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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '256.6663', 'ambient_counter': 6663, 'ambient_order': 256, 'ambient_tex': '(C_4\\times C_8).D_4', 'central': True, 'central_factor': False, 'centralizer_order': 256, 'characteristic': True, 'core_order': 2, 'counter': 197, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '256.6663.128.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '128.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '128.928', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 928, 'quotient_metabelian': False, 'quotient_nilpotent': True, 'quotient_order': 128, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\wr D_4', 'simple': True, 'solvable': True, 'special_labels': ['Z', 'S', 'U4', 'L4', 'C7'], 'split': False, 'standard_generators': False, 'stem': True, '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': '256.6663', 'aut_centralizer_order': 1024, 'aut_label': '128.a1', 'aut_quo_index': 4, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 1024, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['64.a1.a1', '64.b1.a1', '64.b1.b1', '64.c1.a1', '64.c1.b1', '64.d1.a1', '64.e1.a1', '64.f1.a1', '64.f1.b1', '64.g1.a1'], 'contains': ['256.a1.a1'], 'core': '128.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4615, 4857, 3903, 4120, 4601, 4680, 3895, 4097], 'generators': [128], 'label': '256.6663.128.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '128.a1.a1', 'normal_contained_in': ['64.a1.a1'], 'normal_contains': ['256.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '128.a1.a1', 'projective_image': '128.928', 'quotient_action_image': '1.1', 'quotient_action_kernel': '128.928', 'quotient_action_kernel_order': 128, 'quotient_fusion': None, 'short_label': '128.a1.a1', '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 8, 32], [193, 78, 200, 224], [1, 2, 8, 160], [213, 198, 24, 184], [61, 46, 8, 104], [129, 2, 8, 32], [1, 150, 88, 32]], 'aut_group': '1024.djj', 'aut_hash': 354343342895453903, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 1024, 'aut_permdeg': 18, 'aut_perms': [127905523125535, 732740287131631, 3846775457832768, 362807548126200, 1, 732744162851209], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 8, 2, 2], [2, 16, 1, 1], [4, 4, 1, 3], [4, 8, 2, 2], [4, 16, 1, 1], [4, 32, 2, 1], [8, 4, 2, 1], [8, 8, 1, 1], [8, 16, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'D_4^2.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': '128.928', 'autcentquo_hash': 928, 'autcentquo_nilpotent': True, 'autcentquo_order': 128, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\wr D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 8, 4], [2, 16, 1], [4, 4, 3], [4, 8, 4], [4, 16, 1], [4, 32, 2], [8, 4, 2], [8, 8, 1], [8, 16, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '128.928', 'commutator_count': 1, 'commutator_label': '32.34', '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', '2.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 6663, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 8, 1, 4], [2, 16, 1, 1], [4, 4, 1, 3], [4, 8, 1, 2], [4, 8, 2, 1], [4, 16, 1, 1], [4, 32, 1, 2], [8, 4, 2, 1], [8, 8, 1, 1], [8, 16, 1, 2], [8, 16, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 5376, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, 0, 4], [8, 1, 1]], 'familial': False, 'frattini_label': '32.34', 'frattini_quotient': '8.5', 'hash': 6663, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 4, 4, 4], 'inner_gens': [[1, 134, 200, 224], [133, 2, 88, 56], [65, 210, 8, 32], [65, 10, 8, 32]], 'inner_hash': 928, 'inner_nilpotent': True, 'inner_order': 128, 'inner_split': True, 'inner_tex': 'C_2\\wr D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 6], [4, 10], [8, 1]], 'label': '256.6663', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': '(C4*C8).D4', 'ngens': 3, 'nilpotency_class': 5, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 17, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 25, 'number_divisions': 22, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 135, 'number_subgroup_classes': 204, 'number_subgroups': 831, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 51], [4, 124], [8, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 16], 'outer_gens': [[129, 2, 8, 32], [1, 2, 8, 160], [189, 174, 8, 232]], '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': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 6], [8, 1], [16, 1]], 'representations': {'PC': {'code': 46212741270981503544712202162364517670647733994, 'gens': [1, 2, 4, 6], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 2145, 41, 6403, 1419, 211, 91, 5764, 2892, 10757, 1357, 2709, 141, 10758, 2254, 2710, 166]}, 'Perm': {'d': 16, 'gens': [2796019828553, 11219175435023, 11212821043200, 8402294102383, 5606234750016, 4097467152142, 5167, 1313941673647]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_4\\times C_8).D_4', '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[23616, 5183, 5047, 11536, 121, 5160, 5167], [40319, 16, 120, 11536, 126, 5160, 5167], [23616, 11527, 6, 16703, 5046, 7, 5167], [40193, 120, 16, 126, 16696, 7, 5167], [28783, 16680, 1, 16703, 121, 7, 5167], [28783, 5183, 127, 11536, 5046, 5160, 5167], [35278, 5183, 5047, 16696, 126, 5160, 5167], [28783, 23, 5040, 16703, 5046, 5160, 5167], [28783, 5176, 127, 16703, 5046, 5160, 5167], [28783, 5183, 5047, 11536, 121, 5160, 5167]], 'aut_group': '512.406983', 'aut_hash': 7717662186163716498, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 12, 'aut_perms': [167410080, 1174325, 40683007, 262091536, 263265840, 85800, 806416, 806400, 40723320], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 2, 1], [2, 4, 4, 1], [2, 8, 1, 2], [4, 4, 1, 1], [4, 4, 2, 1], [4, 8, 1, 1], [4, 8, 2, 1], [4, 16, 2, 1], [8, 16, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^6:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '64.138', 'autcentquo_hash': 138, 'autcentquo_nilpotent': True, 'autcentquo_order': 64, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\wr C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 6], [2, 8, 2], [4, 4, 3], [4, 8, 3], [4, 16, 2], [8, 16, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '64.138', 'commutator_count': 1, 'commutator_label': '16.11', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 928, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 6], [2, 8, 1, 2], [4, 4, 1, 3], [4, 8, 1, 3], [4, 16, 1, 2], [8, 16, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 1344, 'exponent': 8, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, 1, 4]], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '8.5', 'hash': 928, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 2, 2, 2, 1], 'inner_gens': [[23616, 16687, 5166, 11536, 121, 7, 5167], [35152, 5183, 5047, 11536, 126, 5160, 5167], [23737, 5183, 5047, 16696, 121, 5160, 5167], [23616, 5183, 127, 11536, 5046, 5160, 5167], [23616, 5176, 5047, 16703, 121, 5160, 5167], [28783, 5183, 5047, 11536, 121, 5160, 5167], [23616, 5183, 5047, 11536, 121, 5160, 5167]], 'inner_hash': 138, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': True, 'inner_tex': 'C_2\\wr C_2^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 6], [4, 6]], 'label': '128.928', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 4, 'linQ_dim': 4, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2wrD4', 'ngens': 7, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 20, 'number_divisions': 20, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 96, 'number_subgroup_classes': 177, 'number_subgroups': 576, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 43], [4, 68], [8, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[23616, 16, 5047, 11536, 121, 5160, 5167], [23616, 120, 5183, 121, 11536, 7, 5167]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [5, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 6]], 'representations': {'PC': {'code': 1276935575609822601997427050, 'gens': [1, 2, 4, 6], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 85, 36, 2467, 1018, 185, 80, 2532, 1027, 124, 1973]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16566404, 35892680, 17330674]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [34876415, 14921359, 28410960, 30086884, 28816400, 14811013, 28233362]}, 'Perm': {'d': 8, 'gens': [23616, 5183, 5047, 11536, 121, 5160, 5167]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\wr D_4', 'transitive_degree': 8, 'wreath_data': ['64.c1.a1', '16.ba1.b1', '64.c1.b1', '4T3'], 'wreath_product': True}
