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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '640.21479', 'ambient_counter': 21479, 'ambient_order': 640, 'ambient_tex': 'D_{10}.C_2^5', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 10, 'counter': 74, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '640.21479.64.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '64.c1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '64.194', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 194, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 64, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3.D_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '10.1', 'subgroup_hash': 1, 'subgroup_order': 10, 'subgroup_tex': 'D_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '640.21479', 'aut_centralizer_order': 49152, 'aut_label': '64.c1', 'aut_quo_index': 1, 'aut_stab_index': 14, 'aut_weyl_group': '20.3', 'aut_weyl_index': 688128, 'centralizer': '20.a1', 'complements': ['10.a1'], 'conjugacy_class_count': 14, 'contained_in': ['32.b1', '32.f1', '32.g1', '32.h1'], 'contains': ['128.a1', '320.c1'], 'core': '64.c1', 'coset_action_label': None, 'count': 14, 'diagramx': [5047, 2906, 4540, 3319], 'generators': [96029, 65281], 'label': '640.21479.64.c1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '64.c1', 'normal_contained_in': ['32.b1', '32.f1', '32.g1', '32.h1'], 'normal_contains': ['128.a1'], 'normalizer': '1.a1', 'old_label': '64.c1', 'projective_image': '640.21479', 'quotient_action_image': '4.1', 'quotient_action_kernel': '16.10', 'quotient_action_kernel_order': 16, 'quotient_fusion': None, 'short_label': '64.c1', 'subgroup_fusion': None, 'weyl_group': '20.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 5], 'aut_gens': [[1, 2], [3, 6], [5, 2]], 'aut_group': '20.3', 'aut_hash': 3, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20, 'aut_permdeg': 5, 'aut_perms': [17, 89], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_5', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [5, 2, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 10, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 1, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '10.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 8], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 2]], 'label': '10.1', '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': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D5', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 8, 'old_label': None, 'order': 10, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 5], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [1], 'outer_gens': [[1, 6]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 1]], 'representations': {'PC': {'code': 227, 'gens': [1, 2], 'pres': [2, -2, -5, 33]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [23707240, 26329372]}, 'Lie': [{'d': 2, 'q': 4, 'gens': [81, 94], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 129]}, 'Perm': {'d': 5, 'gens': [23, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_5', 'transitive_degree': 5, '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}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [4, 4, 8], 'aut_gens': [[28815, 30735, 14465, 36929], [61583, 30863, 14465, 28751], [30863, 28815, 53263, 4161], [61583, 38913, 12353, 6337]], 'aut_group': '49152.bbg', 'aut_hash': 6677038305809647454, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 49152, 'aut_permdeg': 32, 'aut_perms': [54733909724066174318939810536271038, 133328814881071808031695848748382700, 89351392007089843615340833574597896], 'aut_phi_ratio': 1536.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 12, 1], [4, 2, 8, 1], [4, 2, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5.C_2^6:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': None, 'autcent_hash': 4923965932110249069, 'autcent_nilpotent': False, 'autcent_order': 24576, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^8.C_2^4.S_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, 15], [4, 2, 24]], 'center_label': '16.14', 'center_order': 16, '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', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 194, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.4', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [4, 2, 1, 8], [4, 2, 2, 8]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 105, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '16.14', 'hash': 194, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 1, 2, 2], 'inner_gens': [[28815, 30735, 14465, 36929], [28815, 30735, 14465, 36929], [28815, 30735, 14465, 37057], [28815, 30735, 14337, 36929]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8]], 'label': '64.194', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'C2^3.D4', 'ngens': 4, 'nilpotency_class': 2, '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': 7, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 40, 'number_divisions': 32, 'number_normal_subgroups': 169, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 209, 'number_subgroups': 249, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15], [4, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [6, 12, 4], 'outer_gen_pows': [36929, 4097, 4097], 'outer_gens': [[6273, 61455, 14529, 63695], [6145, 28687, 20559, 28879], [63503, 28687, 12417, 63695]], 'outer_group': None, 'outer_hash': 7369186840958827619, 'outer_nilpotent': False, 'outer_order': 12288, 'outer_permdeg': 20, 'outer_perms': [1914777894302795528, 528030223163185819, 1169523366875584682], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4.C_2^6.D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16]], 'representations': {'PC': {'code': 4366303252, 'gens': [1, 2, 3, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 50, 376, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101750005091126, 25038659807093174, 125101736067519466, 24992906222183252]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [224019788, 1083311105, 27132992, 1087570945, 2338096842, 2203649218]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [39041, 12289, 63503, 4161, 4225, 6273]}, 'Perm': {'d': 12, 'gens': [7626367, 43920016, 11543, 87102727, 87091200, 11520]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3.D_4', 'transitive_degree': 64, 'wreath_data': None, 'wreath_product': False}
