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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '640.21479', 'ambient_counter': 21479, 'ambient_order': 640, 'ambient_tex': 'D_{10}.C_2^5', 'central': False, 'central_factor': False, 'centralizer_order': 128, 'characteristic': False, 'core_order': 1, 'counter': 103, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '640.21479.320.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '320.c1', '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': 320, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '640.21479', 'aut_centralizer_order': 196608, 'aut_label': '320.c1', 'aut_quo_index': None, 'aut_stab_index': 70, 'aut_weyl_group': '1.1', 'aut_weyl_index': 13762560, 'centralizer': '5.a1', 'complements': None, 'conjugacy_class_count': 14, 'contained_in': ['64.c1', '160.d1', '160.g1', '160.h1', '160.i1'], 'contains': ['640.a1'], 'core': '640.a1', 'coset_action_label': None, 'count': 70, 'diagramx': [4920, -1, 4650, -1], 'generators': [96029], 'label': '640.21479.320.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '64.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1', 'old_label': '320.c1', 'projective_image': '640.21479', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '320.c1', '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': '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}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '64.260', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 840, 'aut_gen_orders': [4, 6, 140], 'aut_gens': [[1121239, 65521, 1856029, 608029, 1248019], [865051, 1376561, 1856829, 704031, 1248019], [449049, 1857549, 608029, 1248819, 1856829], [864651, 704751, 1248819, 1376801, 704031]], 'aut_group': None, 'aut_hash': 8168337809177570170, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 13762560, 'aut_permdeg': 160, 'aut_perms': [454917196633867308392420738526385169513529639145412635318157268779069783083170409004256411611134537690954080563138630918501277929348824491094727919976730292027279599667153931120816478001797183087796138950429289652948471801044130788625807512990352357358888493224676171021443092129777581, 335567588479576165392314537757606382880481446453431554877461713818080993590246293682125246149533027806751469940555729035751038542511195475114967962330239407955350077202645278228499598661591801590498258613298302607599691327883226299025574982098221946006697671512225142115696366796095216, 73204143791857422973282119386139821250724094373441592560780556727607927318417817102936116276651893436742920755911348023152372038852475458351992138544967991976393103637942649893510081504185747404942238200487189943960031216647259854024466734635630911534616761096224463214594013650848916], 'aut_phi_ratio': 53760.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [2, 5, 1, 2], [2, 5, 14, 1], [4, 2, 8, 1], [4, 10, 8, 1], [4, 10, 16, 2], [5, 4, 1, 1], [10, 4, 1, 1], [10, 4, 14, 1], [20, 4, 16, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2^5\\times D_5).C_2^6.C_2^2.\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': None, 'autcent_hash': 6173500181054804164, 'autcent_nilpotent': False, 'autcent_order': 344064, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^5.C_2^6.\\PSL(2,7)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 15], [2, 5, 16], [4, 2, 8], [4, 10, 40], [5, 4, 1], [10, 4, 15], [20, 4, 16]], 'center_label': '16.14', 'center_order': 16, 'central_product': True, 'central_quotient': '40.12', 'commutator_count': 1, 'commutator_label': '10.2', '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', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 21479, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['80.31', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [2, 5, 1, 16], [4, 2, 1, 8], [4, 10, 1, 8], [4, 10, 2, 16], [5, 4, 1, 1], [10, 4, 1, 15], [20, 4, 2, 8]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 2321280, 'exponent': 20, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '320.1638', 'hash': 21479, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 10, 1, 1, 1], 'inner_gens': [[1121239, 65361, 1856029, 608029, 1248019], [1121079, 65521, 1856029, 608029, 1248019], [1121239, 65521, 1856029, 608029, 1248019], [1121239, 65521, 1856029, 608029, 1248019], [1121239, 65521, 1856029, 608029, 1248019]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': False, 'inner_tex': 'C_2\\times F_5', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 64], [2, 16], [4, 32]], 'label': '640.21479', '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': 'D10.C2^5', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 112, 'number_divisions': 88, 'number_normal_subgroups': 1106, 'number_subgroup_autclasses': 106, 'number_subgroup_classes': 2488, 'number_subgroups': 8528, 'old_label': None, 'order': 640, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 95], [4, 416], [5, 4], [10, 60], [20, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [12, 12], 'outer_gen_pows': [64001, 64001], 'outer_gens': [[2368839, 704751, 608829, 1376801, 704831], [1504021, 1377521, 1856829, 2496019, 1376001]], 'outer_group': None, 'outer_hash': 8833623613039832737, 'outer_nilpotent': False, 'outer_order': 344064, 'outer_permdeg': 32, 'outer_perms': [109877263647741730747748397736691626, 118950470453212818104671736778912790], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^8.C_2^3.\\PSL(2,7)', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 4], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 32], [4, 16], [8, 8]], 'representations': {'PC': {'code': '1524352326526132982645303675627672240320', 'gens': [1, 3, 6, 7, 8], 'pres': [8, -2, -2, -2, -2, -5, -2, -2, -2, 16, 290, 442, 66, 771, 1163, 91, 1924, 1292]}, 'GLZN': {'d': 2, 'p': 40, 'gens': [1248019, 1760009, 192021, 64401, 64801, 1856029, 576001, 64321]}, 'Perm': {'d': 15, 'gens': [6746307247, 10207, 16, 362903, 379576, 16680, 13412044800, 94363315200]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{10}.C_2^5', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
