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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1088.13', 'ambient_counter': 13, 'ambient_order': 1088, 'ambient_tex': 'C_{68}.D_8', 'central': True, 'central_factor': False, 'centralizer_order': 1088, 'characteristic': True, 'core_order': 17, 'counter': 36, 'cyclic': True, 'direct': True, 'hall': 17, 'label': '1088.13.64.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '64.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '64.13', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 13, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 64, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4.D_8', 'simple': True, 'solvable': True, 'special_labels': ['C6'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '17.1', 'subgroup_hash': 1, 'subgroup_order': 17, 'subgroup_tex': 'C_{17}', 'supersolvable': True, 'sylow': 17}
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gps_subgroup_data • Show schema
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{'ambient': '1088.13', 'aut_centralizer_order': 256, 'aut_label': '64.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '16.1', 'aut_weyl_index': 256, 'centralizer': '1.a1.a1', 'complements': ['17.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['32.a1.a1', '32.b1.a1', '32.c1.a1'], 'contains': ['1088.a1.a1'], 'core': '64.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4183, 2287, 7519, 2032, 5896, 4250, 6210, 5767], 'generators': [64], 'label': '1088.13.64.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '64.a1.a1', 'normal_contained_in': ['32.c1.a1', '32.a1.a1', '32.b1.a1'], 'normal_contains': ['1088.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '64.a1.a1', 'projective_image': '64.13', 'quotient_action_image': '1.1', 'quotient_action_kernel': '64.13', 'quotient_action_kernel_order': 64, 'quotient_fusion': None, 'short_label': '64.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '17.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 16, 'aut_gen_orders': [16], 'aut_gens': [[1], [3]], 'aut_group': '16.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 16, 'aut_perms': [1401602636313], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [17, 1, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{16}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 16, 'autcent_group': '16.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{16}', '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], [17, 1, 16]], 'center_label': '17.1', 'center_order': 17, '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': ['17.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [17, 1, 16, 1]], 'element_repr_type': 'PC', 'elementary': 17, 'eulerian_function': 1, 'exponent': 17, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [17], 'faithful_reps': [[1, 0, 16]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '17.1', 'hash': 1, 'hyperelementary': 17, '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': 16, 'irrQ_dim': 16, 'irrR_degree': 2, 'irrep_stats': [[1, 17]], 'label': '17.1', 'linC_count': 16, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 8, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C17', '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': 17, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 17, 'order_factorization_type': 1, 'order_stats': [[1, 1], [17, 16]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [16], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '16.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 16, 'outer_perms': [1401602636313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{16}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 17, 'pgroup': 17, 'primary_abelian_invariants': [17], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [16, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -17]}, 'Lie': [{'d': 1, 'q': 17, 'gens': [22324392524313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 17, 'gens': [4931]}, 'Perm': {'d': 17, 'gens': [334764638208000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [17], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{17}', 'transitive_degree': 17, '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': '136.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 16, 'aut_gen_orders': [2, 4, 2, 2, 2, 4, 16], 'aut_gens': [[1, 2, 16], [1, 6, 16], [5, 2, 16], [1, 554, 16], [545, 2, 16], [1, 278, 560], [829, 2, 560], [1, 2, 592]], 'aut_group': None, 'aut_hash': 990274229788671091, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 48, 'aut_perms': [2079504906572258367455086767076635235144870867953847429741933, 2873662193140552394539038578941320473226435189418669456089, 4662954158045232579437967107752901189328414814980404392480359, 4548381113148498941107119515766265337308197738255342182649, 8541906688848022970032393194215144107453248272962263948255865, 9834985593457462479925762570857259646918364222120731522493993, 225730214837532741425157471095420927621932465996742078520992], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 4], [4, 4, 1, 1], [4, 8, 2, 1], [8, 4, 4, 2], [17, 1, 16, 1], [34, 1, 16, 3], [68, 2, 16, 4], [68, 4, 16, 1], [68, 8, 32, 1], [136, 4, 64, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_{16}\\times D_4^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '256.55608', 'autcent_hash': 55608, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\times C_{16}', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '16.14', 'autcentquo_hash': 14, 'autcentquo_nilpotent': True, 'autcentquo_order': 16, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 4], [4, 4, 1], [4, 8, 2], [8, 4, 8], [17, 1, 16], [34, 1, 48], [68, 2, 64], [68, 4, 16], [68, 8, 32], [136, 4, 128]], 'center_label': '68.5', 'center_order': 68, 'central_product': True, 'central_quotient': '16.3', 'commutator_count': 1, 'commutator_label': '8.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '17.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['64.13', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 4], [4, 4, 1, 1], [4, 8, 1, 2], [8, 4, 4, 2], [17, 1, 16, 1], [34, 1, 16, 3], [68, 2, 16, 4], [68, 4, 16, 1], [68, 8, 16, 2], [136, 4, 64, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 108, 'exponent': 136, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 17], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.2', 'frattini_quotient': '68.5', 'hash': 13, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2], 'inner_gens': [[1, 278, 560], [829, 2, 560], [545, 546, 16]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'C_2^2:C_4', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 136], [2, 170], [4, 17]], 'label': '1088.13', 'linC_count': 4608, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 32, 'linQ_dim': 24, 'linQ_dim_count': 8, 'linR_count': 64, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C68.D8', 'ngens': 7, 'nilpotency_class': 3, 'nilpotent': True, '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': 24, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 323, 'number_divisions': 26, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 58, 'number_subgroup_classes': 64, 'number_subgroups': 114, 'old_label': None, 'order': 1088, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 28], [8, 32], [17, 16], [34, 48], [68, 448], [136, 512]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [2, 2, 2, 2, 16], 'outer_gen_pows': [0, 4, 0, 0, 0], 'outer_gens': [[1, 6, 16], [5, 2, 16], [1, 554, 16], [545, 2, 16], [1, 2, 592]], 'outer_group': '256.55608', 'outer_hash': 55608, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 24, 'outer_perms': [25852016738884976640000, 51090942171709440000, 121645100408832000, 355687428096000, 1401602636313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 33, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 17], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5], [16, 4], [32, 4], [64, 5]], 'representations': {'PC': {'code': 1770194213734471108854705678716392527888399, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -17, 56, 3893, 36, 254, 58, 19604, 9811, 102, 124]}, 'Perm': {'d': 33, 'gens': [25747232704157566206339563372544000, 280929608716061267013396802609152000, 551487852391275464609578252050432000, 1320910334403966738432000, 823345791522074881148683026432000000, 823345791524045261729585570930688000, 334764638208000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 68], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{68}.D_8', 'transitive_degree': 1088, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 4, 4], 'aut_gens': [[1, 4, 16], [1, 36, 16], [51, 12, 24], [33, 4, 16], [3, 6, 56], [1, 46, 16], [59, 62, 16]], 'aut_group': '256.55683', 'aut_hash': 55683, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 256, 'aut_permdeg': 12, 'aut_perms': [5176, 123742096, 11520, 7, 9, 80213176], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 4], [4, 4, 1, 1], [4, 8, 2, 1], [8, 4, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times D_4^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '16.14', 'autcentquo_hash': 14, 'autcentquo_nilpotent': True, 'autcentquo_order': 16, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 4], [4, 4, 1], [4, 8, 2], [8, 4, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '16.3', 'commutator_count': 1, 'commutator_label': '8.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, '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, 3], [4, 2, 1, 4], [4, 4, 1, 1], [4, 8, 1, 2], [8, 4, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 8, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.2', 'frattini_quotient': '4.2', 'hash': 13, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 2, 2], 'inner_gens': [[1, 20, 24], [49, 4, 48], [9, 36, 16]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'C_2^2:C_4', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 10], [4, 1]], 'label': '64.13', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 6, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4.D8', 'ngens': 2, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 17, 'number_conjugacy_classes': 19, 'number_divisions': 13, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 29, 'number_subgroup_classes': 32, 'number_subgroups': 57, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 28], [8, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [2, 0, 0, 0], 'outer_gens': [[1, 46, 16], [3, 4, 56], [1, 36, 16], [33, 4, 16]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5]], 'representations': {'PC': {'code': 4976513065026618711, 'gens': [1, 3, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 12, 247, 362, 404, 50, 724, 376, 88]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [34876415, 3397304, 15977911, 28816400, 28233362, 33722916]}, 'Perm': {'d': 16, 'gens': [287408223658, 1514642981612, 2890861759793, 12593, 4296764102400, 4296764121019]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4.D_8', 'transitive_degree': 64, 'wreath_data': None, 'wreath_product': False}
