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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1344.2349', 'ambient_counter': 2349, 'ambient_order': 1344, 'ambient_tex': 'C_{24}:D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 24, 'counter': 115, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.2349.28.j1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '28.j1.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': 28, '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': '48.9', 'subgroup_hash': 9, 'subgroup_order': 48, 'subgroup_tex': 'C_6:C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.2349', 'aut_centralizer_order': 96, 'aut_label': '28.j1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '48.51', 'aut_weyl_index': 672, 'centralizer': '168.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.j1.a1', '14.d1.a1', '14.e1.a1', '14.g1.a1'], 'contains': ['56.a1.a1', '56.n1.a1', '84.j1.a1'], 'core': '56.a1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [7574, -1, 7291, -1, 7988, -1, 8875, -1], 'generators': [59, 448, 28, 336, 672], 'label': '1344.2349.28.j1.a1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.j1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '28.j1.a1', 'projective_image': '168.50', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '28.j1.a1', 'subgroup_fusion': None, 'weyl_group': '24.14'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 3], 'aut_gens': [[1, 8], [1, 40], [3, 8], [3, 12], [27, 8], [5, 8], [17, 8]], 'aut_group': '96.209', 'aut_hash': 209, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 9, 'aut_perms': [5160, 120, 1, 136, 7, 45360], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 2, 1, 1], [4, 1, 2, 2], [6, 2, 1, 1], [6, 2, 2, 1], [8, 3, 8, 1], [12, 2, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.11', 'autcent_hash': 11, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 1], [4, 1, 4], [6, 2, 3], [8, 3, 8], [12, 2, 4]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['24.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 1, 2, 2], [6, 2, 1, 3], [8, 3, 4, 2], [12, 2, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 24, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '12.4', 'hash': 9, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 40], [17, 8]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 8]], 'label': '48.9', 'linC_count': 48, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 4, 'linQ_dim': 5, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6:C8', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 24, 'number_divisions': 14, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 22, 'number_subgroups': 28, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [8, 24], [12, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[25, 8], [1, 12], [7, 8], [5, 8]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [6, 289, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 4]], 'representations': {'PC': {'code': 2900468177249, 'gens': [1, 4], 'pres': [5, -2, -2, -2, -2, -3, 10, 26, 803, 58, 804]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [706304316034, 140954895519]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [106892, 1692709, 518956, 1449557, 1565004]}, 'Perm': {'d': 13, 'gens': [39923583, 375196, 18550, 12316, 518918400]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6:C_8', 'transitive_degree': 48, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [6, 6, 2, 6, 6, 14, 6, 42, 6, 2], 'aut_gens': [[1, 2, 56], [713, 1158, 56], [33, 918, 392], [53, 278, 616], [25, 10, 1316], [697, 690, 616], [53, 674, 952], [29, 722, 280], [21, 254, 1064], [45, 722, 308], [41, 1146, 308]], 'aut_group': None, 'aut_hash': 9164245504634031933, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 32256, 'aut_permdeg': 48, 'aut_perms': [6946112789889413734956109168394882903331451377873078665611814, 11446615763050536010691426490775854677620519012195743923974226, 1317295907598383288067702861823508896426518266942928681789959, 11696046400915500340261360942456522413893662682071466677966838, 6407228161531974283113051153456353965730634561276535571732769, 3644658638809762063969039859642290258985456620379099296775616, 1542843054279244500984106874960177051366012830065054189036476, 1539437329285131042329463614548765301452109764383449031913665, 2588239466429093072822342760505780231377307814755865431418258, 1106837839833238597796681923886865079517619141836817650327687], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 14, 2, 1], [2, 84, 1, 1], [3, 2, 1, 1], [4, 1, 2, 2], [4, 12, 1, 2], [4, 14, 2, 1], [4, 84, 1, 1], [6, 2, 1, 3], [6, 14, 4, 1], [7, 2, 3, 1], [8, 2, 4, 1], [8, 12, 2, 1], [8, 14, 4, 1], [8, 84, 2, 1], [12, 2, 2, 2], [12, 14, 4, 1], [14, 2, 3, 3], [21, 4, 3, 1], [24, 2, 8, 1], [24, 14, 8, 1], [28, 2, 6, 2], [28, 12, 6, 2], [42, 4, 3, 3], [56, 4, 12, 1], [56, 12, 12, 1], [84, 4, 6, 2], [168, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^7\\times S_3\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 14, 2], [2, 84, 1], [3, 2, 1], [4, 1, 4], [4, 12, 2], [4, 14, 2], [4, 84, 1], [6, 2, 3], [6, 14, 4], [7, 2, 3], [8, 2, 4], [8, 12, 2], [8, 14, 4], [8, 84, 2], [12, 2, 4], [12, 14, 4], [14, 2, 9], [21, 4, 3], [24, 2, 8], [24, 14, 8], [28, 2, 12], [28, 12, 12], [42, 4, 9], [56, 4, 12], [56, 12, 12], [84, 4, 12], [168, 4, 24]], 'center_label': '8.2', 'center_order': 8, 'central_product': False, 'central_quotient': '168.50', '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': 2349, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 14, 1, 2], [2, 84, 1, 1], [3, 2, 1, 1], [4, 1, 2, 2], [4, 12, 1, 2], [4, 14, 2, 1], [4, 84, 1, 1], [6, 2, 1, 3], [6, 14, 2, 2], [7, 2, 3, 1], [8, 2, 2, 2], [8, 12, 2, 1], [8, 14, 4, 1], [8, 84, 2, 1], [12, 2, 2, 2], [12, 14, 4, 1], [14, 2, 3, 3], [21, 4, 3, 1], [24, 2, 4, 2], [24, 14, 8, 1], [28, 2, 6, 2], [28, 12, 6, 2], [42, 4, 3, 3], [56, 4, 6, 2], [56, 12, 12, 1], [84, 4, 6, 2], [168, 4, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 21504, 'exponent': 168, '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': '8.2', 'frattini_quotient': '168.50', 'hash': 2349, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 14, 6], 'inner_gens': [[1, 54, 56], [5, 2, 280], [1, 1122, 56]], 'inner_hash': 50, 'inner_nilpotent': False, 'inner_order': 168, 'inner_split': True, 'inner_tex': 'S_3\\times D_{14}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 92], [4, 60]], 'label': '1344.2349', 'linC_count': 384, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 80, 'linQ_dim': 14, 'linQ_dim_count': 80, 'linR_count': 96, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C24: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': 43, 'number_characteristic_subgroups': 70, 'number_conjugacy_classes': 168, 'number_divisions': 49, 'number_normal_subgroups': 82, 'number_subgroup_autclasses': 224, 'number_subgroup_classes': 248, 'number_subgroups': 1844, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 115], [3, 2], [4, 140], [6, 62], [7, 6], [8, 256], [12, 64], [14, 18], [21, 12], [24, 128], [28, 168], [42, 36], [56, 192], [84, 48], [168, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 952], [673, 2, 56], [1, 2, 728], [1, 30, 56], [1, 2, 84], [1, 38, 1064]], 'outer_group': '192.1543', 'outer_hash': 1543, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 15, 'outer_perms': [3628800, 87178291200, 479001600, 24, 40320, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 7], [6, 4], [8, 4], [12, 8], [16, 1], [24, 5], [48, 2]], 'representations': {'PC': {'code': 1493818518567778240135077704069728784005063348834129, 'gens': [1, 2, 5], 'pres': [8, -2, -2, -2, -7, -2, -2, -2, -3, 865, 41, 1250, 66, 1539, 5612, 116, 13453, 141, 31374, 166, 28687]}, 'GLZN': {'d': 2, 'p': 105, 'gens': [74088064, 80963931, 32035032, 9261008, 1159201, 120858296, 50481271, 49777918]}, 'Perm': {'d': 22, 'gens': [2439325317609018744, 93884399305, 8443704, 12506424, 186810624000, 3719544, 3, 55969571858264064000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{24}:D_{28}', 'transitive_degree': 672, 'wreath_data': None, 'wreath_product': False}
