'aut_hash': 1333562389757447279, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 47520, 'aut_permdeg': 168, 'aut_perms': [98014462987519498002958815154255670548097921317422024972115107317625877978818382526767328532443896769752952675785493845272450858065065973435419456085295317910541791323763129615140434150952989817382801801410908877139262143438147972000945930074791986527359342623663882868000978019093493326397233321883304, 188404594889563905105001553554719256168906208007069787523467086719056377023225250854022597044955493017554961614841353676176759224442577038906982187097281421175128651020346078835163598550790061587052637727563740502612312051395836513148338590916592541490275860511637156406843220825120521528189440483688566], 'aut_phi_ratio': 4.125, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 165, 1, 1], [2, 495, 1, 1], [3, 2, 1, 1], [3, 440, 1, 1], [3, 880, 1, 1], [4, 990, 1, 1], [4, 2970, 1, 1], [5, 1584, 1, 1], [6, 330, 1, 1], [6, 1320, 1, 2], [6, 2640, 1, 1], [6, 3960, 1, 1], [8, 990, 1, 2], [8, 2970, 1, 2], [10, 4752, 1, 1], [11, 720, 1, 2], [12, 1980, 1, 1], [15, 3168, 1, 1], [22, 2160, 1, 2], [24, 1980, 1, 2], [33, 1440, 1, 2]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times M_{11}', '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': 1320, 'autcentquo_group': '47520.a', 'autcentquo_hash': 1333562389757447279, 'autcentquo_nilpotent': False, 'autcentquo_order': 47520, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times M_{11}', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 165, 1], [2, 495, 1], [3, 2, 1], [3, 440, 1], [3, 880, 1], [4, 990, 1], [4, 2970, 1], [5, 1584, 1], [6, 330, 1], [6, 1320, 2], [6, 2640, 1], [6, 3960, 1], [8, 990, 2], [8, 2970, 2], [10, 4752, 1], [11, 720, 2], [12, 1980, 1], [15, 3168, 1], [22, 2160, 2], [24, 1980, 2], [33, 1440, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '47520.a', 'commutator_count': 1, 'commutator_label': '23760.b', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '7920.a'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['6.1', 1], ['7920.a', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 165, 1, 1], [2, 495, 1, 1], [3, 2, 1, 1], [3, 440, 1, 1], [3, 880, 1, 1], [4, 990, 1, 1], [4, 2970, 1, 1], [5, 1584, 1, 1], [6, 330, 1, 1], [6, 1320, 1, 2], [6, 2640, 1, 1], [6, 3960, 1, 1], [8, 990, 2, 1], [8, 2970, 2, 1], [10, 4752, 1, 1], [11, 720, 2, 1], [12, 1980, 1, 1], [15, 3168, 1, 1], [22, 2160, 2, 1], [24, 1980, 2, 1], [33, 1440, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 19434, 'exponent': 1320, 'exponents_of_order': [5, 3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[20, 0, 2], [20, 1, 1], [22, 1, 1], [32, 0, 2], [88, 1, 1], [90, 1, 1], [110, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '47520.a', 'hash': 1333562389757447279, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 1320, 'inner_gen_orders': [4, 6], 'inner_gens': [[7320831841, 64688986828], [44188830869, 14061657723]], 'inner_hash': 1333562389757447279, 'inner_nilpotent': False, 'inner_order': 47520, 'inner_split': True, 'inner_tex': 'S_3\\times M_{11}', 'inner_used': [1, 2], 'irrC_degree': 20, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 20, 'irrep_stats': [[1, 2], [2, 1], [10, 6], [11, 2], [16, 4], [20, 3], [22, 1], [32, 2], [44, 2], [45, 2], [55, 2], [88, 1], [90, 1], [110, 1]], 'label': '47520.a', 'linC_count': 6, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 2, 'linQ_dim': 12, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'S3*M11', 'ngens': 2, '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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 30, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 30, 'number_divisions': 24, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 242, 'number_subgroup_classes': 242, 'number_subgroups': 118962, 'old_label': None, 'order': 47520, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 663], [3, 1322], [4, 3960], [5, 1584], [6, 9570], [8, 7920], [10, 4752], [11, 1440], [12, 1980], [15, 3168], [22, 4320], [24, 3960], [33, 2880]], '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': None, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [10, 2], [11, 2], [20, 3], [22, 1], [32, 2], [40, 1], [44, 2], [45, 2], [55, 2], [64, 1], [88, 1], [90, 1], [110, 1]], 'representations': {'Perm': {'d': 14, 'gens': [7320831841, 14061657723]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times M_{11}', 'transitive_degree': 33, 'wreath_data': None, 'wreath_product': False}
