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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1344.8572', 'ambient_counter': 8572, 'ambient_order': 1344, 'ambient_tex': 'D_{56}:D_6', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 4, 'counter': 401, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.8572.84.z1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '84.z1.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': 84, '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': '16.5', 'subgroup_hash': 5, 'subgroup_order': 16, 'subgroup_tex': 'C_2\\times C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.8572', 'aut_centralizer_order': 96, 'aut_label': '84.z1', 'aut_quo_index': None, 'aut_stab_index': 21, 'aut_weyl_group': '8.5', 'aut_weyl_index': 2016, 'centralizer': '84.z1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.bb1.a1', '28.ba1.a1', '42.e1.a1', '42.i1.a1', '42.l1.a1'], 'contains': ['168.t1.a1', '168.bo1.a1', '168.bp1.a1'], 'core': '336.a1.a1', 'coset_action_label': None, 'count': 21, 'diagramx': [9363, -1, 9765, -1, 1346, -1, 8718, -1], 'generators': [85, 170], 'label': '1344.8572.84.z1.a1', 'mobius_quo': None, 'mobius_sub': 2, 'normal_closure': '4.r1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '21.a1.a1', 'old_label': '84.z1.a1', 'projective_image': '672.1150', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '84.z1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2], 'aut_gens': [[1, 2], [1, 3], [9, 14], [1, 6], [1, 10]], 'aut_group': '16.11', 'aut_hash': 11, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 6, 'aut_perms': [126, 55, 289, 288], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 2, 2], [8, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4', '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': 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], [4, 1, 4], [8, 1, 8]], 'center_label': '16.5', 'center_order': 16, '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', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [8, 1, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 8, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '4.2', '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, 16]], 'label': '16.5', 'linC_count': 48, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 4, 'linQ_dim': 5, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C8', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 16, 'number_divisions': 8, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 11, 'number_subgroups': 11, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4], [8, 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': [[1, 3], [9, 14], [1, 6], [1, 10]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 2]], 'representations': {'PC': {'code': 9222, 'gens': [1, 2], 'pres': [4, -2, 2, -2, -2, 21, 34]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706304316034]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [19665, 44226]}, 'Perm': {'d': 10, 'gens': [40176, 362880, 16582, 5167]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], '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_8', 'transitive_degree': 16, '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': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 12, 42, 6, 4, 42, 6, 2], 'aut_gens': [[1, 2, 4, 56], [1, 698, 372, 616], [337, 690, 356, 952], [1009, 714, 900, 392], [1009, 682, 940, 616], [1009, 674, 564, 952], [337, 714, 900, 1064], [337, 698, 916, 392], [673, 722, 500, 952]], 'aut_group': None, 'aut_hash': 7058334312241270196, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 28, 'aut_perms': [8215573116303713124379483635, 97278350225508586151133620726, 191239624680093416444555132328, 193734662399727099750878656743, 195265094757675500272386651040, 92836508163241759078905098208, 90073191052243883424754084175, 260712067719383772577160624542], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 2], [2, 6, 1, 1], [2, 12, 1, 1], [2, 14, 1, 1], [2, 28, 1, 2], [2, 42, 1, 1], [2, 84, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 12, 1, 1], [4, 14, 1, 1], [4, 42, 1, 1], [4, 84, 1, 1], [6, 2, 1, 1], [6, 8, 1, 2], [6, 14, 2, 1], [6, 56, 1, 2], [7, 2, 3, 1], [8, 4, 1, 1], [8, 12, 1, 1], [8, 14, 2, 1], [8, 84, 1, 1], [12, 4, 1, 1], [12, 28, 1, 1], [14, 2, 3, 1], [14, 8, 3, 2], [14, 12, 3, 1], [14, 24, 3, 1], [21, 4, 3, 1], [24, 4, 2, 1], [24, 28, 2, 1], [28, 4, 3, 1], [28, 12, 3, 1], [28, 24, 3, 1], [42, 4, 3, 1], [42, 16, 3, 2], [56, 8, 3, 1], [56, 24, 3, 1], [84, 8, 3, 1], [168, 8, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '1008.906', 'autcentquo_hash': 906, 'autcentquo_nilpotent': False, 'autcentquo_order': 1008, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 2], [2, 6, 1], [2, 12, 1], [2, 14, 1], [2, 28, 2], [2, 42, 1], [2, 84, 1], [3, 2, 1], [4, 2, 1], [4, 6, 1], [4, 12, 1], [4, 14, 1], [4, 42, 1], [4, 84, 1], [6, 2, 1], [6, 8, 2], [6, 14, 2], [6, 56, 2], [7, 2, 3], [8, 4, 1], [8, 12, 1], [8, 14, 2], [8, 84, 1], [12, 4, 1], [12, 28, 1], [14, 2, 3], [14, 8, 6], [14, 12, 3], [14, 24, 3], [21, 4, 3], [24, 4, 2], [24, 28, 2], [28, 4, 3], [28, 12, 3], [28, 24, 3], [42, 4, 3], [42, 16, 6], [56, 8, 3], [56, 24, 3], [84, 8, 3], [168, 8, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '672.1150', 'commutator_count': 1, 'commutator_label': '84.6', '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': 8572, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 2], [2, 6, 1, 1], [2, 12, 1, 1], [2, 14, 1, 1], [2, 28, 1, 2], [2, 42, 1, 1], [2, 84, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 12, 1, 1], [4, 14, 1, 1], [4, 42, 1, 1], [4, 84, 1, 1], [6, 2, 1, 1], [6, 8, 1, 2], [6, 14, 2, 1], [6, 56, 1, 2], [7, 2, 3, 1], [8, 4, 1, 1], [8, 12, 1, 1], [8, 14, 2, 1], [8, 84, 1, 1], [12, 4, 1, 1], [12, 28, 1, 1], [14, 2, 3, 1], [14, 8, 3, 2], [14, 12, 3, 1], [14, 24, 3, 1], [21, 4, 3, 1], [24, 4, 2, 1], [24, 28, 2, 1], [28, 4, 3, 1], [28, 12, 3, 1], [28, 24, 3, 1], [42, 4, 3, 1], [42, 16, 3, 2], [56, 8, 3, 1], [56, 24, 3, 1], [84, 8, 3, 1], [168, 8, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 59754240, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, 0, 6]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '336.219', 'hash': 8572, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 14, 12], 'inner_gens': [[1, 2, 1012, 392], [1, 2, 724, 56], [337, 682, 4, 616], [1009, 2, 788, 56]], 'inner_hash': 1150, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': True, 'inner_tex': 'D_{12}:D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 36], [4, 26], [8, 12]], 'label': '1344.8572', 'linC_count': 96, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 64, 'linQ_dim': 16, 'linQ_dim_count': 64, 'linR_count': 408, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D56:D6', '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': 49, 'number_characteristic_subgroups': 126, 'number_conjugacy_classes': 90, 'number_divisions': 49, 'number_normal_subgroups': 126, 'number_subgroup_autclasses': 536, 'number_subgroup_classes': 536, 'number_subgroups': 5300, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 223], [3, 2], [4, 160], [6, 158], [7, 6], [8, 128], [12, 32], [14, 162], [21, 12], [24, 64], [28, 120], [42, 108], [56, 96], [84, 24], [168, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 2, 4, 952], [673, 2, 4, 56], [1, 2, 12, 280]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [40320, 24, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 26, '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, 2], [6, 8], [8, 1], [12, 6], [16, 1], [24, 2], [48, 1]], 'representations': {'PC': {'code': 2007696531538044882013484319184082456053874468412067153, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -7, -2, -2, -2, -3, 24290, 8698, 66, 779, 15684, 6180, 116, 37637, 14805, 141, 15702, 166, 14359]}, 'Perm': {'d': 26, 'gens': [621574835363380683752551, 357034988101053, 47511470780, 3055849103693, 4491430513087, 5633060522407, 6758061133824000, 16754357281155028254720000]}}, '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': 'D_{56}:D_6', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
