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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '192.1315', 'ambient_counter': 1315, 'ambient_order': 192, 'ambient_tex': 'D_8:D_6', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 8, 'counter': 51, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '192.1315.12.i1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '12.i1', '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': 12, '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': '16.13', 'subgroup_hash': 13, 'subgroup_order': 16, 'subgroup_tex': 'D_4:C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '192.1315', 'aut_centralizer_order': 16, 'aut_label': '12.i1', 'aut_quo_index': None, 'aut_stab_index': 24, 'aut_weyl_group': '16.11', 'aut_weyl_index': 384, 'centralizer': '24.f1', 'complements': None, 'conjugacy_class_count': 8, 'contained_in': ['4.f1', '6.b1', '6.f1'], 'contains': ['24.b1', '24.h1', '24.i1', '24.k1', '24.m1'], 'core': '24.b1', 'coset_action_label': None, 'count': 24, 'diagramx': None, 'generators': [13, 2, 144], 'label': '192.1315.12.i1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.f1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '12.i1', 'projective_image': '96.209', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.i1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 3, 2, 2], 'aut_gens': [[30, 377, 501], [120, 105, 65], [120, 377, 129], [65, 90, 120], [120, 253, 129], [120, 377, 501]], 'aut_group': '48.48', 'aut_hash': 48, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 6, 'aut_perms': [415, 450, 233, 403, 444], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [4, 1, 2, 1], [4, 2, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', '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], [2, 2, 3], [4, 1, 2], [4, 2, 3]], 'center_label': '4.1', 'center_order': 4, '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', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [4, 1, 2, 1], [4, 2, 1, 3]], 'element_repr_type': 'GLFp', 'elementary': 2, 'eulerian_function': 28, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 13, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[30, 253, 129], [120, 377, 501], [120, 377, 501]], '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': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.13', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4:C2', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 10, 'number_divisions': 9, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 20, 'number_subgroups': 23, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [377, 126], 'outer_gens': [[65, 377, 501], [65, 90, 30]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 1]], 'representations': {'PC': {'code': 8716, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 81, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [25793872, 22766861, 817574]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [30, 377, 501]}, 'Perm': {'d': 8, 'gens': [35278, 23616, 16582, 5167]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4:C_2', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 24, 'aut_gen_orders': [4, 8, 12, 8, 4, 8, 24], 'aut_gens': [[1, 2, 4, 24], [97, 10, 189, 169], [1, 115, 188, 24], [97, 114, 53, 169], [1, 10, 45, 24], [97, 10, 165, 169], [1, 98, 44, 25], [97, 11, 124, 120]], 'aut_group': None, 'aut_hash': 6013115784330334307, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6144, 'aut_permdeg': 28, 'aut_perms': [174802018960489145955200534829, 58621991895705691680174235744, 7375088195761032245969327146, 294575228134472589246342373807, 196579241896205691726436095789, 301813997397208227120894047286, 285721830975677211930650931420], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 4, 4, 1], [2, 6, 2, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 3, 4, 1], [4, 12, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 8, 4, 1], [8, 2, 4, 1], [8, 6, 4, 1], [12, 4, 1, 2], [24, 4, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2.C_2^6.C_2^4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5.C_2^2', 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': 48, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 4], [2, 6, 2], [3, 2, 1], [4, 2, 2], [4, 3, 4], [4, 12, 4], [6, 2, 3], [6, 8, 4], [8, 2, 4], [8, 6, 4], [12, 4, 2], [24, 4, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '48.38', 'commutator_count': 1, 'commutator_label': '12.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1315, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['96.119', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 4], [2, 6, 1, 2], [3, 2, 1, 1], [4, 2, 1, 2], [4, 3, 2, 2], [4, 12, 1, 4], [6, 2, 1, 3], [6, 8, 1, 4], [8, 2, 2, 2], [8, 6, 2, 2], [12, 4, 1, 2], [24, 4, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 65520, 'exponent': 24, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '48.51', 'hash': 1315, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [1, 2, 6, 4], 'inner_gens': [[1, 2, 4, 24], [1, 2, 116, 24], [1, 106, 4, 168], [1, 2, 52, 24]], 'inner_hash': 38, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': None, 'inner_tex': 'S_3\\times D_4', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 20], [4, 6]], 'label': '192.1315', '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': 'D8:D6', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 23, 'number_conjugacy_classes': 42, 'number_divisions': 34, 'number_normal_subgroups': 103, 'number_subgroup_autclasses': 98, 'number_subgroup_classes': 266, 'number_subgroups': 664, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 31], [3, 2], [4, 64], [6, 38], [8, 32], [12, 8], [24, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 2, 2, 2], 'outer_gen_pows': [0, 48, 48, 24, 24], 'outer_gens': [[1, 99, 101, 121], [1, 2, 29, 25], [1, 99, 52, 121], [1, 2, 28, 24], [97, 98, 28, 24]], 'outer_group': '128.1135', 'outer_hash': 1135, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 32, 'outer_perms': [92902115138128388376567419245092479, 84404905712009632381082545681488592, 119529273469064081153232440528339887, 25207954330807493600598406934260432, 16976374818804423968495230841917661], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 2], [8, 4]], 'representations': {'PC': {'code': 874287809008755684153864, 'gens': [1, 2, 3, 5], 'pres': [7, -2, -2, -2, -3, -2, -2, -2, 1227, 58, 234, 1488, 102, 1531, 124]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [111169, 111361, 111745, 110767, 4536559, 5197871, 4535177]}, 'Perm': {'d': 21, 'gens': [525747976439212807, 1, 2480629983870246481, 5477347138783050720, 8081060682102912720, 10260160285904161920, 30]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_8:D_6', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
