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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '576.2132', 'ambient_counter': 2132, 'ambient_order': 576, 'ambient_tex': 'C_4^2.S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 48, 'counter': 55, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '576.2132.12.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '12.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '12.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 12, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '48.11', 'subgroup_hash': 11, 'subgroup_order': 48, 'subgroup_tex': 'C_{12}:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '576.2132', 'aut_centralizer_order': 96, 'aut_label': '12.b1', 'aut_quo_index': 2, 'aut_stab_index': 2, 'aut_weyl_group': '48.51', 'aut_weyl_index': 192, 'centralizer': '12.g1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '6.a1.a1', '6.e1.a1', '6.f1.a1'], 'contains': ['24.a1.a1', '24.d1.a1', '24.e1.a1', '36.b1.a1'], 'core': '12.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4933, 6236, 5138, 4170, 4471, 4786, 5092, 6655], 'generators': [2, 192, 12, 312, 288], 'label': '576.2132.12.b1.a1', 'mobius_quo': 0, 'mobius_sub': -6, 'normal_closure': '12.b1.a1', 'normal_contained_in': ['4.b1.a1', '6.a1.a1'], 'normal_contains': ['24.a1.a1', '24.d1.a1', '24.e1.a1'], 'normalizer': '1.a1.a1', 'old_label': '12.b1.a1', 'projective_image': '72.46', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.b1.a1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 12], 'aut_gens': [[1, 4], [27, 4], [15, 4], [1, 20], [25, 4], [31, 30]], 'aut_group': '192.1147', 'aut_hash': 1147, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 11, 'aut_perms': [10824, 12501360, 3719545, 3719544, 12511683], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 2, 1, 1], [4, 1, 4, 1], [4, 3, 8, 1], [6, 2, 1, 1], [6, 2, 2, 1], [12, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', '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], [3, 2, 1], [4, 1, 4], [4, 3, 8], [6, 2, 3], [12, 2, 4]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 1, 2, 2], [4, 3, 2, 4], [6, 2, 1, 3], [12, 2, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '12.4', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 20], [33, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 8]], 'label': '48.11', 'linC_count': 48, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 8, 'linQ_dim': 6, 'linQ_dim_count': 40, 'linR_count': 40, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C12:C4', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 24, 'number_divisions': 16, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 30, 'number_subgroups': 44, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 28], [6, 6], [12, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[25, 30], [27, 30], [3, 28], [15, 30]], 'outer_group': '32.27', 'outer_hash': 27, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [23, 16, 16703, 40303], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\wr C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 2]], 'representations': {'PC': {'code': 139216212204961, 'gens': [1, 3], 'pres': [5, -2, -2, -2, -2, -3, 10, 302, 42, 803, 58, 804]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101740316382986, 91655872471932365]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [1173753, 635912, 72975, 1440589, 1565004]}, 'Perm': {'d': 11, 'gens': [368770, 13, 11543, 23, 3991680]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}:C_4', 'transitive_degree': 48, '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': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 6, 6, 6, 6, 12, 6, 2, 4], 'aut_gens': [[1, 4, 48], [155, 452, 360], [329, 212, 360], [443, 460, 336], [465, 284, 336], [331, 220, 240], [185, 436, 72], [27, 500, 336], [3, 140, 528], [443, 476, 48]], 'aut_group': None, 'aut_hash': 1609739320405297462, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9216, 'aut_permdeg': 72, 'aut_perms': [24724805921113586222024596191312426510679559898937294596624835922701730571625208788870174883445530271488, 5223024514767853899704787056351667791324550171018167673212285017470009866658891457284758323803676828660, 25252653296887664586029157417445434144196329638792097023148420212229292828842951937294006610791475167016, 37870521287056057790367227697732965941118084028804070507423299294435268889697833170815573029633675635206, 58842023021206000440308100463062481875000704360494142423160420402786243064131359226190280660244206622642, 35151181205594122679059389635926973737605546482077528019097816193773851410247384144063443177981089523911, 5185704859922526854165836621863100386285251183553843381443402603787839270339644703314371207833266137717, 15356189333506491933188203438086694136572433276121111457833285255838917368029002464095453708197818739584, 50777794181403380090203586328515887399750033406093584284050202818017382916036873088264062429307265857568], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 2], [3, 4, 1, 1], [4, 1, 2, 2], [4, 2, 2, 2], [4, 6, 4, 2], [6, 2, 1, 6], [6, 4, 1, 3], [8, 6, 8, 1], [8, 9, 8, 1], [8, 18, 4, 1], [12, 2, 2, 4], [12, 4, 2, 6], [12, 4, 4, 2], [12, 12, 4, 2], [24, 12, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6^2.C_2^6.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '72.46', 'autcentquo_hash': 46, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 2], [3, 4, 1], [4, 1, 4], [4, 2, 4], [4, 6, 8], [6, 2, 6], [6, 4, 3], [8, 6, 8], [8, 9, 8], [8, 18, 4], [12, 2, 8], [12, 4, 20], [12, 12, 8], [24, 12, 8]], 'center_label': '8.2', 'center_order': 8, 'central_product': False, 'central_quotient': '72.46', 'commutator_count': 1, 'commutator_label': '18.5', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2132, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 2], [3, 4, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 6, 1, 4], [4, 6, 2, 2], [6, 2, 1, 6], [6, 4, 1, 3], [8, 6, 4, 2], [8, 9, 4, 2], [8, 18, 4, 1], [12, 2, 2, 4], [12, 4, 1, 4], [12, 4, 2, 8], [12, 12, 1, 4], [12, 12, 2, 2], [24, 12, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5376, 'exponent': 24, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '72.46', 'hash': 2132, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 6], 'inner_gens': [[1, 20, 336], [33, 4, 528], [289, 100, 48]], 'inner_hash': 46, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'S_3\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 40], [4, 24]], 'label': '576.2132', 'linC_count': 128, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 40, 'linQ_dim': 10, 'linQ_dim_count': 48, 'linR_count': 36, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 40, 'number_characteristic_subgroups': 56, 'number_conjugacy_classes': 96, 'number_divisions': 56, 'number_normal_subgroups': 106, 'number_subgroup_autclasses': 154, 'number_subgroup_classes': 226, 'number_subgroups': 564, 'old_label': None, 'order': 576, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [3, 8], [4, 60], [6, 24], [8, 192], [12, 192], [24, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[289, 332, 48], [1, 20, 528], [27, 316, 336], [1, 316, 552], [3, 20, 240], [169, 164, 72]], 'outer_group': '128.2320', 'outer_hash': 2320, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [16680, 87107880, 127387567, 40284841, 11520, 87096382], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [4, 19], [8, 11]], 'representations': {'PC': {'code': 11772575952342122912478170723414721005075001097, 'gens': [1, 3, 6], 'pres': [8, -2, -2, -2, -2, -3, -2, -2, -3, 16, 2505, 482, 66, 1283, 91, 1284, 16133, 6357, 141, 6742, 166, 6167]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [189007, 40966, 640637, 297011, 630313, 626113, 513019, 32773]}, 'Perm': {'d': 22, 'gens': [58408856625034631545, 102438021137715221304, 163511332286129543544, 209376870451689434424, 209376870451676928000, 3719544, 6706022400, 3]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.S_3^2', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, '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': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [1, 10], [3, 2]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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, 1], [2, 3, 2], [3, 2, 1], [6, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', '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': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 10], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.4', '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': 'D6', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, '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': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 16, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[7, 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': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 43377, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 61, 16, 74]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [17, 35]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 55, 56]}, 'Perm': {'d': 5, 'gens': [6, 1, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_6', 'transitive_degree': 6, 'wreath_data': None, 'wreath_product': False}
