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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '128.930', 'ambient_counter': 930, 'ambient_order': 128, 'ambient_tex': 'C_4^2.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 64, 'characteristic': True, 'core_order': 4, 'counter': 96, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '128.930.32.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '32.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '32.27', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 27, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 32, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2\\wr C_2', 'simple': False, 'solvable': True, 'special_labels': ['U2', 'L2', 'C5'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '128.930', 'aut_centralizer_order': 256, 'aut_label': '32.a1', 'aut_quo_index': 24, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 256, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.a1.a1', '16.a1.b1', '16.b1.a1', '16.c1.a1', '16.d1.a1', '16.e1.a1', '16.f1.a1', '16.f1.b1', '16.g1.a1', '16.g1.b1'], 'contains': ['64.a1.a1', '64.b1.a1'], 'core': '32.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6324, 5144, 6702, 5507, 6323, 5410, 6681, 5389], 'generators': [64, 16], 'label': '128.930.32.a1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '32.a1.a1', 'normal_contained_in': ['16.a1.a1', '16.a1.b1', '16.b1.a1'], 'normal_contains': ['64.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '32.a1.a1', 'projective_image': '64.138', 'quotient_action_image': '2.1', 'quotient_action_kernel': '16.14', 'quotient_action_kernel_order': 16, 'quotient_fusion': None, 'short_label': '32.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', '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]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': [[1, 2, 8, 32], [17, 6, 88, 32], [85, 86, 24, 56], [117, 46, 24, 40], [85, 70, 8, 120], [1, 18, 24, 96], [1, 90, 24, 96], [65, 82, 8, 32], [81, 66, 8, 32], [1, 18, 8, 32]], 'aut_group': '512.406983', 'aut_hash': 7717662186163716498, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 12, 'aut_perms': [87102720, 1987343, 134, 214910656, 216080040, 5160, 363016, 363000, 40284840], '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, 8, 1, 2], [4, 4, 1, 1], [4, 4, 2, 1], [4, 4, 4, 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, 2], [2, 8, 2], [4, 4, 7], [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': 930, '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, 2], [2, 8, 1, 2], [4, 4, 1, 3], [4, 4, 2, 2], [4, 8, 1, 3], [4, 16, 1, 2], [8, 16, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1344, 'exponent': 8, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, 0, 4]], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '8.5', 'hash': 930, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2, 4], 'inner_gens': [[1, 22, 88, 32], [21, 2, 72, 120], [81, 66, 8, 32], [1, 74, 8, 32]], '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, 4], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 6], [4, 6]], 'label': '128.930', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.D4', 'ngens': 3, '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': 18, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 83, 'number_subgroup_classes': 123, 'number_subgroups': 336, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 27], [4, 84], [8, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [24, 0], 'outer_gens': [[121, 34, 8, 32], [121, 34, 8, 48]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [6, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', '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, 2], [8, 2]], 'representations': {'PC': {'code': 163447753678201408244186587499, 'gens': [1, 2, 4, 6], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 112, 309, 36, 2467, 1018, 185, 80, 2532, 1027, 124, 1973]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 115142802501, 61179005627, 30594431879, 30704310001, 77580274917, 122244534693]}, 'Perm': {'d': 16, 'gens': [11212821043200, 9703741437382, 5612501704198, 5606234750016, 2789832700823, 1313941668480, 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_4^2.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': 12, 'aut_gen_orders': [2, 2, 2, 3, 2, 2, 2, 2], 'aut_gens': [[233, 235, 195, 231], [67, 235, 73, 231], [99, 227, 195, 239], [107, 97, 195, 101], [235, 67, 203, 207], [225, 97, 195, 231], [99, 105, 195, 101], [233, 235, 195, 109], [233, 235, 195, 101]], 'aut_group': '384.20100', 'aut_hash': 20100, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 12, 'aut_perms': [12785662, 302392930, 147357745, 52098429, 25832449, 125220487, 98166967, 136110610], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 2, 6, 1], [2, 4, 1, 1], [4, 4, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\wr S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', '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], [2, 2, 6], [2, 4, 1], [4, 4, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 27, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 6], [2, 4, 1, 1], [4, 4, 1, 3]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 28, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '8.5', 'hash': 27, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 2], 'inner_gens': [[233, 235, 195, 109], [233, 235, 195, 101], [233, 235, 195, 231], [107, 97, 195, 231]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 6]], 'label': '32.27', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 12, 'linQ_dim': 4, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2wrC2', 'ngens': 3, 'nilpotency_class': 2, '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 14, 'number_divisions': 14, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 19, 'number_subgroup_classes': 65, 'number_subgroups': 106, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 19], [4, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [65, 65, 65, 65, 65], 'outer_gens': [[67, 235, 73, 231], [99, 227, 195, 231], [235, 67, 203, 207], [99, 105, 195, 231], [233, 105, 195, 231]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [136, 316, 430, 269, 82], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_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]], 'representations': {'PC': {'code': 18891336, 'gens': [1, 2, 3, 4], 'pres': [5, 2, 2, 2, 2, 2, 243, 248, 58]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7233260, 16563328]}, 'GLFp': {'d': 4, 'p': 2, 'gens': [33837, 18465, 27183, 18659, 34029]}, 'GLZq': {'d': 2, 'q': 4, 'gens': [67, 235, 69, 195, 73]}, 'Perm': {'d': 8, 'gens': [23616, 16687, 11543, 11536, 5167]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\wr C_2', 'transitive_degree': 8, 'wreath_data': ['C_2^2', 'C_2', '2T1'], 'wreath_product': True}
