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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1344.9733', 'ambient_counter': 9733, 'ambient_order': 1344, 'ambient_tex': '(C_3\\times Q_8):D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 1, 'counter': 215, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1344.9733.672.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '672.e1', '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': 672, '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': '1344.9733', 'aut_centralizer_order': 1536, 'aut_label': '672.e1', 'aut_quo_index': None, 'aut_stab_index': 252, 'aut_weyl_group': '1.1', 'aut_weyl_index': 387072, 'centralizer': '84.g1', 'complements': None, 'conjugacy_class_count': 3, 'contained_in': ['96.e1', '224.e1', '336.d1', '336.h1', '336.k1'], 'contains': ['1344.a1'], 'core': '1344.a1', 'coset_action_label': None, 'count': 252, 'diagramx': [1312, -1, 3039, -1], 'generators': [3], 'label': '1344.9733.672.e1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '84.g1', 'old_label': '672.e1', 'projective_image': '1344.9733', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '672.e1', '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 6, 6, 6, 6, 12, 6], 'aut_gens': [[1, 2, 56, 112], [433, 466, 420, 532], [1073, 902, 392, 504], [1069, 250, 1064, 1176], [29, 930, 1064, 784], [769, 38, 1036, 840], [21, 486, 56, 1232], [433, 258, 1036, 1176], [753, 1130, 1036, 868]], 'aut_group': None, 'aut_hash': 6474015548063029219, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 387072, 'aut_permdeg': 52, 'aut_perms': [55433197966524655169460500141427370662010404711011572902768411682272, 4852055011527230721051729932063943515729315794466317443850294655530, 2588311741403813652598826529630599931151367917649252561785065715199, 34858535384740672839894892300121326589421263200466203199463489828069, 63438605217736978051780099025042447061993287764323599446386319626527, 5567007761477504387419379017321928684263907777897701212578493600919, 2609607593732605249341117657860607957690190076682836565437856058395, 38802671448339347231477285305077223197942382737800973770034019037933], 'aut_phi_ratio': 1008.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 28, 3, 1], [2, 84, 3, 1], [3, 2, 1, 1], [4, 2, 6, 1], [4, 6, 2, 1], [4, 12, 3, 1], [4, 14, 2, 1], [4, 42, 2, 1], [6, 2, 1, 3], [6, 28, 6, 1], [7, 2, 3, 1], [12, 4, 6, 1], [12, 14, 4, 1], [14, 2, 3, 3], [21, 4, 3, 1], [28, 4, 18, 1], [28, 6, 12, 1], [28, 12, 18, 1], [42, 4, 3, 3], [84, 8, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.(C_3\\times C_{21}).C_6.C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '1512.792', 'autcentquo_hash': 792, 'autcentquo_nilpotent': False, 'autcentquo_order': 1512, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 28, 3], [2, 84, 3], [3, 2, 1], [4, 2, 6], [4, 6, 2], [4, 12, 3], [4, 14, 2], [4, 42, 2], [6, 2, 3], [6, 28, 6], [7, 2, 3], [12, 4, 6], [12, 14, 4], [14, 2, 9], [21, 4, 3], [28, 4, 18], [28, 6, 12], [28, 12, 18], [42, 4, 9], [84, 8, 18]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '336.219', 'commutator_count': 1, 'commutator_label': '84.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 9733, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 28, 1, 3], [2, 84, 1, 3], [3, 2, 1, 1], [4, 2, 1, 6], [4, 6, 1, 2], [4, 12, 1, 3], [4, 14, 2, 1], [4, 42, 2, 1], [6, 2, 1, 3], [6, 28, 2, 3], [7, 2, 3, 1], [12, 4, 1, 6], [12, 14, 4, 1], [14, 2, 3, 3], [21, 4, 3, 1], [28, 4, 3, 6], [28, 6, 6, 2], [28, 12, 6, 3], [42, 4, 3, 3], [84, 8, 3, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2489760, 'exponent': 84, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '336.219', 'hash': 9733, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 14, 2, 6], 'inner_gens': [[1, 54, 728, 112], [5, 2, 56, 560], [673, 2, 56, 784], [1, 898, 728, 112]], 'inner_hash': 219, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': True, 'inner_tex': 'D_6\\times D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 72], [4, 41], [8, 6]], 'label': '1344.9733', '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': '(C3*Q8):D28', '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, 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': 30, 'number_characteristic_subgroups': 36, 'number_conjugacy_classes': 135, 'number_divisions': 62, 'number_normal_subgroups': 164, 'number_subgroup_autclasses': 216, 'number_subgroup_classes': 624, 'number_subgroups': 5908, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 339], [3, 2], [4, 172], [6, 174], [7, 6], [12, 80], [14, 18], [21, 12], [28, 360], [42, 36], [84, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 2, 3, 3, 2, 2], 'outer_gen_pows': [56, 0, 0, 0, 0, 337, 0, 0, 0], 'outer_gens': [[729, 30, 56, 168], [1, 2, 56, 560], [673, 30, 56, 112], [673, 2, 56, 112], [673, 674, 56, 560], [1, 6, 56, 112], [1065, 2, 392, 504], [1, 2, 84, 112], [1, 2, 56, 588]], 'outer_group': '1152.157869', 'outer_hash': 1697699593468017772, 'outer_nilpotent': False, 'outer_order': 1152, 'outer_permdeg': 15, 'outer_perms': [5165, 87657297960, 186810624000, 11520, 274467916800, 3991680, 3, 7, 16], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times C_2^3\\times S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 7], [6, 8], [8, 2], [12, 14], [24, 3]], 'representations': {'PC': {'code': 8020324940447728095235799053576379471889648444714241873, 'gens': [1, 2, 5, 6], 'pres': [8, -2, -2, -2, -7, -2, -2, -2, -3, 865, 41, 1250, 66, 1539, 29124, 516, 13453, 709, 141, 31374, 166, 28687]}, 'Perm': {'d': 22, 'gens': [2439325304673764405, 6227020809, 49114800, 91578240, 135732240, 16, 93405312000, 55969571858264064000]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_3\\times Q_8):D_{28}', 'transitive_degree': 672, 'wreath_data': None, 'wreath_product': False}
