-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '729.494', 'ambient_counter': 494, 'ambient_order': 729, 'ambient_tex': 'C_3^2.C_3^4', 'central': False, 'central_factor': False, 'centralizer_order': 243, 'characteristic': False, 'core_order': 1, 'counter': 65, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '729.494.243.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '243.c1', 'outer_equivalence': True, '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': 243, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '729.494', 'aut_centralizer_order': 8748, 'aut_label': '243.c1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '1.1', 'aut_weyl_index': 78732, 'centralizer': '3.a1', 'complements': None, 'conjugacy_class_count': 3, 'contained_in': ['81.b1', '81.j1', '81.m1', '81.n1'], 'contains': ['729.a1'], 'core': '729.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [2274, -1, 2142, -1], 'generators': [9], 'label': '729.494.243.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '81.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '243.c1', 'projective_image': '729.494', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '243.c1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'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}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '81.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 18, 'aut_gen_orders': [18, 6, 9], 'aut_gens': [[1, 3, 9, 27, 81], [245, 332, 279, 513, 405], [29, 39, 9, 297, 711], [271, 392, 63, 270, 153]], 'aut_group': None, 'aut_hash': 2716519217131416509, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 78732, 'aut_permdeg': 180, 'aut_perms': [192232112061192237952382103945020761819219836633653314311328246802558937852496742750954159794973763544074348508381354147558260044977724984294624440382257466500749277447767961075420767828293331361139949284873727046182992988429031876534758729814190054424006695217834086207009910909202949634112260957053475423894662426074601076733423, 148173244434815155760694206271524549714633042667301543893650171781857051214793903902000924772507970060688337351978725360968783607370440369418271864475874070036705112784022687085557836181718294681142496777897963134095732822740604730011405009384401284315641925790072538771296260752541049508287703420383987322968847533333325079607467, 65435134458750126701506302042558497787290826419517417208147196194055045292735778706449410678887666545688837742785495254278516628759700526787633223965456194376027380727187157964554195226698420163669778301389505614268739500036957857225913558981884625420507075445423947845388920571034430540656248634294303387087180475298180371723077], 'aut_phi_ratio': 162.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 3, 3, 2], [3, 3, 6, 1], [3, 9, 2, 2], [3, 9, 18, 1], [9, 3, 18, 1], [9, 9, 6, 2], [9, 9, 36, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3\\times C_3^4.\\He_3.D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '81.15', 'autcent_hash': 15, 'autcent_nilpotent': True, 'autcent_order': 81, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '972.480', 'autcentquo_hash': 480, 'autcentquo_nilpotent': False, 'autcentquo_order': 972, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^3:S_3^2', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 3, 14], [3, 9, 22], [9, 3, 18], [9, 9, 48]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '243.62', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 494, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 7], [3, 9, 2, 11], [9, 3, 2, 9], [9, 9, 2, 24]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 2021760, 'exponent': 9, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [[9, 0, 6]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '81.15', 'hash': 494, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 3, 3, 3], 'inner_gens': [[1, 57, 9, 27, 81], [28, 3, 9, 270, 81], [1, 3, 9, 27, 567], [1, 489, 9, 27, 81], [1, 3, 252, 27, 81]], 'inner_hash': 62, 'inner_nilpotent': True, 'inner_order': 243, 'inner_split': False, 'inner_tex': 'C_3^2\\times \\He_3', 'inner_used': [1, 2, 3, 5], 'irrC_degree': 9, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 81], [3, 18], [9, 6]], 'label': '729.494', 'linC_count': None, 'linC_degree': None, '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': False, 'monomial': True, 'name': 'C3^2.C3^4', 'ngens': 4, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 105, 'number_divisions': 53, 'number_normal_subgroups': 235, 'number_subgroup_autclasses': 69, 'number_subgroup_classes': 490, 'number_subgroups': 1228, 'old_label': None, 'order': 729, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 242], [9, 486]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [6, 6, 3, 3], 'outer_gen_pows': [0, 0, 54, 0], 'outer_gens': [[488, 6, 495, 513, 162], [1, 6, 9, 297, 576], [1, 572, 63, 27, 81], [1, 12, 9, 27, 108]], 'outer_group': '324.117', 'outer_hash': 117, 'outer_nilpotent': False, 'outer_order': 324, 'outer_permdeg': 12, 'outer_perms': [8356347, 49367520, 25967544, 91990224], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^3:D_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 27, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 40], [6, 9], [18, 3]], 'representations': {'PC': {'code': 813050670123056, 'gens': [1, 2, 3, 4, 5], 'pres': [6, 3, 3, 3, 3, 3, 3, 685, 2169, 1906, 118]}, 'Perm': {'d': 27, 'gens': [7550102686338507196661760000, 806634631153204606767213507, 3323121639686209319422821120, 51706588025052938880243, 711549292653507, 806634631153204606767248884]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3, 3], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2.C_3^4', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
