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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '14880.a', 'ambient_counter': 1, 'ambient_order': 14880, 'ambient_tex': '\\PSL(2,31)', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 1, 'counter': 13, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '14880.a.930.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '930.b1.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': 930, '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': '16.7', 'subgroup_hash': 7, 'subgroup_order': 16, 'subgroup_tex': 'D_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '14880.a', 'aut_centralizer_order': None, 'aut_label': '930.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '7440.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['465.a1.a1'], 'contains': ['1860.a1.a1', '1860.b1.a1'], 'core': '14880.a1.a1', 'coset_action_label': None, 'count': 465, 'diagramx': [9522, -1, 7764, -1, 8320, -1, 7746, -1], 'generators': [121330, 97771, 366059, 460898], 'label': '14880.a.930.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '465.a1.a1', 'old_label': '930.b1.a1', 'projective_image': '14880.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '930.b1.a1', 'subgroup_fusion': None, 'weyl_group': '16.7'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 8], 'aut_gens': [[1, 2], [1, 10], [1, 14], [3, 2]], 'aut_group': '32.43', 'aut_hash': 43, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [10824, 18366, 7679], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 2, 1], [4, 2, 1, 1], [8, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_8:C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 2], [4, 2, 1], [8, 2, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '8.3', 'commutator_count': 1, 'commutator_label': '4.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 2], [4, 2, 1, 1], [8, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 8, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 2]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '4.2', 'hash': 7, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4], 'inner_gens': [[1, 14], [5, 2]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 3]], 'label': '16.7', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D8', 'ngens': 2, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 7, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 11, 'number_subgroups': 19, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 9], [4, 2], [8, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 2], 'outer_gens': [[1, 10], [3, 2]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1]], 'representations': {'PC': {'code': 2499614, 'gens': [1, 2], 'pres': [4, -2, 2, -2, -2, 113, 21, 146, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20182740, 20401702]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [2059, 2042]}, 'Perm': {'d': 8, 'gens': [23616, 143, 16577, 5167]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_8', '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': '1.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 14880, 'aut_gen_orders': [30, 3], 'aut_gens': [[625614, 28891], [476586, 826217], [218722, 389749]], 'aut_group': '29760.b', 'aut_hash': 3439321479274393560, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 29760, 'aut_permdeg': 465, 'aut_perms': [99739083677262090663323012692693596214509122027449598797993575907799528771014843575541047405677579501122894164106655130106740643689905795666734695214668354640720348624073390273187518505244901910908276945789718474006222349603532001973120854298786762630654743093928275079381874575833832369782866815105189098729562368170030002047409517200665885985946621664669780764244156212905631533087567969800102143709321528301849736260162129765805583094252951398333923047741309346321818743569798186142457878427815638221624554226611576337884777589680478866074366833824318371434860820578324122996836396564107843099208032612777310968132847559283567122977819752355538395829426972265314936389141821455648908300785034971393746852262727723292332695422007532906875326753834256124446878734914581538905911316980173098188417559748414517140942299555027738393922451820223567578982494125173299162791965118414788658488343371465793977076674251852396964302785620430207811339394536964497285729277496726485965743594137168386426291951783699049935538217186881624216297316696630, 80249114119569950538037255454936204735754884794616388276394063285249845471651794638001904693670917372199087526941623760951464825027456933408239528929280732112632115275884655534074960062866277293728873118867511144212424826335381582357065870803313278383763425768920336721162050132914174307620715192531697691833972965423176216565278332034168752481786340924093934240503781502102381226977079483141344983956072766978416744401898885143241526339147137788149714008785100234351267547681288100776191267183432372556218957500733224480687754451631277818552583633591109131245649969788241225294069235806725463322427031087608625843159015841081270621052065134105687623715712028972438612302924415883089831505100851753744056803583273858781347555772156603360388016019172729617692109323449975245002288585156351166335370608615712491047737289226397188762383543573359765097318578662456377474298981593719841850118195479448042387078831833538865249274865299732326493142720504680692672659569869029846008130177487477769337155021979749209216514845012604369596743697628430], 'aut_phi_ratio': 7.75, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 465, 1, 1], [3, 992, 1, 1], [4, 930, 1, 1], [5, 992, 1, 2], [8, 930, 1, 2], [15, 992, 1, 4], [16, 930, 1, 4], [31, 480, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PGL(2,31)', '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': 14880, 'autcentquo_group': '29760.b', 'autcentquo_hash': 3439321479274393560, 'autcentquo_nilpotent': False, 'autcentquo_order': 29760, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PGL(2,31)', 'cc_stats': [[1, 1, 1], [2, 465, 1], [3, 992, 1], [4, 930, 1], [5, 992, 2], [8, 930, 2], [15, 992, 4], [16, 930, 4], [31, 480, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '14880.a', 'commutator_count': 1, 'commutator_label': '14880.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['14880.a'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 465, 1, 1], [3, 992, 1, 1], [4, 930, 1, 1], [5, 992, 2, 1], [8, 930, 2, 1], [15, 992, 4, 1], [16, 930, 4, 1], [31, 480, 2, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': 7135, 'exponent': 7440, 'exponents_of_order': [5, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 31], 'factors_of_order': [2, 3, 5, 31], 'faithful_reps': [[15, 0, 2], [30, 1, 7], [31, 1, 1], [32, 1, 7]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '14880.a', 'hash': 7311360894803160256, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 7440, 'inner_gen_orders': [15, 3], 'inner_gens': [[625614, 21389], [106692, 28891]], 'inner_hash': 7311360894803160256, 'inner_nilpotent': False, 'inner_order': 14880, 'inner_split': True, 'inner_tex': '\\PSL(2,31)', 'inner_used': [1, 2], 'irrC_degree': 15, 'irrQ_degree': 30, 'irrQ_dim': 30, 'irrR_degree': 30, 'irrep_stats': [[1, 1], [15, 2], [30, 7], [31, 1], [32, 7]], 'label': '14880.a', 'linC_count': 2, 'linC_degree': 15, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 30, 'linQ_degree_count': 2, 'linQ_dim': 30, 'linQ_dim_count': 2, 'linR_count': 8, 'linR_degree': 30, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'PSL(2,31)', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 17, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 18, 'number_divisions': 9, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 23, 'number_subgroup_classes': 29, 'number_subgroups': 15413, 'old_label': None, 'order': 14880, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 465], [3, 992], [4, 930], [5, 1984], [8, 1860], [15, 3968], [16, 3720], [31, 960]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [666265], 'outer_gens': [[884983, 130600]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': None, 'perfect': True, 'permutation_degree': 32, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': True, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [30, 2], [31, 1], [32, 1], [60, 1], [64, 1], [120, 1], [128, 1]], 'representations': {'Lie': [{'d': 2, 'q': 31, 'gens': [625614, 28891], 'family': 'PSL'}, {'d': 2, 'q': 31, 'family': 'PSU'}, {'d': 3, 'q': 31, 'gens': [7676020260497, 26653738683], 'family': 'Omega'}, {'d': 3, 'q': 31, 'family': 'POmega'}, {'d': 2, 'q': 31, 'family': 'PSigmaL'}], 'Perm': {'d': 32, 'gens': [539991602744554192829015036719637, 8488419390677809895742993284734362]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\PSL(2,31)', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
