-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1320.22', 'ambient_counter': 22, 'ambient_order': 1320, 'ambient_tex': 'C_{11}:C_5\\times \\SL(2,3)', 'central': True, 'central_factor': False, 'centralizer_order': 1320, 'characteristic': True, 'core_order': 2, 'counter': 27, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1320.22.660.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '660.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '660.16', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 16, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 660, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'A_4\\times C_{11}:C_5', 'simple': True, 'solvable': True, 'special_labels': ['Z', 'Phi', 'U0', 'D2', 'C4'], 'split': False, 'standard_generators': False, 'stem': True, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1320.22', 'aut_centralizer_order': 2640, 'aut_label': '660.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 2640, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['60.a1.a1', '132.a1.a1', '220.a1.a1', '330.a1.a1'], 'contains': ['1320.a1.a1'], 'core': '660.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6106, 4514, 4479, 4514, 4967, 5088, 5563, 4923], 'generators': [660], 'label': '1320.22.660.a1.a1', 'mobius_quo': -1, 'mobius_sub': 44, 'normal_closure': '660.a1.a1', 'normal_contained_in': ['60.a1.a1', '165.a1.a1'], 'normal_contains': ['1320.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '660.a1.a1', 'projective_image': '660.16', 'quotient_action_image': '1.1', 'quotient_action_kernel': '660.16', 'quotient_action_kernel_order': 660, 'quotient_fusion': None, 'short_label': '660.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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, 1]], 'center_label': '2.1', 'center_order': 2, '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': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], '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': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [20, 30, 2, 22], 'aut_gens': [[1, 3, 30], [347, 1083, 585], [1006, 393, 855], [1006, 3, 690], [676, 1143, 30]], 'aut_group': '2640.bv', 'aut_hash': 5844094576270439126, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2640, 'aut_permdeg': 19, 'aut_perms': [70031724642321618, 3706582075859460, 3720671916596940, 44077477931845776], 'aut_phi_ratio': 8.25, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 4, 2, 1], [4, 6, 1, 1], [5, 11, 1, 4], [6, 4, 2, 1], [10, 11, 1, 4], [11, 5, 2, 1], [15, 44, 2, 4], [20, 66, 1, 4], [22, 5, 2, 1], [30, 44, 2, 4], [33, 20, 4, 1], [44, 30, 2, 1], [66, 20, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times F_{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': 660, 'autcentquo_group': '2640.bv', 'autcentquo_hash': 5844094576270439126, 'autcentquo_nilpotent': False, 'autcentquo_order': 2640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 4, 2], [4, 6, 1], [5, 11, 4], [6, 4, 2], [10, 11, 4], [11, 5, 2], [15, 44, 8], [20, 66, 4], [22, 5, 2], [30, 44, 8], [33, 20, 4], [44, 30, 2], [66, 20, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '660.16', 'commutator_count': 1, 'commutator_label': '88.10', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 22, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['24.3', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 4, 2, 1], [4, 6, 1, 1], [5, 11, 4, 1], [6, 4, 2, 1], [10, 11, 4, 1], [11, 5, 2, 1], [15, 44, 8, 1], [20, 66, 4, 1], [22, 5, 2, 1], [30, 44, 8, 1], [33, 20, 4, 1], [44, 30, 2, 1], [66, 20, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 384, 'exponent': 660, 'exponents_of_order': [3, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[10, 0, 6]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '660.16', 'hash': 22, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [3, 10, 22], 'inner_gens': [[1, 348, 705], [1006, 3, 450], [676, 903, 30]], 'inner_hash': 16, 'inner_nilpotent': False, 'inner_order': 660, 'inner_split': True, 'inner_tex': 'A_4\\times C_{11}:C_5', 'inner_used': [1, 2, 3], 'irrC_degree': 10, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 20, 'irrep_stats': [[1, 15], [2, 15], [3, 5], [5, 6], [10, 6], [15, 2]], 'label': '1320.22', 'linC_count': 90, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 1, 'linQ_dim': 14, 'linQ_dim_count': 2, 'linR_count': 24, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C11:C5*SL(2,3)', '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, 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': 12, 'number_conjugacy_classes': 49, 'number_divisions': 15, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 28, 'number_subgroups': 210, 'old_label': None, 'order': 1320, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 6], [5, 44], [6, 8], [10, 44], [11, 10], [15, 352], [20, 264], [22, 10], [30, 352], [33, 80], [44, 60], [66, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 12], 'outer_gens': [[2, 663, 645], [1, 3, 510]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 2], [3, 1], [4, 2], [8, 2], [10, 1], [12, 1], [16, 1], [20, 2], [30, 1], [40, 1]], 'representations': {'PC': {'code': 88948306651217626019010249891444754086078103009, 'gens': [1, 2, 4], 'pres': [6, 3, 2, 5, 2, 2, 11, 4177, 31, 1994, 16923, 3609, 615, 69, 9010, 1516, 88, 5771, 3617]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [35962, 826574, 60644, 36037, 48302, 36301]}, 'Perm': {'d': 19, 'gens': [2187, 400434839844480, 7039, 12593, 7159885125120000, 18619]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}:C_5\\times \\SL(2,3)', 'transitive_degree': 88, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [20, 30, 2, 22], 'aut_gens': [[1, 15, 30], [266, 330, 255], [616, 330, 525], [16, 15, 30], [526, 15, 30]], 'aut_group': '2640.bv', 'aut_hash': 5844094576270439126, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2640, 'aut_permdeg': 19, 'aut_perms': [97255114079389952, 5325314711042161, 59454991622689944, 24501861178669824], 'aut_phi_ratio': 16.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 11, 1, 4], [10, 33, 1, 4], [11, 5, 2, 1], [15, 44, 2, 4], [22, 15, 2, 1], [33, 20, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times F_{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': 660, 'autcentquo_group': '2640.bv', 'autcentquo_hash': 5844094576270439126, 'autcentquo_nilpotent': False, 'autcentquo_order': 2640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 4, 2], [5, 11, 4], [10, 33, 4], [11, 5, 2], [15, 44, 8], [22, 15, 2], [33, 20, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '660.16', 'commutator_count': 1, 'commutator_label': '44.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 16, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 11, 4, 1], [10, 33, 4, 1], [11, 5, 2, 1], [15, 44, 8, 1], [22, 15, 2, 1], [33, 20, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 96, 'exponent': 330, 'exponents_of_order': [2, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[15, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '660.16', 'hash': 16, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [15, 2, 22], 'inner_gens': [[1, 345, 135], [331, 15, 30], [586, 15, 30]], 'inner_hash': 16, 'inner_nilpotent': False, 'inner_order': 660, 'inner_split': True, 'inner_tex': 'A_4\\times C_{11}:C_5', 'inner_used': [1, 2, 3], 'irrC_degree': 15, 'irrQ_degree': 30, 'irrQ_dim': 30, 'irrR_degree': 30, 'irrep_stats': [[1, 15], [3, 5], [5, 6], [15, 2]], 'label': '660.16', 'linC_count': 30, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 13, 'linQ_degree_count': 1, 'linQ_dim': 13, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 13, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'A4*C11:C5', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 28, 'number_divisions': 9, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 20, 'number_subgroups': 140, 'old_label': None, 'order': 660, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 8], [5, 44], [10, 132], [11, 10], [15, 352], [22, 30], [33, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[11, 330, 315], [346, 15, 630]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 1], [8, 1], [10, 1], [12, 1], [20, 1], [30, 1]], 'representations': {'PC': {'code': 1084183530999503755850513009, 'gens': [1, 3, 4], 'pres': [5, -3, -5, -2, 2, -11, 15, 5177, 2703, 1808, 58, 6004, 4509]}, 'GLZN': {'d': 2, 'p': 44, 'gens': [2001847, 766693, 85361, 1044979, 128745]}, 'Perm': {'d': 15, 'gens': [4, 7277200680, 7, 16, 100725742080]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_4\\times C_{11}:C_5', 'transitive_degree': 44, 'wreath_data': None, 'wreath_product': False}
