-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '864.4669', 'ambient_counter': 4669, 'ambient_order': 864, 'ambient_tex': 'C_6^2:D_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 9, 'counter': 56, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '864.4669.24.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '24.d1.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': 24, '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': '36.13', 'subgroup_hash': 13, 'subgroup_order': 36, 'subgroup_tex': 'C_6:S_3', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '864.4669', 'aut_centralizer_order': 4, 'aut_label': '24.d1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '288.1027', 'aut_weyl_index': 12, 'centralizer': '216.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.a1.a1', '12.f1.a1', '12.f1.b1'], 'contains': ['48.b1.a1', '48.d1.a1', '72.c1.a1', '72.c1.b1'], 'core': '96.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [4996, -1, 6568, -1, 4073, -1, 6373, -1], 'generators': [16703, 1855560, 16, 126000], 'label': '864.4669.24.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '24.d1.a1', 'projective_image': '864.4669', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.d1.a1', 'subgroup_fusion': None, 'weyl_group': '72.40'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 4, 2, 4, 3, 2, 6, 6], 'aut_gens': [[1, 2, 6], [19, 2, 6], [35, 26, 32], [25, 4, 30], [33, 14, 34], [5, 2, 6], [19, 28, 6], [33, 2, 6], [23, 16, 30]], 'aut_group': '864.4661', 'aut_hash': 4661, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 11, 'aut_perms': [1, 10889689, 24054000, 20507935, 14605950, 2245249, 19496545, 16180705], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [3, 2, 4, 1], [6, 2, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_3^2:\\GL(2,3)', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 2, 4], [6, 2, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 2, 1, 4], [6, 2, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 21, 'exponent': 6, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '36.13', 'hash': 13, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[1, 4, 30], [5, 2, 6], [13, 2, 6]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 8]], 'label': '36.13', 'linC_count': 18, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 18, 'linQ_dim': 4, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6:S3', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 12, 'number_divisions': 12, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 30, 'number_subgroups': 78, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 19], [3, 8], [6, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 1, 1], 'outer_gens': [[1, 24, 20], [19, 2, 6], [1, 24, 34], [1, 28, 8], [1, 16, 10]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [143, 127, 576, 126, 121], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8]], 'representations': {'PC': {'code': 405218245, 'gens': [1, 2, 3], 'pres': [4, -2, -3, -2, -3, 33, 362, 34, 387]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [12016555, 35931151, 11780031]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8156, 8101, 13286]}, 'Perm': {'d': 8, 'gens': [721, 24, 5763, 5760]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6:S_3', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [24, 6, 4], 'aut_gens': [[722, 369360, 16680, 4, 823080, 862680, 7, 16], [1174446, 1537320, 1861320, 19, 782040, 126000, 16, 23], [734, 373080, 16680, 15, 862680, 823080, 23, 7], [374402, 373080, 16680, 12, 126000, 782040, 16, 7]], 'aut_group': '3456.jg', 'aut_hash': 5305840276193666630, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3456, 'aut_permdeg': 72, 'aut_perms': [9507268148402333116358126752452520381383056294146753897583136046185450624031797507023677349874853757476, 54433421111448532870037929592091425890257188376218057020060586832333267931509371914395733859568194756408, 22047094041939682301680336340057255343365598533904043135622417239691009666841818725207582108101670683400], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [2, 36, 2, 1], [3, 4, 2, 1], [3, 8, 1, 1], [3, 32, 2, 1], [4, 18, 1, 1], [4, 36, 2, 1], [4, 54, 1, 1], [6, 12, 2, 1], [6, 72, 1, 1], [6, 72, 2, 1], [12, 72, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'F_9:C_2\\times S_4', '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': 24, 'autcentquo_group': '3456.jg', 'autcentquo_hash': 5305840276193666630, 'autcentquo_nilpotent': False, 'autcentquo_order': 3456, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9:C_2\\times S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 1], [2, 36, 2], [3, 4, 2], [3, 8, 1], [3, 32, 2], [4, 18, 1], [4, 36, 2], [4, 54, 1], [6, 12, 2], [6, 72, 3], [12, 72, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '864.4669', 'commutator_count': 1, 'commutator_label': '216.167', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4669, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [2, 36, 1, 2], [3, 4, 1, 2], [3, 8, 1, 1], [3, 32, 1, 2], [4, 18, 1, 1], [4, 36, 1, 2], [4, 54, 1, 1], [6, 12, 1, 2], [6, 72, 1, 3], [12, 72, 1, 2], [12, 72, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 36, 'exponent': 12, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '864.4669', 'hash': 4669, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 2, 3, 3, 3, 2, 2], 'inner_gens': [[722, 373080, 16680, 3, 862680, 823080, 16, 7], [15962, 369360, 16680, 4, 1936800, 823080, 7, 16], [722, 369360, 16680, 4, 1975800, 1936800, 7, 16], [725, 369360, 16680, 4, 823080, 862680, 23, 7], [126722, 534960, 1987320, 4, 823080, 862680, 7, 16], [85682, 1100280, 1946280, 4, 823080, 862680, 7, 16], [741, 369360, 16680, 15, 823080, 862680, 7, 16], [741, 369360, 16680, 19, 823080, 862680, 7, 16]], 'inner_hash': 4669, 'inner_nilpotent': False, 'inner_order': 864, 'inner_split': False, 'inner_tex': 'C_6^2:D_{12}', 'inner_used': [1, 2, 4, 5, 7], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 5], [3, 4], [4, 4], [6, 1], [8, 2], [12, 4]], 'label': '864.4669', 'linC_count': 16, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 16, 'linQ_dim': 7, 'linQ_dim_count': 16, 'linR_count': 16, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C6^2:D12', 'ngens': 8, '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': 16, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 24, 'number_divisions': 23, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 123, 'number_subgroup_classes': 194, 'number_subgroups': 2876, 'old_label': None, 'order': 864, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 111], [3, 80], [4, 144], [6, 240], [12, 288]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [369360, 0], 'outer_gens': [[374402, 369360, 16680, 4, 782040, 126000, 7, 16], [722, 369360, 16680, 3, 1975800, 1936800, 16, 7]], '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': 4, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [3, 4], [4, 5], [6, 1], [8, 2], [12, 4]], 'representations': {'PC': {'code': 13237150843795853279720875387965386456695083520211888927134335001513940501, 'gens': [1, 2, 5, 7], 'pres': [8, 2, 2, 2, 3, 2, 3, 2, 3, 353, 41, 482, 66, 515, 25924, 4332, 4820, 2308, 116, 13829, 6925, 1173, 25542, 3374, 11110, 4566, 166, 3079, 3087, 9239]}, 'Perm': {'d': 10, 'gens': [722, 369360, 16680, 4, 823080, 862680, 7, 16]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2:D_{12}', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
