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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.6795', 'ambient_counter': 6795, 'ambient_order': 1344, 'ambient_tex': 'D_8:C_2\\times F_7', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 4, 'counter': 531, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.6795.168.bf1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '168.bf1.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': 168, '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': '8.3', 'subgroup_hash': 3, 'subgroup_order': 8, 'subgroup_tex': 'D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.6795', 'aut_centralizer_order': 48, 'aut_label': '168.bf1', 'aut_quo_index': None, 'aut_stab_index': 14, 'aut_weyl_group': '8.3', 'aut_weyl_index': 672, 'centralizer': '56.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['24.k1.a1', '56.bd1.a1', '84.l1.a1', '84.r1.a1', '84.t1.a1', '84.v1.b1', '84.bc1.a1', '84.bd1.a1', '84.be1.a1'], 'contains': ['336.c1.a1', '336.i1.a1'], 'core': '336.c1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [8770, -1, 8260, -1, 5557, -1, 7350, -1], 'generators': [38, 336], 'label': '1344.6795.168.bf1.a1', 'mobius_quo': None, 'mobius_sub': -8, 'normal_closure': '24.k1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '168.bf1.a1', 'projective_image': '672.1093', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '168.bf1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], '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': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '48.52', '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, 4, 6, 4, 12, 6], 'aut_gens': [[1, 2, 4, 24], [1, 2, 772, 793], [673, 2, 196, 793], [1, 1010, 676, 1321], [1, 2, 772, 120], [1, 338, 484, 312], [673, 1010, 1156, 121], [1, 2, 676, 1273]], 'aut_group': None, 'aut_hash': 356946386062274584, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5376, 'aut_permdeg': 26, 'aut_perms': [126262422142581860715280214, 157299152692238798517272813, 69838712310673592767322759, 304709046457693382948227159, 118160415334265071565177909, 149788526300404293312377927, 305639930364278252368927124], 'aut_phi_ratio': 14.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 4, 2, 1], [2, 7, 2, 1], [2, 14, 1, 1], [2, 28, 1, 1], [2, 28, 2, 1], [3, 7, 1, 2], [4, 2, 1, 2], [4, 4, 1, 1], [4, 14, 1, 2], [4, 28, 1, 1], [6, 7, 1, 2], [6, 7, 2, 2], [6, 14, 1, 4], [6, 28, 1, 4], [6, 28, 2, 4], [7, 6, 1, 1], [8, 4, 2, 1], [8, 28, 2, 1], [12, 14, 1, 8], [12, 28, 1, 4], [14, 6, 1, 1], [14, 12, 1, 1], [14, 24, 1, 1], [14, 24, 2, 1], [24, 28, 2, 4], [28, 12, 1, 2], [28, 24, 1, 1], [56, 24, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_7.(C_3\\times D_4^2).C_2^2', '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': '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, 1], [2, 2, 1], [2, 4, 3], [2, 7, 2], [2, 14, 1], [2, 28, 3], [3, 7, 2], [4, 2, 2], [4, 4, 1], [4, 14, 2], [4, 28, 1], [6, 7, 6], [6, 14, 4], [6, 28, 12], [7, 6, 1], [8, 4, 2], [8, 28, 2], [12, 14, 8], [12, 28, 4], [14, 6, 1], [14, 12, 1], [14, 24, 3], [24, 28, 8], [28, 12, 2], [28, 24, 1], [56, 24, 2]], 'center_label': '2.1', 'center_order': 2, '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', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 6795, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['32.43', 1], ['42.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 3], [2, 7, 1, 2], [2, 14, 1, 1], [2, 28, 1, 3], [3, 7, 2, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 14, 1, 2], [4, 28, 1, 1], [6, 7, 2, 3], [6, 14, 2, 2], [6, 28, 2, 6], [7, 6, 1, 1], [8, 4, 1, 2], [8, 28, 1, 2], [12, 14, 2, 4], [12, 28, 2, 2], [14, 6, 1, 1], [14, 12, 1, 1], [14, 24, 1, 3], [24, 28, 2, 4], [28, 12, 1, 2], [28, 24, 1, 1], [56, 24, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 183859200, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[24, 1, 1]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '336.216', 'hash': 6795, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 6, 28], 'inner_gens': [[1, 674, 4, 696], [673, 2, 4, 360], [1, 2, 4, 792], [673, 1010, 580, 24]], 'inner_hash': 1093, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': True, 'inner_tex': 'C_2\\times D_4\\times F_7', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 48], [2, 12], [4, 6], [6, 8], [12, 2], [24, 1]], 'label': '1344.6795', 'linC_count': 48, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 16, 'linQ_dim': 10, 'linQ_dim_count': 16, 'linR_count': 16, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D8:C2*F7', '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': 60, 'number_characteristic_subgroups': 90, 'number_conjugacy_classes': 77, 'number_divisions': 55, 'number_normal_subgroups': 182, 'number_subgroup_autclasses': 428, 'number_subgroup_classes': 596, 'number_subgroups': 3892, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 127], [3, 14], [4, 64], [6, 434], [7, 6], [8, 64], [12, 224], [14, 90], [24, 224], [28, 48], [56, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 2, 4, 697], [1, 2, 676, 24], [673, 2, 4, 24]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 6], [6, 8], [8, 2], [12, 2], [24, 1]], 'representations': {'PC': {'code': 14618703011278059950120291402852419296055082688720814798525266197, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -3, -2, -2, -2, -7, 10785, 66, 27844, 7212, 7940, 3028, 116, 17293, 2901, 7229, 141, 6742, 7422, 166, 15383, 6175]}, 'GLZN': {'d': 2, 'p': 56, 'gens': [175625, 175643, 176065, 176009, 176105, 177185, 176401, 2637027]}, 'Perm': {'d': 15, 'gens': [6267305809, 5094, 572, 12341, 13499136000, 18523, 5329, 99752083200]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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': 'D_8:C_2\\times F_7', 'transitive_degree': 56, 'wreath_data': None, 'wreath_product': False}
