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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '704.312', 'ambient_counter': 312, 'ambient_order': 704, 'ambient_tex': '(C_2\\times C_8).D_{22}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 4, 'counter': 90, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '704.312.88.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '88.g1.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': 88, '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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '704.312', 'aut_centralizer_order': 320, 'aut_label': '88.g1', 'aut_quo_index': None, 'aut_stab_index': 22, 'aut_weyl_group': '4.2', 'aut_weyl_index': 7040, 'centralizer': '88.g1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.h1.a1', '44.g1.a1', '44.i1.a1', '44.k1.a1'], 'contains': ['176.a1.a1', '176.i1.a1'], 'core': '176.a1.a1', 'coset_action_label': None, 'count': 22, 'diagramx': [6140, -1, 5585, -1, 5525, -1, 5827, -1], 'generators': [9, 4], 'label': '704.312.88.g1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.e1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '22.d1.a1', 'old_label': '88.g1.a1', 'projective_image': '176.31', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '88.g1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', '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, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', '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], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, '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', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], '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': [], 'familial': True, 'frattini_label': '2.1', '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', 'ngens': 2, 'nilpotency_class': 1, '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': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', 'transitive_degree': 8, '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': 2, 'aut_exponent': 220, 'aut_gen_orders': [10, 10, 10, 10, 10, 10, 10, 10, 22], 'aut_gens': [[1, 2, 8, 64], [21, 2, 632, 576], [33, 54, 344, 384], [49, 2, 444, 256], [17, 2, 188, 512], [37, 2, 396, 512], [5, 18, 124, 384], [37, 6, 492, 512], [17, 38, 252, 320], [33, 34, 76, 64]], 'aut_group': None, 'aut_hash': 7247618788880966071, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 28160, 'aut_permdeg': 60, 'aut_perms': [4905908072690918260754374808972231103325022462139762676678228211090487951521586826, 6366639753930197897978844483667241888467229724701966021985307558193833087810482725, 4037235230423632293409511229372234355655978957936517562244312700213509087380265468, 7712358131806636942283438501429681600887159186766043929585089852287223598124320088, 234024911282301074786936870006413149997887215777480131098303975931923523423115892, 7890957843943771511182141451256269040937720797558011264659542694385422532813840886, 4892837452028818575657243120025516867217808789879997875354989444159940332198303626, 2489410856154158340522945292052824150968902249960128930017476168937296553548878479, 845089038774626539149866198890031494881344497357061189454857882315639347560179896], 'aut_phi_ratio': 88.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 8, 1, 1], [2, 44, 1, 1], [4, 2, 1, 2], [4, 8, 1, 1], [4, 44, 1, 1], [4, 88, 1, 2], [8, 4, 2, 1], [8, 44, 2, 1], [11, 2, 5, 1], [22, 2, 5, 3], [22, 8, 10, 1], [44, 4, 5, 2], [44, 8, 10, 1], [88, 4, 20, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2^3\\times C_{11}:C_5).C_2^6', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', '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], [2, 8, 1], [2, 44, 1], [4, 2, 2], [4, 8, 1], [4, 44, 1], [4, 88, 2], [8, 4, 2], [8, 44, 2], [11, 2, 5], [22, 2, 15], [22, 8, 10], [44, 4, 10], [44, 8, 10], [88, 4, 20]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '176.31', 'commutator_count': 1, 'commutator_label': '88.8', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 312, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 8, 1, 1], [2, 44, 1, 1], [4, 2, 1, 2], [4, 8, 1, 1], [4, 44, 1, 1], [4, 88, 1, 2], [8, 4, 2, 1], [8, 44, 1, 2], [11, 2, 5, 1], [22, 2, 5, 3], [22, 8, 10, 1], [44, 4, 5, 2], [44, 8, 10, 1], [88, 4, 20, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4032, 'exponent': 88, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '88.11', 'hash': 312, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 44, 'inner_gen_orders': [2, 2, 4, 11], 'inner_gens': [[1, 22, 24, 64], [53, 2, 56, 64], [49, 18, 8, 640], [1, 2, 136, 64]], 'inner_hash': 31, 'inner_nilpotent': False, 'inner_order': 176, 'inner_split': None, 'inner_tex': 'D_4\\times D_{11}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 46], [4, 32]], 'label': '704.312', 'linC_count': 520, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 16, 'linQ_dim': 16, 'linQ_dim_count': 8, 'linR_count': 55, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*C8).D22', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 23, 'number_characteristic_subgroups': 37, 'number_conjugacy_classes': 86, 'number_divisions': 24, 'number_normal_subgroups': 39, 'number_subgroup_autclasses': 112, 'number_subgroup_classes': 120, 'number_subgroups': 944, 'old_label': None, 'order': 704, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 55], [4, 232], [8, 96], [11, 10], [22, 110], [44, 120], [88, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 2, 2, 10], 'outer_gen_pows': [576, 0, 640, 0, 0], 'outer_gens': [[21, 2, 632, 64], [53, 2, 376, 640], [49, 2, 700, 64], [5, 18, 252, 640], [17, 38, 444, 576]], 'outer_group': '160.238', 'outer_hash': 238, 'outer_nilpotent': True, 'outer_order': 160, 'outer_permdeg': 160, 'outer_perms': [139007132924502405480023605563701584465719565586531017317987930393218048942546657844760462049539667215168241853598637246140078629951641634793711041225900784734308393759917872147251635559001115118458811522252871096237195637823102175110892332215506848521973225788520801222066418544080222, 186886578638254562064794446703779146148512688061344037599514739998814813964396674733341487403427725698107631121936134760089946382119826134446579401085757331627890698786972734248528861098752624496367453889157743254586243113591223203227828883039761037903196486987262739657160583220777630, 160568889369409957289299117438033503989865290829462869506220103349137527479912617623180014564303921604749950870116380685024873703404861127586952481346669273858761278249181239029291451885950988416756624371890460829447067065140386518468814596594124565557176116322604430948428450503791316, 245647248199452106688068089142320521041375498530846623539364547662663670588880297826274612536305505869030758760836805105715322921077378148390750702270786513670049522005814092537445002489496468847775202837350983560003017762416559017911973580388387097744867221847418798233708602671771384, 122549433242030877418630047251071286016359376466698945195927082257294097786788487404859583940032615537847499937166872561312276585151868808217302743694626732279902463642114019185840200751122311419788100540780799663754460034151986120194613141123905107499814958285245450999107528458240000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_{10}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 3], [10, 4], [20, 2], [40, 3]], 'representations': {'PC': {'code': 730113206533597075932278109233540000009, 'gens': [1, 2, 4, 7], 'pres': [7, -2, -2, -2, -2, -2, -2, -11, 309, 36, 675, 794, 80, 1684, 851, 102, 3947]}, 'Perm': {'d': 23, 'gens': [1124250420439788732847, 10116106155411379200, 2401761594863509401600, 3586238433383493888000, 186810624000, 1129109842917568512000, 4364893]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times C_8).D_{22}', 'transitive_degree': 352, 'wreath_data': None, 'wreath_product': False}
