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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '256.26347', 'ambient_counter': 26347, 'ambient_order': 256, 'ambient_tex': 'C_4^2.(C_2^2\\times C_4)', 'central': False, 'central_factor': False, 'centralizer_order': 64, 'characteristic': False, 'core_order': 1, 'counter': 240, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '256.26347.128.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '128.e1', 'outer_equivalence': True, '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': 128, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, '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': '256.26347', 'aut_centralizer_order': 2048, 'aut_label': '128.e1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '1.1', 'aut_weyl_index': 8192, 'centralizer': '4.j1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['64.e1', '64.q1', '64.x1', '64.y1'], 'contains': ['256.a1'], 'core': '256.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [3020, -1, 3024, -1], 'generators': [236], 'label': '256.26347.128.e1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '32.d1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.j1', 'old_label': '128.e1', 'projective_image': '256.26347', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '128.e1', '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': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 4, 4, 2, 8, 2, 4, 2, 2], 'aut_gens': [[1, 4, 8, 16, 64], [115, 8, 4, 208, 192], [45, 168, 164, 208, 224], [241, 132, 136, 16, 64], [1, 8, 4, 16, 64], [101, 40, 132, 144, 64], [197, 132, 168, 144, 64], [45, 36, 40, 16, 64], [237, 164, 168, 16, 64], [205, 164, 168, 144, 64]], 'aut_group': '8192.bcf', 'aut_hash': 2182866901409810851, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 128, 'aut_perms': [19411620143631347324735012901471960606920223132357430256738346128363387710651555697672414381322349278390834414793061464919663370076535323266953508313371310037709011867630745304619860119609561092851701057733825287838, 98082755105236157762965224088745536874195673587026272700499419788258544736529158509505521348967879647408826435678960288755598084564201766408485854254001879188502024209835217876139708217848075645791703502242221561539, 170504244880239976799467051188143946551279495478574967136805011668642084922729563551904019887196651240373832178122548202217842224856171563650156262874840271241118444503818286416183565912756474878952578720651077414137, 382637198169289589834933120177758958014617639034845389051538233160700651505786858817297189874469588760305431920726629572769252485645929265519640549304229748768907777651100522382938389446365470909488547826822431570087, 174064295798751875137220575434623125533234989066342694007493939372454776740156131522250531472977054381513037113240477571108351467905047983696313645109866037093284410339708672419897229696286486505303486847225790759860, 189945795904867535591253092890543653566274810383683746617479201406674774486115630522476620860585093373495419702678007295407195317626600467067434463594177377676056915255311708693828688062463272055561332619734935113888, 384308805492888594338886340261146360032088156154326996120859451090010646677809800816211158128177643582383618354713711058569738067594257790875641661347457772463760592211899227620082749906479414240169975699040223330023, 284950733969181038488655315961103088656139709668246333588689554295556285643507384388307901664152900136301491161561478653040610357081769396694111311740883313735007175204435154750082817125033478687204969248570952607164, 90634068784476351853406491435371586559481873225752054215865871541746564600338793028821171410157667662785632105956251063308705264021758665266480132764241161116708188104673980206530589707900987735463592180532377825963], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 2, 4, 1], [2, 4, 1, 1], [2, 4, 2, 1], [2, 4, 4, 1], [2, 8, 1, 1], [2, 8, 2, 1], [4, 2, 2, 1], [4, 4, 1, 1], [4, 4, 2, 3], [4, 8, 1, 2], [4, 8, 2, 1], [4, 8, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^7.D_4^2', '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': 4, 'autcentquo_group': '512.6480905', 'autcentquo_hash': 411620650426100056, 'autcentquo_nilpotent': True, 'autcentquo_order': 512, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3:D_4^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 5], [2, 4, 7], [2, 8, 3], [4, 2, 2], [4, 4, 7], [4, 8, 20]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '128.1613', 'commutator_count': 1, 'commutator_label': '8.2', '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': 26347, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 5], [2, 4, 1, 7], [2, 8, 1, 3], [4, 2, 1, 2], [4, 4, 1, 7], [4, 8, 1, 4], [4, 8, 2, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 161280, 'exponent': 4, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[8, 1, 2]], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '16.14', 'hash': 26347, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 2, 2, 4, 2], 'inner_gens': [[1, 4, 8, 112, 96], [1, 4, 136, 16, 64], [1, 132, 8, 16, 64], [225, 4, 8, 16, 64], [33, 4, 8, 16, 64]], 'inner_hash': 1613, 'inner_nilpotent': True, 'inner_order': 128, 'inner_split': False, 'inner_tex': 'C_2^5:C_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 32], [2, 8], [4, 4], [8, 2]], 'label': '256.26347', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.(C2^2*C4)', 'ngens': 4, 'nilpotency_class': 4, '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': 18, 'number_characteristic_subgroups': 35, 'number_conjugacy_classes': 46, 'number_divisions': 38, 'number_normal_subgroups': 181, 'number_subgroup_autclasses': 244, 'number_subgroup_classes': 678, 'number_subgroups': 2111, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 63], [4, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 190, 32, 0, 32], 'outer_gens': [[51, 4, 8, 112, 64], [29, 164, 168, 208, 224], [47, 164, 168, 144, 224], [5, 4, 168, 144, 64], [9, 8, 164, 16, 64]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [368056, 1174336, 136, 134, 806529], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 4], [8, 2]], 'representations': {'PC': {'code': 636340017085675587547094654033, 'gens': [1, 3, 4, 5, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 16, 1107, 4484, 972, 116, 7685, 5382, 5390, 166]}, 'Perm': {'d': 16, 'gens': [14002653732480, 4103693833552, 2789792421136, 8402294108878, 6920136138823, 2789832700823, 1313941668480, 1313941673647]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.(C_2^2\\times C_4)', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
