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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.5044', 'ambient_counter': 5044, 'ambient_order': 256, 'ambient_tex': 'Q_8:\\OD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': True, 'core_order': 128, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '256.5044.2.b1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.b1.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': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '128.901', 'subgroup_hash': 901, 'subgroup_order': 128, 'subgroup_tex': 'C_{16}:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.5044', 'aut_centralizer_order': 4, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '256.55630', 'aut_weyl_index': 4, 'centralizer': '16.c1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.b1.a1', '4.g1.a1', '4.j1.a1', '4.l1.a1', '4.m1.a1'], 'core': '2.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2410, 9788, 7514, 7749, 2411, 9555, 7508, 7875], 'generators': [1, 2, 72], 'label': '256.5044.2.b1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.b1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.b1.a1', '4.g1.a1', '4.j1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.b1.a1', 'projective_image': '16.11', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2.b1.a1', 'subgroup_fusion': None, 'weyl_group': '16.11'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 2, 4, 2, 2, 2, 2], 'aut_gens': [[1, 2, 8], [103, 70, 40], [5, 6, 76], [103, 2, 56], [65, 2, 76], [65, 2, 8], [65, 66, 124], [5, 2, 56]], 'aut_group': '512.7532443', 'aut_hash': 3711438873667065943, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 16, 'aut_perms': [8139322201103, 355459478423, 9448353745943, 4377642796807, 5773413905280, 4377642796816, 529823318423], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 2, 1], [4, 1, 2, 2], [4, 2, 2, 2], [4, 4, 2, 1], [8, 1, 4, 2], [8, 2, 4, 1], [8, 4, 4, 1], [16, 2, 8, 2], [16, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3.C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.56082', 'autcent_hash': 56082, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\\times C_4', '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, 3], [2, 4, 2], [4, 1, 4], [4, 2, 4], [4, 4, 2], [8, 1, 8], [8, 2, 4], [8, 4, 4], [16, 2, 16], [16, 4, 8]], 'center_label': '16.5', 'center_order': 16, 'central_product': False, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 901, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 2], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 2], [8, 1, 4, 2], [8, 2, 4, 1], [8, 4, 2, 2], [16, 2, 4, 2], [16, 2, 8, 1], [16, 4, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1344, 'exponent': 16, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.5', 'frattini_quotient': '8.5', 'hash': 901, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[1, 70, 72], [69, 2, 8], [65, 2, 8]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24]], 'label': '128.901', 'linC_count': 192, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 4, 'linQ_dim': 10, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C16:D4', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 17, 'number_characteristic_subgroups': 26, 'number_conjugacy_classes': 56, 'number_divisions': 23, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 54, 'number_subgroup_classes': 81, 'number_subgroups': 116, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 11], [4, 20], [8, 32], [16, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 2, 0, 0, 0], 'outer_gens': [[1, 66, 8], [35, 2, 8], [1, 2, 76], [1, 66, 56], [1, 2, 40]], 'outer_group': '64.260', 'outer_hash': 260, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 12, 'outer_perms': [120, 39916800, 362880, 5040, 5049], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 5], [8, 3], [16, 1]], 'representations': {'PC': {'code': 5224461511639957882, 'gens': [1, 2, 4], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 981, 36, 2019, 80, 102, 124]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [110881, 148241, 111745, 1880081, 2617663, 111169, 2545799]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [1032787, 33041, 32905, 836105, 33281, 910177, 557073]}, 'Perm': {'d': 20, 'gens': [143432391521639645, 34902003964435207, 282927500369715616, 399803830646734096, 534792313166081880, 622670279737449000, 399803830646734080]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{16}:D_4', 'transitive_degree': 64, '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': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 2, 4, 2, 2, 2, 2, 2, 4], 'aut_gens': [[1, 2, 32], [1, 6, 56], [1, 146, 32], [1, 10, 32], [1, 2, 48], [1, 18, 32], [17, 2, 32], [1, 18, 104], [17, 2, 160], [217, 130, 32]], 'aut_group': None, 'aut_hash': 4195678848728926071, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 1024, 'aut_permdeg': 32, 'aut_perms': [67930080481415379227643239431530118, 191272958099747304909294678913352454, 89812999811568298974699496009965988, 141959256443155463974018099346896, 127609530976473962762791153760380848, 141959029925434366888000614673920, 127706997698017638503483447377801969, 212617258739727060359236440252304, 80410281366315652957816912415424384], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 8, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 2, 2], [4, 8, 1, 1], [8, 1, 4, 2], [8, 2, 4, 1], [8, 4, 2, 2], [8, 4, 4, 2], [8, 8, 2, 1], [16, 2, 8, 1], [16, 4, 4, 1], [16, 4, 8, 2], [16, 8, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4.C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.56082', 'autcent_hash': 56082, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\\times C_4', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '4.2', 'autcentquo_hash': 2, 'autcentquo_nilpotent': True, 'autcentquo_order': 4, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 8, 1], [4, 1, 4], [4, 2, 4], [4, 4, 4], [4, 8, 1], [8, 1, 8], [8, 2, 4], [8, 4, 12], [8, 8, 2], [16, 2, 8], [16, 4, 20], [16, 8, 4]], 'center_label': '16.5', 'center_order': 16, 'central_product': False, 'central_quotient': '16.11', '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', '2.1', '2.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 5044, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 8, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 2], [4, 4, 2, 1], [4, 8, 1, 1], [8, 1, 4, 2], [8, 2, 4, 1], [8, 4, 2, 4], [8, 4, 4, 1], [8, 8, 2, 1], [16, 2, 8, 1], [16, 4, 4, 1], [16, 4, 8, 2], [16, 8, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 5376, 'exponent': 16, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '32.3', 'frattini_quotient': '8.5', 'hash': 5044, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 4], 'inner_gens': [[1, 18, 104], [17, 2, 160], [217, 130, 32]], 'inner_hash': 11, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'C_2\\times D_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 40], [4, 4]], 'label': '256.5044', 'linC_count': 256, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 2, 'linR_count': 80, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'Q8:OD32', 'ngens': 3, 'nilpotency_class': 3, 'nilpotent': True, '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': 26, 'number_characteristic_subgroups': 55, 'number_conjugacy_classes': 76, 'number_divisions': 28, 'number_normal_subgroups': 57, 'number_subgroup_autclasses': 106, 'number_subgroup_classes': 111, 'number_subgroups': 203, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 11], [4, 36], [8, 80], [16, 128]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 48], [17, 2, 32], [1, 142, 56], [1, 130, 32], [1, 10, 32]], 'outer_group': '64.260', 'outer_hash': 260, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 12, 'outer_perms': [5040, 120, 40279696, 39916816, 362889], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 24, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 7], [8, 4], [16, 3]], 'representations': {'PC': {'code': 656282608519475986284780067054, 'gens': [1, 2, 6], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 289, 41, 66, 91, 4997, 3853, 141, 11654, 166]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [1033043, 32803, 33041, 107147, 32905, 33281, 311817, 557073]}, 'Perm': {'d': 24, 'gens': [26061380763030716123225, 3432837894059256973921, 52837894219812429436990, 82312508452279217954527, 5799938813995, 57334146464739336192000, 7202601115327, 4103708450527]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_8:\\OD_{32}', 'transitive_degree': 128, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'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}
