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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '2997.51', 'ambient_counter': 51, 'ambient_order': 2997, 'ambient_tex': 'C_{333}:C_9', 'central': False, 'central_factor': True, 'centralizer_order': 9, 'characteristic': False, 'core_order': 333, 'counter': 13, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '2997.51.9.d1.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '9.d1.d1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '9.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 9, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_9', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '333.3', 'subgroup_hash': 3, 'subgroup_order': 333, 'subgroup_tex': 'C_{37}:C_9', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2997.51', 'aut_centralizer_order': 6, 'aut_label': '9.d1', 'aut_quo_index': 1, 'aut_stab_index': 9, 'aut_weyl_group': '1332.31', 'aut_weyl_index': 54, 'centralizer': '333.a1.a1', 'complements': ['333.a1.a1', '333.c1.a1', '333.c1.b1', '333.d1.g1', '333.d1.i1', '333.d1.h1', '333.d1.b1', '333.d1.c1', '333.d1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['3.b1.b1'], 'contains': ['27.b1.b1', '333.d1.d1'], 'core': '9.d1.d1', 'coset_action_label': None, 'count': 1, 'diagramx': [1852, 1153, 1090, 3007, 805, 1208, 3327, 1158], 'generators': [10, 1488, 81], 'label': '2997.51.9.d1.d1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '9.d1.d1', 'normal_contained_in': ['3.b1.b1'], 'normal_contains': ['27.b1.b1'], 'normalizer': '1.a1.a1', 'old_label': '9.d1.d1', 'projective_image': '2997.51', 'quotient_action_image': '1.1', 'quotient_action_kernel': '9.1', 'quotient_action_kernel_order': 9, 'quotient_fusion': None, 'short_label': '9.d1.d1', 'subgroup_fusion': None, 'weyl_group': '333.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '9.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1332, 'aut_gen_orders': [36, 37], 'aut_gens': [[1, 9], [1, 171], [10, 9]], 'aut_group': '1332.31', 'aut_hash': 31, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1332, 'aut_permdeg': 37, 'aut_perms': [13205615616340170960925241054103261394129333, 13391759764436443828847980133430067200000000], 'aut_phi_ratio': 6.166666666666667, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 37, 1, 2], [9, 37, 1, 6], [37, 9, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_{37}', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1332, 'autcentquo_group': '1332.31', 'autcentquo_hash': 31, 'autcentquo_nilpotent': False, 'autcentquo_order': 1332, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{37}', 'cc_stats': [[1, 1, 1], [3, 37, 2], [9, 37, 6], [37, 9, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '333.3', 'commutator_count': 1, 'commutator_label': '37.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '37.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 37, 2, 1], [9, 37, 6, 1], [37, 9, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 333, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 37], 'factors_of_order': [3, 37], 'faithful_reps': [[9, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '333.3', 'hash': 3, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 333, 'inner_gen_orders': [9, 37], 'inner_gens': [[1, 81], [262, 9]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 333, 'inner_split': True, 'inner_tex': 'C_{37}:C_9', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 18, 'irrep_stats': [[1, 9], [9, 4]], 'label': '333.3', 'linC_count': 4, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 36, 'linQ_degree_count': 1, 'linQ_dim': 36, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C37:C9', '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': 10, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 13, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 78, 'old_label': None, 'order': 333, 'order_factorization_type': 22, 'order_stats': [[1, 1], [3, 74], [9, 222], [37, 36]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [7], 'outer_gens': [[1, 18]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 37, 'pgroup': 0, 'primary_abelian_invariants': [9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [6, 1], [36, 1]], 'representations': {'PC': {'code': 4142943486359, 'gens': [1, 3], 'pres': [3, -3, -3, -37, 9, 731, 707]}, 'GLFp': {'d': 2, 'p': 37, 'gens': [50691, 1316990]}, 'Perm': {'d': 37, 'gens': [163947397675923090080217383295284717941066, 262494983102074267962526693511783514921893, 13391759764436443828847980133430067200000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [9], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{37}:C_9', 'transitive_degree': 37, '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': '81.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1332, 'aut_gen_orders': [6, 36, 18], 'aut_gens': [[1, 9], [1045, 1422], [2017, 1179], [469, 2583]], 'aut_group': None, 'aut_hash': 1638811925615564483, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 71928, 'aut_permdeg': 333, 'aut_perms': [10300061016832145349431795212254172128039361751878120520530066651391594131270944476425257168421303893320417383942779439955899540515392467249418779986585762209374581636231054125402958521229427482659578026447430976561704534413071875268009208951508521410800381032598272717636055066455274596302861665977984401262053358889602788995289499259543127451767944810076509671774852017919797061361549886226120363661776017477679071823542691533413568602996863565995025116763476985798040061654518556514943810784847902722394067364398188449538356113988700892213680030945122894658933822345186393185540951536682943146727626895231944686528803521582347630337435246069130203937666120895460987823106374382302627739587183066, 10145827579469484593113567963713758752887742261055696346125789795675742706356947010692194307027146098225724779592859534442766256186693304891947975725579193958384239346105663685993455596505118505218282425475405887226140133475116289268057370935897226016869162385791043882588609754274395957952706102692390204118010858932959656873928273141832986479836331774212932736572888499574793152070072259146050406812009635911279794320712960364111264823023515112808204533992731743677063830136130164907930651478282259799162663834926813445367694870780771638234882949041467751930491459388445955811347434816281522793894493630133514798598728971405572391300597118661102100893452843442613249804937822612806195638964783413, 1430487831244205707338629659986595022236101544007605347622726912409257690204751721684066718107632186130866749313585123166226155915087779008448439176710315097622081333187414545442333651997369529134921502024226784055167817298824697824359134044942578309255201476976411886177499814084671693197119981296324983339451476202202746301308041591015871254714141571690861390745398101779386320420141218635674151352670560932257879362667312272511622099090614269887225699914614648008470935704497932672987296058271892718629841609587406029028956661467693569957457762906140369798567411817134559897206442866688143457076839945127242565194580790417349966579161146478613976771951931317793063198477714366183708258823054823], 'aut_phi_ratio': 37.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 37, 3, 2], [9, 1, 6, 1], [9, 37, 6, 2], [9, 37, 9, 6], [37, 9, 4, 1], [111, 9, 8, 1], [333, 9, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{333}.(C_6\\times C_{36})', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 18, 'autcent_group': '54.6', 'autcent_hash': 6, 'autcent_nilpotent': False, 'autcent_order': 54, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_9:C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1332, 'autcentquo_group': '1332.31', 'autcentquo_hash': 31, 'autcentquo_nilpotent': False, 'autcentquo_order': 1332, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{37}', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 37, 6], [9, 1, 6], [9, 37, 66], [37, 9, 4], [111, 9, 8], [333, 9, 24]], 'center_label': '9.1', 'center_order': 9, 'central_product': True, 'central_quotient': '333.3', 'commutator_count': 1, 'commutator_label': '37.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1', '37.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 51, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['333.3', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 37, 2, 3], [9, 1, 6, 1], [9, 37, 6, 11], [37, 9, 4, 1], [111, 9, 8, 1], [333, 9, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 333, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 37], 'factors_of_order': [3, 37], 'faithful_reps': [[9, 0, 24]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '999.13', 'hash': 51, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 333, 'inner_gen_orders': [9, 37], 'inner_gens': [[1, 414], [2593, 9]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 333, 'inner_split': True, 'inner_tex': 'C_{37}:C_9', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 216, 'irrQ_dim': 216, 'irrR_degree': 18, 'irrep_stats': [[1, 81], [9, 36]], 'label': '2997.51', 'linC_count': 24, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 42, 'linQ_degree_count': 9, 'linQ_dim': 42, 'linQ_dim_count': 9, 'linR_count': 12, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C333:C9', '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': 16, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 117, 'number_divisions': 20, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 46, 'number_subgroups': 766, 'old_label': None, 'order': 2997, 'order_factorization_type': 31, 'order_stats': [[1, 1], [3, 224], [9, 2448], [37, 36], [111, 72], [333, 216]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 36, 'outer_gen_orders': [6, 36], 'outer_gen_pows': [0, 0], 'outer_gens': [[1999, 342], [667, 2115]], 'outer_group': '216.53', 'outer_hash': 53, 'outer_nilpotent': False, 'outer_order': 216, 'outer_permdeg': 13, 'outer_perms': [142621920, 3006264977], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{36}:C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 46, 'pgroup': 0, 'primary_abelian_invariants': [9, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 12], [36, 1], [72, 1], [216, 1]], 'representations': {'PC': {'code': '161271511141369186433649048693427576778903', 'gens': [1, 3], 'pres': [5, -3, -3, -3, -3, -37, 15, 6212, 4507, 57, 24843, 18008, 78, 18229, 17559]}, 'GLFp': {'d': 2, 'p': 37, 'gens': [50691, 810449, 354587]}, 'Perm': {'d': 46, 'gens': [52023, 2843934061013125658966551704738547863063046281486722356, 98356, 5565470115197013320507207356385968389645534653310796800, 127907817012681192628757648731259649999920391389042112000]}}, 'schur_multiplier': [9], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [9, 9], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{333}:C_9', 'transitive_degree': 333, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '9.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [6], 'aut_gens': [[1], [2]], 'aut_group': '6.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 6, 'aut_permdeg': 5, 'aut_perms': [27], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', '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], [3, 1, 2], [9, 1, 6]], 'center_label': '9.1', 'center_order': 9, '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': ['3.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 9, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 6]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 9]], 'label': '9.1', 'linC_count': 6, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C9', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 9, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 9, 'order_factorization_type': 2, 'order_stats': [[1, 1], [3, 2], [9, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [9], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [6, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -3, -3, 6]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41907234971425459]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [78572]}, 'Perm': {'d': 9, 'gens': [357120, 80884]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [9], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9', 'transitive_degree': 9, 'wreath_data': None, 'wreath_product': False}
