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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1336.5', 'ambient_counter': 5, 'ambient_order': 1336, 'ambient_tex': 'C_2^2\\times C_{334}', 'central': True, 'central_factor': False, 'centralizer_order': 1336, 'characteristic': True, 'core_order': 167, 'counter': 16, 'cyclic': True, 'direct': True, 'hall': 167, 'label': '1336.5.8.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '8.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3', 'simple': True, 'solvable': True, 'special_labels': ['C3'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '167.1', 'subgroup_hash': 1, 'subgroup_order': 167, 'subgroup_tex': 'C_{167}', 'supersolvable': True, 'sylow': 167}
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gps_subgroup_data • Show schema
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{'ambient': '1336.5', 'aut_centralizer_order': 168, 'aut_label': '8.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '166.2', 'aut_weyl_index': 168, 'centralizer': '1.a1.a1', 'complements': ['167.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.a1.b1', '4.a1.c1', '4.a1.d1', '4.a1.e1', '4.a1.f1', '4.a1.g1'], 'contains': ['1336.a1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1543, 1543, 4747, 4747, 3263, 3263, 4110, 4110], 'generators': [8], 'label': '1336.5.8.a1.a1', 'mobius_quo': -1, 'mobius_sub': -8, 'normal_closure': '8.a1.a1', 'normal_contained_in': ['4.a1.e1', '4.a1.c1', '4.a1.a1', '4.a1.b1', '4.a1.f1', '4.a1.g1', '4.a1.d1'], 'normal_contains': ['1336.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.a1.a1', 'projective_image': '8.5', 'quotient_action_image': '1.1', 'quotient_action_kernel': '8.5', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '8.a1.a1', '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': '167.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 166, 'aut_gen_orders': [166], 'aut_gens': [[1], [67]], 'aut_group': '166.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 166, 'aut_permdeg': 85, 'aut_perms': [3353220009928858569336442912103262493625458976355624940065453228977352138859978594316152307777012044859965440000000000000000000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [167, 1, 166, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{166}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 166, 'autcent_group': '166.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 166, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{166}', '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], [167, 1, 166]], 'center_label': '167.1', 'center_order': 167, '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': ['167.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [167, 1, 166, 1]], 'element_repr_type': 'PC', 'elementary': 167, 'eulerian_function': 1, 'exponent': 167, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 83], 'factors_of_order': [167], 'faithful_reps': [[1, 0, 166]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '167.1', 'hash': 1, 'hyperelementary': 167, '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': 166, 'irrQ_dim': 166, 'irrR_degree': 2, 'irrep_stats': [[1, 167]], 'label': '167.1', 'linC_count': 166, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 166, 'linQ_degree_count': 1, 'linQ_dim': 166, 'linQ_dim_count': 1, 'linR_count': 83, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C167', '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': 167, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 167, 'order_factorization_type': 1, 'order_stats': [[1, 1], [167, 166]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 166, 'outer_gen_orders': [166], 'outer_gen_pows': [0], 'outer_gens': [[67]], 'outer_group': '166.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 166, 'outer_permdeg': 85, 'outer_perms': [3353220009928858569336442912103262493625458976355624940065453228977352138859978594316152307777012044859965440000000000000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{166}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 167, 'pgroup': 167, 'primary_abelian_invariants': [167], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [166, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -167]}, 'Lie': [{'d': 1, 'q': 167, 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 167, 'gens': [4657631]}, 'Perm': {'d': 167, 'gens': [1494612823159220602834727446116490698483472390775123871527783668178402860636296923089193082650613568595129745165679275491241374525405329019506357391567179908243866583487687583237455390476159062872416219779074004057956107619164239470402730489650248203178564648960000000000000000000000000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [167], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{167}', 'transitive_degree': 167, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '1336.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 6972, 'aut_gen_orders': [332, 332], 'aut_gens': [[1, 2, 4], [2, 668, 543], [1, 669, 690]], 'aut_group': None, 'aut_hash': 221315993921501907, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 27888, 'aut_permdeg': 173, 'aut_perms': [860054680900566837436981483837440137907374861835005471316781269307041004506049997484512173545914203721998223252262272196488195462441646167977729926069016168957431686938977674825992525345386247699857657701360486770791460315824580256049847779687709199201769047859860795965060464947232768000000000000000000000000000, 1255532902951954906069927790264889980107866801063127784349073541805519205384159542543131533188387913060923177734775814277222461369675097098055622906459237344242263884428294891164797969004379489182940028925589767386266235683635306733794789055443886833839177782815127896244323593105039052556442336528920420940313], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [167, 1, 166, 1], [334, 1, 1162, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{166}\\times \\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6972, 'autcent_group': None, 'autcent_hash': 221315993921501907, 'autcent_nilpotent': False, 'autcent_order': 27888, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_{166}\\times \\PSL(2,7)', '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, 7], [167, 1, 166], [334, 1, 1162]], 'center_label': '1336.5', 'center_order': 1336, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '167.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['167.1', 1], ['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [167, 1, 166, 1], [334, 1, 166, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 28057, 'exponent': 334, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 7, 83], 'factors_of_order': [2, 167], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1336.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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': None, 'irrep_stats': [[1, 1336]], 'label': '1336.5', 'linC_count': None, 'linC_degree': 3, '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': 'C2^2*C334', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 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': 1336, 'number_divisions': 16, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 32, 'number_subgroups': 32, 'old_label': None, 'order': 1336, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [167, 166], [334, 1162]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6972, 'outer_gen_orders': [1162, 249], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 669, 161], [670, 2, 389]], 'outer_group': None, 'outer_hash': 221315993921501907, 'outer_nilpotent': False, 'outer_order': 27888, 'outer_permdeg': 173, 'outer_perms': [858784504844743927773455147587488467679696188460553907147394946775310838909966548564937593222499356808050200897163224966081878157690107810627181152177211856613377522845351891954715919545013377435884900122421248657324780188852323412253150994292006723355072724728465217513783486085857280000000000000000000000000000, 1283227242379590957271834886010265320064651581530279566506970721040954511381426344308192804398274779844829400828226405078049201382991162369160676120835331451896276452822380889641965637546516429807635370307997287105536677331877397689092770651021795535962949876289659643153510869621700616007974912000000000000000000], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_{166}\\times \\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 173, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 167], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [166, 8]], 'representations': {'PC': {'code': 391815898341, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -167, 34]}, 'Perm': {'d': 173, 'gens': [213455108077438865629072570145733886730056159330291227886899710221263324938130981514753340236723864719151973034287306573083301055694802251980973629541579310661401455397074590303866009781148657954570396550703618437210885875866741044575478989978191912006970522334798649753600000000000000000000000000000000000000000, 7257415615307998967396728211129263114716991681296451376543577798900561843401706157852350749242617459511490991237838520776666022565442753025328900773207510902400430280058295603966612599658257104398558294257568966313439612262571094946806711205568880457193340212661452800000000000000000000000000000000000000000, 252607574497319838753801886917134114678628321660161899636045172255501630469951484784279524860515748677740723676917456344471493998101035608260664837215715659672830848592352788164518364067464570288846442542901808782229015393754650015551921726612213033664926565007360000000000000000000000000000000000000000, 1494612823159220602834727446116490698483472390775123871527783668178402860636296923089193082650613568595129745165679275491241374525405329019506357391567179908243866583487687583237455390476159062872416219779074004057956107619164239470402730489650248203178564648960000000000000000000000000000000000000000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 334], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{334}', 'transitive_degree': 1336, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 84, 'aut_gen_orders': [3, 3], 'aut_gens': [[1, 2, 4], [4, 5, 3], [2, 4, 1]], 'aut_group': '168.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 7, 'aut_perms': [4361, 244], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 168, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\PSL(2,7)', '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, 7]], 'center_label': '8.5', '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': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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': '8.5', 'linC_count': 28, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 28, 'linQ_dim': 3, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3', 'ngens': 3, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 8, 'number_divisions': 8, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 5, 3], [2, 4, 1]], 'outer_group': '168.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 168, 'outer_permdeg': 7, 'outer_perms': [4361, 244], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -2, 2, 2]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16482, 16322, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 13286, 13933]}, 'Perm': {'d': 6, 'gens': [120, 6, 1]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
