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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '544.38', 'ambient_counter': 38, 'ambient_order': 544, 'ambient_tex': 'C_{34}.C_4^2', 'central': True, 'central_factor': False, 'centralizer_order': 544, 'characteristic': False, 'core_order': 2, 'counter': 74, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '544.38.272.d1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '272.d1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '272.12', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 12, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 272, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{34}.D_4', 'simple': True, 'solvable': True, 'special_labels': [], 'split': False, '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': '544.38', 'aut_centralizer_order': 17408, 'aut_label': '272.d1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '1.1', 'aut_weyl_index': 34816, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.d1.b1', '136.b1.a1', '136.d1.a1', '136.e1.b1'], 'contains': ['544.a1.a1'], 'core': '272.d1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [4183, 5242, 3615, 5426, 4435, 4742, 4295, 4578], 'generators': [6], 'label': '544.38.272.d1.b1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '272.d1.b1', 'normal_contained_in': ['16.d1.b1', '136.e1.b1', '136.d1.a1', '136.b1.a1'], 'normal_contains': ['544.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '272.d1.b1', 'projective_image': '272.12', 'quotient_action_image': '1.1', 'quotient_action_kernel': '272.12', 'quotient_action_kernel_order': 272, 'quotient_fusion': None, 'short_label': '272.d1.b1', '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': '16.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 272, 'aut_gen_orders': [4, 16, 16, 16, 16, 2], 'aut_gens': [[1, 4, 8], [465, 4, 380], [231, 4, 184], [487, 4, 222], [9, 4, 334], [93, 4, 460], [279, 4, 264]], 'aut_group': None, 'aut_hash': 2808966716487297098, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 34816, 'aut_permdeg': 272, 'aut_perms': 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100055977950626833736984933437795958305621666846390004942547829688213842263526257342767144352456539238508731039242831549840698355802176808516642939922287362560390754209411279285304034579488516385275095056729790225052694068380079634742846324209675957536071666690150582173324369160516988555108723534740792854522569059289066119322782199002176470960376739024979538910266027315565885957520542302878264904337799134845562650794005207669100946262154808713522574622215941478718766806611761895591237489077765466809559077829495829245776593506721522405649721, 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19928042545361325401374425452428582897970117717587107218271830136857326591496523188919912279168448180414348513789103121547054238726211485438223719479100217885966871519715892158025851504010171136890207371363948664300844911693505955774964141852133888421245136412994126413243110296474938759265777387381678398805430612392325466385905486127604035883190054131349773833242258465625185700113328040380485831141642917135729377761354709326452404849515351305405696905409261898343835557025779082163562649405569987562890274835317786004067057061830694638978483], 'aut_phi_ratio': 136.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 2, 2], [4, 2, 4, 1], [4, 34, 8, 1], [17, 2, 8, 1], [34, 2, 8, 3], [34, 2, 16, 2], [68, 2, 64, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{17}:((C_2^6\\times C_{16}).C_2)', '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': 272, 'autcentquo_group': '544.242', 'autcentquo_hash': 242, 'autcentquo_nilpotent': False, 'autcentquo_order': 544, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_{17}', 'cc_stats': [[1, 1, 1], [2, 1, 7], [4, 2, 4], [4, 34, 8], [17, 2, 8], [34, 2, 56], [68, 2, 64]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '68.4', 'commutator_count': 1, 'commutator_label': '34.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '17.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 38, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 2, 2, 2], [4, 34, 2, 4], [17, 2, 8, 1], [34, 2, 8, 7], [68, 2, 32, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 68, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 17], 'factors_of_order': [2, 17], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '68.4', 'hash': 38, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 34, 'inner_gen_orders': [2, 1, 34], 'inner_gens': [[1, 4, 268], [1, 4, 8], [293, 4, 8]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 68, 'inner_split': True, 'inner_tex': 'D_{34}', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 132]], 'label': '544.38', 'linC_count': 3072, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 40, 'linQ_dim': 22, 'linQ_dim_count': 98, 'linR_count': 336, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C34.C4^2', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 15, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 148, 'number_divisions': 24, 'number_normal_subgroups': 45, 'number_subgroup_autclasses': 44, 'number_subgroup_classes': 76, 'number_subgroups': 468, 'old_label': None, 'order': 544, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 280], [17, 16], [34, 112], [68, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [8, 8, 16, 16], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[141, 4, 346], [411, 4, 474], [55, 4, 220], [143, 4, 492]], 'outer_group': '512.402863', 'outer_hash': 8883592558033294494, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 128, 'outer_perms': [72693269162008703661597282970999808355487279169616796923998365539998047951298180008409582618719216489358474358178788164327398049590244387970034881024570622946660707673342810416685999136893913790006656617011495755076, 221405980054765349494906919802924553957113975979995972419104520261273524368065031114597439378342730006431051266199390517103419020116946431019513623706546956731774403879834569426320481970853539185999697876686364857500, 376524461071708393840294941904378984576635549530022347831989590551972086702340474846264542743586459746219303043350416174743188124357317119629414861426924856036317450048657799502904930977906115003856841874854283036233, 12242504610095167509024361326512567017193216280861587399054361006457440354630489207453067463291441571850161229871794247105624768714427284088013309449653048931586579445777982841497000423632310026243678366046808328927], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{16}\\times C_2^2\\wr C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 29, 'pgroup': 0, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [16, 4], [32, 6]], 'representations': {'PC': {'code': 12596900873214323014433243151, 'gens': [1, 3, 4], 'pres': [6, -2, -2, -2, 2, -2, -17, 12, 6435, 69, 15844, 88, 18437]}, 'GLZN': {'d': 2, 'p': 68, 'gens': [10537791, 17454499, 316745, 10376289, 21067011, 314705]}, 'Perm': {'d': 29, 'gens': [10904406563691366305224787289, 80722080, 139725376, 167368440, 223347720, 326697157739096593105629772800]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{34}.C_4^2', 'transitive_degree': 544, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 272, 'aut_gen_orders': [2, 2, 2, 2, 16, 17], 'aut_gens': [[1, 4, 16], [3, 4, 256], [9, 4, 256], [1, 6, 256], [1, 12, 256], [1, 4, 224], [1, 20, 16]], 'aut_group': '4352.1118910', 'aut_hash': 1118910, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4352, 'aut_permdeg': 72, 'aut_perms': [206235239362721879672198132124634701297019142010093433451018589053563653248259098037559337835134975996, 206235239362721879672198132124634733870512961045121318160135347943366944282476663584209263881160447999, 13158025603360200545396099761605271274069085093921339335537448949214383537658611611088785836084690355686, 31260944030349588966054043356589512761621230437066618083309363555595342293444201147187568338877836723686, 157450163011474601217175462735674963276040863777779609760052785269345392919875094049423073988392326806, 862630799749635196843940186374865537409077938542591285035002939803107148495335601245687954072183372312], 'aut_phi_ratio': 34.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 2, 1], [4, 34, 2, 2], [17, 2, 8, 1], [34, 2, 8, 3], [68, 2, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{17}:(C_2^4\\times C_{16})', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 272, 'autcentquo_group': '272.50', 'autcentquo_hash': 50, 'autcentquo_nilpotent': False, 'autcentquo_order': 272, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{17}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 34, 4], [17, 2, 8], [34, 2, 24], [68, 2, 32]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '68.4', 'commutator_count': 1, 'commutator_label': '34.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '17.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 2, 1], [4, 34, 1, 2], [4, 34, 2, 1], [17, 2, 8, 1], [34, 2, 8, 3], [68, 2, 32, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 68, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 17], 'factors_of_order': [2, 17], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '68.4', 'hash': 12, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 34, 'inner_gen_orders': [2, 2, 17], 'inner_gens': [[1, 12, 16], [9, 4, 256], [1, 36, 16]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 68, 'inner_split': True, 'inner_tex': 'D_{34}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 66]], 'label': '272.12', 'linC_count': 128, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 10, 'linQ_dim': 20, 'linQ_dim_count': 4, 'linR_count': 104, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C34.D4', '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': 12, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 74, 'number_divisions': 13, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 26, 'number_subgroups': 142, 'old_label': None, 'order': 272, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 140], [17, 16], [34, 48], [68, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [2, 2, 16], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[3, 4, 256], [1, 6, 256], [1, 4, 80]], 'outer_group': '64.183', 'outer_hash': 183, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 64, 'outer_perms': [1984156435056817326162859149689132941394637456308417490687424990423955817003656436344041, 4091138822449015848424150938351050036728008234743804319927120894555378700079582362694884, 8122865399503344912607853765958373050463187103950844845153619861479822452871360928430674], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [16, 2], [32, 3]], 'representations': {'PC': {'code': 5207512816961978383, 'gens': [1, 3, 5], 'pres': [5, -2, -2, -2, -2, -17, 10, 182, 42, 1614]}, 'GLZN': {'d': 2, 'p': 51, 'gens': [132805, 2122432, 4069728, 6632600, 2434161]}, 'Perm': {'d': 25, 'gens': [25903229683158464682367, 1577, 7, 11520, 672254861099556368350080]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{34}.D_4', 'transitive_degree': 272, 'wreath_data': None, 'wreath_product': False}
