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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1088.154', 'ambient_counter': 154, 'ambient_order': 1088, 'ambient_tex': 'C_{136}.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 136, 'characteristic': False, 'core_order': 8, 'counter': 67, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1088.154.68.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '68.g1.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': 68, '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': '16.6', 'subgroup_hash': 6, 'subgroup_order': 16, 'subgroup_tex': '\\OD_{16}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1088.154', 'aut_centralizer_order': 256, 'aut_label': '68.g1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '8.5', 'aut_weyl_index': 512, 'centralizer': '8.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.g1.a1', '34.b1.a1'], 'contains': ['136.d1.a1', '136.h1.a1', '136.j1.a1'], 'core': '136.d1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [2837, -1, 1884, -1, 7125, -1, 7053, -1], 'generators': [1, 140], 'label': '1088.154.68.g1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '34.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.a1', 'old_label': '68.g1.a1', 'projective_image': '544.183', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '68.g1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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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], 'aut_gens': [[1, 2], [9, 15], [9, 2], [9, 14], [1, 10]], 'aut_group': '16.11', 'aut_hash': 11, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 6, 'aut_perms': [126, 55, 289, 288], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [8, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [4, 1, 2], [4, 2, 1], [8, 2, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, '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': 6, '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, 1], [4, 1, 2, 1], [4, 2, 1, 1], [8, 2, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 8, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 2]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '4.2', 'hash': 6, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 10], [9, 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': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.6', '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': True, 'monomial': True, 'name': 'OD16', 'ngens': 2, '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': 6, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 10, 'number_divisions': 7, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 10, 'number_subgroups': 11, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4], [8, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 6], [1, 3]], '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': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 1]], 'representations': {'PC': {'code': 140302, 'gens': [1, 2], 'pres': [4, -2, 2, -2, -2, 81, 21, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [41537776, 817574]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [377, 35, 501]}, 'Perm': {'d': 8, 'gens': [40176, 7, 16582, 5167]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\OD_{16}', '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': '136.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 16, 'aut_gen_orders': [8, 2, 2, 2, 4, 4, 16], 'aut_gens': [[1, 2, 8], [1, 138, 8], [1, 2, 824], [1, 2, 552], [1, 6, 8], [549, 2, 824], [1, 274, 8], [1, 2, 840]], 'aut_group': None, 'aut_hash': 8154153396567325628, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 36, 'aut_perms': [8829913627211761792558109532929197271160, 599148915698358366629039754365003138046, 15511210043395149206323326, 7517438241352511418593062398211614359328, 599280759502468219771307341968081640446, 5231334709789732267624229383678248962040, 248299738964940088005284074813290481138864], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 8, 2, 1], [8, 2, 2, 1], [8, 4, 1, 1], [8, 4, 2, 1], [8, 8, 2, 1], [17, 1, 16, 1], [34, 1, 16, 1], [34, 2, 16, 1], [34, 4, 16, 1], [68, 2, 16, 2], [68, 4, 16, 1], [68, 8, 32, 1], [136, 2, 32, 1], [136, 4, 16, 1], [136, 4, 32, 1], [136, 8, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{16}\\times C_4.C_2^5.C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '128.2136', 'autcent_hash': 2136, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_{16}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '32.46', 'autcentquo_hash': 46, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [4, 2, 2], [4, 4, 1], [4, 8, 2], [8, 2, 2], [8, 4, 3], [8, 8, 2], [17, 1, 16], [34, 1, 16], [34, 2, 16], [34, 4, 16], [68, 2, 32], [68, 4, 16], [68, 8, 32], [136, 2, 32], [136, 4, 48], [136, 8, 32]], 'center_label': '34.2', 'center_order': 34, 'central_product': True, 'central_quotient': '32.28', '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': 154, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['64.154', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 8, 1, 2], [8, 2, 2, 1], [8, 4, 1, 1], [8, 4, 2, 1], [8, 8, 1, 2], [17, 1, 16, 1], [34, 1, 16, 1], [34, 2, 16, 1], [34, 4, 16, 1], [68, 2, 16, 2], [68, 4, 16, 1], [68, 8, 16, 2], [136, 2, 32, 1], [136, 4, 16, 1], [136, 4, 32, 1], [136, 8, 16, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 103152, 'exponent': 136, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 17], 'faithful_reps': [[4, 0, 32]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '136.15', 'hash': 154, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 4], 'inner_gens': [[1, 6, 8], [549, 2, 824], [1, 274, 8]], 'inner_hash': 28, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'C_4:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 128, 'irrQ_dim': 128, 'irrR_degree': 8, 'irrep_stats': [[1, 136], [2, 102], [4, 34]], 'label': '1088.154', 'linC_count': 32, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 8, 'linQ_dim': 32, 'linQ_dim_count': 8, 'linR_count': 16, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C136.D4', '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': 272, 'number_divisions': 28, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 78, 'number_subgroup_classes': 100, 'number_subgroups': 162, 'old_label': None, 'order': 1088, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 24], [8, 32], [17, 16], [34, 112], [68, 384], [136, 512]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [2, 2, 2, 16], 'outer_gen_pows': [8, 0, 0, 0], 'outer_gens': [[1, 138, 8], [1, 2, 824], [1, 2, 552], [1, 2, 840]], 'outer_group': '128.2136', 'outer_hash': 2136, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 22, 'outer_perms': [51090942171709440000, 121645100408832000, 355687428096000, 1401602636313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 49, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 17], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 1], [8, 1], [16, 8], [32, 4], [64, 1], [128, 1]], 'representations': {'PC': {'code': 1772662430702322147541926687519647966691343, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -2, -2, -2, -2, -17, 85, 36, 11510, 2872, 11546, 80, 9811, 102, 124]}, 'Perm': {'d': 49, 'gens': [215270041651238110379454230844937855408378102903493099520000000, 139184311744701524200673277893036046963437588005359841124352000, 50712644313715021082405462925974397276814001301185229471744000, 25350585729034648763476669016095062210546921700937795026944000, 37758998699411724737649737926307286770937154910095410315264000, 12419420874373282104691941482291922192923886086820530577408000, 334764638208000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 34], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{136}.D_4', 'transitive_degree': 544, 'wreath_data': None, 'wreath_product': False}
