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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1760.45', 'ambient_counter': 45, 'ambient_order': 1760, 'ambient_tex': '(Q_8\\times C_{11}):C_{20}', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': False, 'core_order': 16, 'counter': 33, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1760.45.22.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '22.a1.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': 22, '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': '80.47', 'subgroup_hash': 47, 'subgroup_order': 80, 'subgroup_tex': 'Q_8\\times C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.45', 'aut_centralizer_order': 40, 'aut_label': '22.a1', 'aut_quo_index': None, 'aut_stab_index': 11, 'aut_weyl_group': '16.11', 'aut_weyl_index': 440, 'centralizer': '88.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '11.a1.a1'], 'contains': ['44.a1.a1', '44.b1.a1', '44.d1.a1', '44.d1.b1', '44.e1.a1', '110.a1.a1'], 'core': '110.a1.a1', 'coset_action_label': None, 'count': 11, 'diagramx': [2216, -1, 7833, -1, 7086, -1, 6215, -1], 'generators': [1, 880, 16, 940, 40], 'label': '1760.45.22.a1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '11.a1.a1', 'old_label': '22.a1.a1', 'projective_image': '440.11', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '22.a1.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': '40.14', '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, 4, 2, 3, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [1, 63, 76], [41, 42, 76], [1, 2, 68], [1, 2, 36], [1, 23, 66], [1, 43, 45], [1, 43, 4], [1, 42, 44], [1, 42, 4]], 'aut_group': '768.1087219', 'aut_hash': 1087219, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 768, 'aut_permdeg': 12, 'aut_perms': [298746840, 299185320, 17, 7, 18235440, 143354280, 167368440, 179788320, 223347720], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 6, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 1, 8, 1], [20, 2, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times C_2^3:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1031', 'autcent_hash': 1031, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3.C_2^4', '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], [4, 2, 6], [5, 1, 4], [10, 1, 12], [20, 2, 24]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, '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', '2.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 47, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 6], [5, 1, 4, 1], [10, 1, 4, 3], [20, 2, 4, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 217, 'exponent': 20, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '40.14', 'hash': 47, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2], 'inner_gens': [[1, 2, 4], [1, 2, 44], [1, 42, 4]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 10]], 'label': '80.47', 'linC_count': 192, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 8, 'linR_count': 16, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'Q8*C10', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 50, 'number_divisions': 20, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 38, 'number_subgroups': 38, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 12], [5, 4], [10, 12], [20, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12, 2, 2], 'outer_gen_pows': [4, 0, 0, 0, 0], 'outer_gens': [[1, 22, 4], [41, 2, 4], [1, 20, 70], [1, 3, 5], [1, 3, 4]], 'outer_group': '192.1469', 'outer_hash': 1469, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 10, 'outer_perms': [5040, 1, 80791, 1174320, 367920], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4.D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 8], [8, 2]], 'representations': {'PC': {'code': 60243744929795, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -5, 206, 337, 42, 58]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [1219466810, 326043207, 2184411956, 1234597583, 2143735230]}, 'Perm': {'d': 15, 'gens': [101671758720, 194163338880, 720, 96, 288848891520]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_8\\times C_{10}', 'transitive_degree': 80, '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': '40.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [10, 20, 2, 10, 10, 2, 20], 'aut_gens': [[1, 2, 80], [881, 462, 400], [21, 1002, 1360], [881, 142, 1680], [881, 922, 1200], [921, 222, 240], [1, 62, 1680], [901, 682, 560]], 'aut_group': None, 'aut_hash': 7590525614158771080, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7040, 'aut_permdeg': 96, 'aut_perms': [277950807090665608176025044783863859337299424518679507655201648146945468518797168341961789631293546513757936968869172644195552036920599388979850272222, 793591299762390839608160187070380338493715165036949272088017499060831667608005418354355228123541268191337737415530624311238884022446031495800276656403, 391653770861133486049344282628042952779299269237940168703873210702066805511651604747769936270272658376357640364543176628419957737657829659201605464630, 618834966139142366943961757057524572431877056022662391105423176976961002523650764805414220590413187093022764309669011645593001659845028329583088844072, 490821892423248782027416907661653675888893812423649832321626786336440424640265895511100378835716429151715480269869334043343855833271027548921805766105, 774943368190312258252597242729436665035570602213234461793655241726355523883589415996515745961191220266352260659344882389833837226358940469219054248768, 38545946246022784165467406151057550227194089043268361994437227566721306711999981159814150488901330711169231469805794424668644786626080659002173927751], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 4, 2, 1], [4, 44, 2, 1], [5, 11, 1, 4], [8, 22, 4, 1], [10, 11, 1, 12], [11, 10, 1, 1], [20, 22, 1, 8], [20, 44, 2, 8], [22, 10, 1, 3], [40, 22, 4, 4], [44, 20, 1, 2], [44, 20, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{22}.(C_2^5\\times C_{10})', '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': 110, 'autcentquo_group': '440.42', 'autcentquo_hash': 42, 'autcentquo_nilpotent': False, 'autcentquo_order': 440, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 4, 2], [4, 44, 2], [5, 11, 4], [8, 22, 4], [10, 11, 12], [11, 10, 1], [20, 22, 8], [20, 44, 16], [22, 10, 3], [40, 22, 16], [44, 20, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '440.11', 'commutator_count': 1, 'commutator_label': '44.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 45, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 4, 1, 2], [4, 44, 2, 1], [5, 11, 4, 1], [8, 22, 4, 1], [10, 11, 4, 3], [11, 10, 1, 1], [20, 22, 4, 2], [20, 44, 4, 2], [20, 44, 8, 1], [22, 10, 1, 3], [40, 22, 16, 1], [44, 20, 1, 2], [44, 20, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 440, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '220.7', 'hash': 45, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 220, 'inner_gen_orders': [2, 20, 11], 'inner_gens': [[1, 942, 80], [901, 2, 1040], [1, 802, 80]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 440, 'inner_split': True, 'inner_tex': 'D_{22}:C_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 30], [10, 8], [20, 2]], 'label': '1760.45', 'linC_count': 80, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 4, 'linQ_dim': 16, 'linQ_dim_count': 6, 'linR_count': 242, 'linR_degree': 16, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(Q8*C11):C20', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 52, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 80, 'number_divisions': 28, 'number_normal_subgroups': 39, 'number_subgroup_autclasses': 76, 'number_subgroup_classes': 84, 'number_subgroups': 530, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [4, 100], [5, 44], [8, 88], [10, 132], [11, 10], [20, 880], [22, 30], [40, 352], [44, 120]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[41, 2, 1680], [1, 42, 1680], [881, 2, 1680], [1, 882, 1680]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 6], [8, 4], [10, 4], [16, 2], [20, 4]], 'representations': {'PC': {'code': 7347521574921282739359503714277699273185588783335016790000309, 'gens': [1, 2, 6], 'pres': [7, -2, -2, -2, -2, -5, -2, -11, 280, 13189, 36, 926, 58, 80, 21852, 12619, 2126, 663, 124, 7853, 7860, 4927, 1504]}, 'Perm': {'d': 23, 'gens': [51353481345463504809, 49114816, 109853078970681446400, 91578240, 16, 135732240, 1277531155775242368000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(Q_8\\times C_{11}):C_{20}', 'transitive_degree': 352, 'wreath_data': None, 'wreath_product': False}
