-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '528.143', 'ambient_counter': 143, 'ambient_order': 528, 'ambient_tex': 'D_{12}:C_{22}', 'central': False, 'central_factor': False, 'centralizer_order': 66, 'characteristic': True, 'core_order': 264, 'counter': 8, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '528.143.2.c1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': ['F'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '264.30', 'subgroup_hash': 30, 'subgroup_order': 264, 'subgroup_tex': 'Q_8\\times C_{33}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '528.143', 'aut_centralizer_order': 6, 'aut_label': '2.c1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '480.1198', 'aut_weyl_index': 6, 'centralizer': '8.a1.a1', 'complements': ['264.b1.a1', '264.b1.b1', '264.b1.c1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.b1.a1', '4.b1.b1', '4.b1.c1', '6.a1.a1', '22.c1.a1'], 'core': '2.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [9337, 1025, 7606, 1625, 1443, 7940, 4406, 8315], 'generators': [6, 268, 48, 132, 264], 'label': '528.143.2.c1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.c1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.b1.b1', '4.b1.a1', '4.b1.c1', '6.a1.a1', '22.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.c1.a1', 'projective_image': '24.14', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.c1.a1', 'subgroup_fusion': None, 'weyl_group': '8.5'}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '132.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [2, 6, 2, 2, 10], 'aut_gens': [[1, 2], [199, 134], [67, 245], [133, 134], [1, 218], [133, 194]], 'aut_group': '480.1198', 'aut_hash': 1198, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 480, 'aut_permdeg': 18, 'aut_perms': [2512186722875766, 5361639331156110, 2867874155643000, 169052162630249, 3183684609987447], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 2, 3, 1], [6, 1, 2, 1], [11, 1, 10, 1], [12, 2, 6, 1], [22, 1, 10, 1], [33, 1, 20, 1], [44, 2, 30, 1], [66, 1, 20, 1], [132, 2, 60, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_{10}\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 10, 'autcent_group': '80.52', 'autcent_hash': 52, 'autcent_nilpotent': True, 'autcent_order': 80, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_{10}', '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, 1, 1], [3, 1, 2], [4, 2, 3], [6, 1, 2], [11, 1, 10], [12, 2, 6], [22, 1, 10], [33, 1, 20], [44, 2, 30], [66, 1, 20], [132, 2, 60]], 'center_label': '66.4', 'center_order': 66, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 30, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['3.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 2, 1, 3], [6, 1, 2, 1], [11, 1, 10, 1], [12, 2, 2, 3], [22, 1, 10, 1], [33, 1, 20, 1], [44, 2, 10, 3], [66, 1, 20, 1], [132, 2, 20, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 48, 'exponent': 132, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[2, 0, 20]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '132.10', 'hash': 30, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 134], [133, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 4, 'irrep_stats': [[1, 132], [2, 33]], 'label': '264.30', 'linC_count': 20, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 20, 'linQ_dim': 14, 'linQ_dim_count': 4, 'linR_count': 10, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'Q8*C33', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 165, 'number_divisions': 20, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 24, 'number_subgroups': 24, 'old_label': None, 'order': 264, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 6], [6, 2], [11, 10], [12, 12], [22, 10], [33, 20], [44, 60], [66, 20], [132, 120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 2, 30], 'outer_gen_pows': [0, 2, 0], 'outer_gens': [[1, 178], [67, 2], [199, 257]], 'outer_group': '120.45', 'outer_hash': 45, 'outer_nilpotent': False, 'outer_order': 120, 'outer_permdeg': 12, 'outer_perms': [40321, 3628800, 43547329], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{10}\\times D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [4, 1], [10, 4], [20, 5], [40, 1]], 'representations': {'PC': {'code': 390001890992131100019, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -3, -11, 660, 1341, 26, 42, 78]}, 'GLFp': {'d': 2, 'p': 67, 'gens': [8530412, 134067]}, 'Perm': {'d': 22, 'gens': [265820153888120832000, 158139385059591936000, 12454041600, 36288000, 51212944267220736000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 66], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_8\\times C_{33}', 'transitive_degree': 264, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '88.12', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [2, 30, 30, 4, 60], 'aut_gens': [[1, 2, 12], [273, 10, 516], [135, 134, 318], [399, 404, 30], [1, 406, 252], [135, 140, 462]], 'aut_group': None, 'aut_hash': 6447933894819248036, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2880, 'aut_permdeg': 22, 'aut_perms': [1054807946722337317438, 788257039087161158745, 1049256562121575287446, 388050850346650370158, 41445231959116189461], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 3, 1], [3, 2, 1, 1], [4, 2, 3, 1], [4, 3, 2, 1], [6, 2, 1, 1], [11, 1, 10, 1], [12, 4, 3, 1], [22, 1, 10, 1], [22, 6, 30, 1], [33, 2, 10, 1], [44, 2, 30, 1], [44, 3, 20, 1], [66, 2, 10, 1], [132, 4, 30, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_{15}\\times A_4).C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 10, 'autcent_group': '80.52', 'autcent_hash': 52, 'autcent_nilpotent': True, 'autcent_order': 80, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 3], [3, 2, 1], [4, 2, 3], [4, 3, 2], [6, 2, 1], [11, 1, 10], [12, 4, 3], [22, 1, 10], [22, 6, 30], [33, 2, 10], [44, 2, 30], [44, 3, 20], [66, 2, 10], [132, 4, 30]], 'center_label': '22.2', 'center_order': 22, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 143, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['48.41', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 3], [3, 2, 1, 1], [4, 2, 1, 3], [4, 3, 2, 1], [6, 2, 1, 1], [11, 1, 10, 1], [12, 4, 1, 3], [22, 1, 10, 1], [22, 6, 10, 3], [33, 2, 10, 1], [44, 2, 10, 3], [44, 3, 20, 1], [66, 2, 10, 1], [132, 4, 10, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 14896, 'exponent': 132, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[4, 0, 10]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '264.37', 'hash': 143, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 2], 'inner_gens': [[1, 274, 12], [5, 2, 276], [1, 266, 12]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 40, 'irrQ_dim': 80, 'irrR_degree': 8, 'irrep_stats': [[1, 88], [2, 66], [4, 11]], 'label': '528.143', 'linC_count': 970, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 8, 'linQ_dim': 16, 'linQ_dim_count': 32, 'linR_count': 80, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D12:C22', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 165, 'number_divisions': 28, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 80, 'number_subgroups': 160, 'old_label': None, 'order': 528, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 19], [3, 2], [4, 12], [6, 2], [11, 10], [12, 12], [22, 190], [33, 20], [44, 120], [66, 20], [132, 120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 2, 30], 'outer_gen_pows': [0, 6, 0], 'outer_gens': [[265, 2, 12], [271, 2, 258], [271, 140, 210]], 'outer_group': '120.45', 'outer_hash': 45, 'outer_nilpotent': False, 'outer_order': 120, 'outer_permdeg': 12, 'outer_perms': [40320, 3669121, 79835329], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{10}\\times D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 11], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 2], [10, 8], [20, 4], [40, 2]], 'representations': {'PC': {'code': 366106181404610250410110001009, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -3, -2, -2, -11, 3289, 31, 4898, 1202, 3321, 69, 88]}, 'GLZN': {'d': 2, 'p': 110, 'gens': [1332299, 1332101, 118459089, 118730041, 116462587, 75207551]}, 'Perm': {'d': 22, 'gens': [51469039215928857600, 112305146080038912000, 153650588794230528000, 4037913, 212155499332288512000, 6706022400]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 22], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{12}:C_{22}', 'transitive_degree': 264, 'wreath_data': None, 'wreath_product': False}
-
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}
