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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '2688.eh', 'ambient_counter': 112, 'ambient_order': 2688, 'ambient_tex': 'C_8:C_2^3\\times F_7', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 1, 'counter': 835, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2688.eh.224.bb1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '224.bb1', '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': 224, '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': '12.5', 'subgroup_hash': 5, 'subgroup_order': 12, 'subgroup_tex': 'C_2\\times C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2688.eh', 'aut_centralizer_order': None, 'aut_label': '224.bb1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '28.t1', 'complements': None, 'conjugacy_class_count': 8, 'contained_in': ['32.bb1', '112.f1', '112.bi1'], 'contains': ['448.e1', '448.f1', '448.i1', '672.bb1'], 'core': '2688.a1', 'coset_action_label': None, 'count': 224, 'diagramx': None, 'generators': [25, 348, 8], 'label': '2688.eh.224.bb1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.bd1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '28.t1', 'old_label': '224.bb1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '224.bb1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [6, 9], [6, 11]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 31], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [6, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.4', 'autcent_hash': 4, 'autcent_nilpotent': False, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_6', '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, 3], [3, 1, 2], [6, 1, 6]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [6, 1, 2, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '12.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 12]], 'label': '12.5', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 6, 'linQ_dim': 3, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C6', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 12, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 9], [6, 11]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4]], 'representations': {'PC': {'code': 273, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16507]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 2064]}, 'Perm': {'d': 7, 'gens': [720, 24, 4]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_6', 'transitive_degree': 12, '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': '96.231', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [12, 6, 12, 12, 2, 12, 4, 6, 2], 'aut_gens': [[1, 2, 4, 24, 48], [217, 2, 316, 24, 432], [2113, 26, 1828, 24, 2448], [1, 1346, 2046, 24, 2162], [217, 1346, 1662, 1368, 2474], [1729, 1370, 1948, 24, 1992], [2497, 26, 2406, 24, 1610], [577, 1370, 222, 24, 626], [577, 2, 220, 24, 432], [1, 26, 1372, 24, 624]], 'aut_group': '7776.kp', 'aut_hash': 7159060724362963568, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 172032, 'aut_permdeg': 52, 'aut_perms': [6095584968410067813344643876559331426998671017995254631650866223, 4837058169795934838829270453870175509791934233174631548708752562192, 29963338402035390165226483851911142360154529912961248004032376597112, 2618386823254431885532957247556309689980873390041711345580261397945, 7425967319106349059186494202569144361928019330549422949697449850951, 17523857646084445613799578450586619559550632761134841887047273773288, 29805805956243931767312265637092647016574051641075500185893630260719, 29818511634488491557695312041386118175440980271635698514818901153, 56691396498727149844073616400504539850850412049740622612507539137519], 'aut_phi_ratio': 224.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 4, 2, 1], [2, 4, 4, 1], [2, 7, 4, 1], [2, 14, 2, 1], [2, 28, 2, 1], [2, 28, 4, 1], [3, 7, 1, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 2, 1], [4, 14, 2, 2], [4, 28, 2, 1], [6, 7, 1, 2], [6, 7, 2, 2], [6, 7, 4, 2], [6, 14, 2, 4], [6, 28, 2, 4], [6, 28, 4, 4], [7, 6, 1, 1], [8, 4, 4, 1], [8, 28, 4, 1], [12, 14, 1, 4], [12, 14, 2, 6], [12, 28, 2, 4], [14, 6, 1, 1], [14, 6, 2, 1], [14, 12, 2, 1], [14, 24, 2, 1], [14, 24, 4, 1], [24, 28, 4, 4], [28, 12, 1, 2], [28, 12, 2, 1], [28, 24, 2, 1], [56, 24, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^4:(D_4\\times D_6)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '512.6249567', 'autcent_hash': 6452065007057432321, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '336.216', 'autcentquo_hash': 216, 'autcentquo_nilpotent': False, 'autcentquo_order': 336, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 6], [2, 7, 4], [2, 14, 2], [2, 28, 6], [3, 7, 2], [4, 2, 4], [4, 4, 2], [4, 14, 4], [4, 28, 2], [6, 7, 14], [6, 14, 8], [6, 28, 24], [7, 6, 1], [8, 4, 4], [8, 28, 4], [12, 14, 16], [12, 28, 8], [14, 6, 3], [14, 12, 2], [14, 24, 6], [24, 28, 16], [28, 12, 4], [28, 24, 2], [56, 24, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '672.1093', 'commutator_count': 1, 'commutator_label': '28.2', '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', '3.1', '7.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 112, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['32.43', 1], ['42.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 6], [2, 7, 1, 4], [2, 14, 1, 2], [2, 28, 1, 6], [3, 7, 2, 1], [4, 2, 1, 4], [4, 4, 1, 2], [4, 14, 1, 4], [4, 28, 1, 2], [6, 7, 2, 7], [6, 14, 2, 4], [6, 28, 2, 12], [7, 6, 1, 1], [8, 4, 1, 4], [8, 28, 1, 4], [12, 14, 2, 8], [12, 28, 2, 4], [14, 6, 1, 3], [14, 12, 1, 2], [14, 24, 1, 6], [24, 28, 2, 8], [28, 12, 1, 4], [28, 24, 1, 2], [56, 24, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 241984512000, 'exponent': 168, 'exponents_of_order': [7, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '672.1266', 'hash': 6630234698194963021, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 6, 1, 28], 'inner_gens': [[1, 2, 4, 24, 1968], [1, 2, 4, 24, 1392], [1, 2, 4, 24, 2256], [1, 2, 4, 24, 48], [769, 1346, 484, 24, 48]], 'inner_hash': 1093, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': False, 'inner_tex': 'C_2\\times D_4\\times F_7', 'inner_used': [1, 2, 3, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 96], [2, 24], [4, 12], [6, 16], [12, 4], [24, 2]], 'label': '2688.eh', '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': 'C8:C2^3*F7', 'ngens': 9, '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 68, 'number_characteristic_subgroups': 99, 'number_conjugacy_classes': 154, 'number_divisions': 110, 'number_normal_subgroups': 953, 'number_subgroup_autclasses': 948, 'number_subgroup_classes': 3440, 'number_subgroups': 23690, 'old_label': None, 'order': 2688, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 255], [3, 14], [4, 128], [6, 882], [7, 6], [8, 128], [12, 448], [14, 186], [24, 448], [28, 96], [56, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 1346, 28, 24, 48], [1, 2, 30, 24, 1394], [25, 1346, 4, 24, 48], [1, 1370, 30, 24, 1418], [1, 1346, 4, 24, 48], [25, 1346, 1348, 1368, 48]], 'outer_group': '256.27633', 'outer_hash': 27633, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 14, 'outer_perms': [367921, 6267300480, 19679350320, 13413214495, 6266937726, 19639433526], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5:D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2, 3], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 40], [4, 12], [6, 16], [8, 4], [12, 4], [24, 2]], 'representations': {'PC': {'code': '3953118781526880276493838637729217589714378590837024575396566605347650041445', 'gens': [1, 2, 3, 5, 6], 'pres': [9, -2, -2, -2, -3, -2, -2, -2, -2, -7, 74, 106277, 37598, 30479, 8132, 158, 78630, 28752, 18933, 186, 179719, 17305, 19042, 214, 186632, 38906, 15587]}, 'Perm': {'d': 17, 'gens': [21103456994063, 969, 45787363816463, 43166190856463, 68349217305743, 45787363776441, 45774829900943, 19239897743]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_8:C_2^3\\times F_7', 'transitive_degree': 112, 'wreath_data': None, 'wreath_product': False}
