-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '800.1157', 'ambient_counter': 1157, 'ambient_order': 800, 'ambient_tex': 'C_{10}^2:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 80, 'characteristic': False, 'core_order': 20, 'counter': 33, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '800.1157.20.g1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '20.g1', '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': 20, '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': '40.14', 'subgroup_hash': 14, 'subgroup_order': 40, 'subgroup_tex': 'C_2^2\\times C_{10}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '800.1157', 'aut_centralizer_order': 128, 'aut_label': '20.g1', 'aut_quo_index': None, 'aut_stab_index': 60, 'aut_weyl_group': '32.25', 'aut_weyl_index': 7680, 'centralizer': '10.f1', 'complements': None, 'conjugacy_class_count': 12, 'contained_in': ['4.a1', '10.e1', '10.f1'], 'contains': ['40.a1', '40.h1', '40.l1', '100.c1'], 'core': '40.a1', 'coset_action_label': None, 'count': 60, 'diagramx': [4351, -1, 5574, -1], 'generators': [39, 400, 160, 20], 'label': '800.1157.20.g1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.b1', 'old_label': '20.g1', 'projective_image': '40.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '20.g1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 84, 'aut_gen_orders': [4, 6], 'aut_gens': [[1, 2, 4], [23, 21, 35], [20, 1, 18]], 'aut_group': '672.1046', 'aut_hash': 1046, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 672, 'aut_permdeg': 11, 'aut_perms': [16062618, 404056], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [5, 1, 4, 1], [10, 1, 28, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times \\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '672.1046', 'autcent_hash': 1046, 'autcent_nilpotent': False, 'autcent_order': 672, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_4\\times \\GL(3,2)', '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], [2, 1, 7], [5, 1, 4], [10, 1, 28]], 'center_label': '40.14', 'center_order': 40, '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': ['2.1', '2.1', '2.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [5, 1, 4, 1], [10, 1, 4, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 31, 'exponent': 10, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '40.14', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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': -1, 'irrep_stats': [[1, 40]], 'label': '40.14', 'linC_count': 3472, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 84, 'linQ_dim': 6, 'linQ_dim_count': 84, 'linR_count': 168, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C10', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 40, 'number_divisions': 16, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 32, 'number_subgroups': 32, 'old_label': None, 'order': 40, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [5, 4], [10, 28]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [4, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[23, 21, 35], [20, 1, 18]], 'outer_group': '672.1046', 'outer_hash': 1046, 'outer_nilpotent': False, 'outer_order': 672, 'outer_permdeg': 11, 'outer_perms': [16062618, 404056], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times \\GL(3,2)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 8]], 'representations': {'PC': {'code': 132099, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -5, 34]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729074815868, 24992906222183252, 25038633859490725]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [1094544393, 510353318, 857494092, 1714988184]}, 'Perm': {'d': 11, 'gens': [3628800, 40320, 720, 96]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{10}', 'transitive_degree': 40, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '80.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 120, 'aut_gen_orders': [12, 8, 20, 2, 4], 'aut_gens': [[1, 2, 40, 80], [475, 74, 460, 520], [11, 54, 420, 520], [455, 402, 400, 680], [51, 418, 460, 720], [447, 38, 40, 280]], 'aut_group': None, 'aut_hash': 2211236357296596235, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 245760, 'aut_permdeg': 84, 'aut_perms': [774067077720693926492013601042879941084684347755117240912140348391598232278722876063560065086445985546970123792268389972138754, 720126599510365802469657920649877998251858998446607858669454855423787885114334070797151762445050087177478865587614900653554732, 2015119846096619488827621620506642355284352956385654023861692698946396632428546195257987222491366846162835890167757526395865624, 2201306299278588356073233456018194038432646425780447091994115245734290880572747362054136796077292230741570725657982823856163237, 1734310839598656397328397743276350967897337566535382611394338867531761552438385960377702427478882548000550025350806475200823094], 'aut_phi_ratio': 768.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 10, 8, 1], [4, 2, 4, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [10, 1, 4, 1], [10, 1, 24, 1], [10, 2, 2, 1], [10, 2, 8, 1], [10, 2, 12, 1], [10, 2, 48, 1], [10, 10, 32, 1], [20, 2, 16, 2], [20, 2, 64, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.C_2^4.C_{30}.C_2.C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': None, 'autcent_hash': 83612722323396385, 'autcent_nilpotent': False, 'autcent_order': 6144, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4.C_2^4.D_6.C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 10, 8], [4, 2, 4], [5, 1, 4], [5, 2, 10], [10, 1, 28], [10, 2, 70], [10, 10, 32], [20, 2, 96]], 'center_label': '40.14', 'center_order': 40, 'central_product': True, 'central_quotient': '20.4', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1157, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['40.6', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 10, 1, 8], [4, 2, 1, 4], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [10, 1, 4, 7], [10, 2, 2, 7], [10, 2, 4, 14], [10, 10, 4, 8], [20, 2, 4, 8], [20, 2, 8, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 507780, 'exponent': 20, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '400.219', 'hash': 1157, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 10, 1, 1], 'inner_gens': [[1, 38, 40, 80], [5, 2, 40, 80], [1, 2, 40, 80], [1, 2, 40, 80]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': False, 'inner_tex': 'D_{10}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 180]], 'label': '800.1157', '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': 'C10^2:D4', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 260, 'number_divisions': 76, 'number_normal_subgroups': 210, 'number_subgroup_autclasses': 81, 'number_subgroup_classes': 526, 'number_subgroups': 1788, 'old_label': None, 'order': 800, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 87], [4, 8], [5, 24], [10, 488], [20, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [12, 12, 4, 12, 12], 'outer_gen_pows': [0, 0, 10, 0, 0], 'outer_gens': [[21, 422, 460, 220], [51, 422, 440, 680], [451, 42, 420, 520], [61, 406, 460, 380], [11, 66, 440, 380]], 'outer_group': None, 'outer_hash': 9138264985037689310, 'outer_nilpotent': False, 'outer_order': 12288, 'outer_permdeg': 28, 'outer_perms': [21780247670826053292656804406, 56941704948321614483856705247, 66620725816413628318920358396, 435557077365247656946341018, 34761940160214530132061332872], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^5.A_4.C_2^4.C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4], [4, 24], [8, 24], [16, 8]], 'representations': {'PC': {'code': 43306816680220700366295630019, 'gens': [1, 2, 5, 6], 'pres': [7, -2, -2, -2, -5, -2, -2, -5, 533, 36, 758, 58, 899, 124]}, 'GLZN': {'d': 2, 'p': 44, 'gens': [85193, 1959255, 766665, 1788885, 944933, 86153, 1831961]}, 'Perm': {'d': 18, 'gens': [21010051641607, 7, 16, 522547216, 46200, 1037836800, 397620186163200]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}^2:D_4', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
