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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '49152.nz', 'ambient_counter': 364, 'ambient_order': 49152, 'ambient_tex': 'C_2^8.C_2^3:S_4', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': True, 'core_order': 24576, 'counter': 2, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '49152.nz.2.A', 'maximal': True, 'maximal_normal': True, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': ['C1'], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': None, 'subgroup_hash': 6391412274801050291, 'subgroup_order': 24576, 'subgroup_tex': 'C_2^8.C_2^4.C_6', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '49152.nz', 'aut_centralizer_order': None, 'aut_label': '2.A', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': '2.A', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [6227383801, 25903229683158464681647, 40279687, 51468682220474400120, 40284847, 79804429772043678105600, 7, 1313901388807, 122000787877207687, 2796019446976, 51091297865491939320, 51091297859137536121, 216288449189794056862080, 355687555466897], 'label': '49152.nz.2.A', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.A', 'normal_contained_in': [], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '2.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2.A', 'subgroup_fusion': None, 'weyl_group': None}
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label None does not appear in gps_groups
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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': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [4, 8, 8, 4, 4, 8, 4, 4], 'aut_gens': [[324406735719760038223312, 432475093854571140547200, 540643971961189621313278], [350207539868583190921559, 432475093501673457674880, 594596007252991883082832], [540643972320980569025153, 270240740219555839250062, 324406870166824380218879], [108169512655792323479633, 296041409922622197674176, 594545294049386820577903], [378358771008778949082942, 458377823439891887592361, 594545037601546968681456], [540643971962503522696918, 296041409921314523664022, 378359027100937472873038], [324406736072657760654719, 270240483770221629744126, 134021272942553337763183], [162121291494169263362633, 242089496277402335390776, 540593137470961203551872], [378307558065819442625153, 432474715762926227158456, 566496366440129423015752]], 'aut_group': None, 'aut_hash': 7829140440281219398, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6291456, 'aut_permdeg': 384, 'aut_perms': [43303151694076089204805970191455462000733437503205691761038341672224970016008334110778416392929119816771756377732307783739486898895836134619290156141289503735209657605841697108473385711055759358807759948312956412991082072773180498141127495237979775553573672392870855961831372476392960369368995115719840113597843475667424903528500055131972599248038656634461293449105342524572703143813972725734887211399987444458682514821268325943881886143278560838797024703031891250636200366124395563987315040717385302138265589372900460238049126734003959026262436452069098121617800937898609599353307966437613944812791053763782300836909620029976411503839570496819408660240201450851051928805120268062228491379712382050413588594394285379955229426195979853102024761862936262596472272401810554232862343442329738824255152369537737157035958571942689373, 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108357593312725036210530889245649585370476869804776550509526524623504218875347065560676234266580553743541436359048106245711615589426026413487465885883893560730567863024991417175589043069893701987838547878655409269318038026278328457690161940704287381888492208820016234189202910935336010104128237671033156164414611707356779816940664742676707706092062815988633140290435755960795105762546439953418596257077373554814681666032654011833413411261515073081792621206548599494273916659007940230861617499483990097918286901849702383386926489652323175278872951430109337112282702827216109781163327082222840434370165007149777015166663880050454381817109442514750031363224746157718388596271993665829591946857214902353137566883503216314078829121374910456053564406189060741040313423258794639042671949282237015465256573882931078726234917470618720111], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 3, 2, 2], [2, 6, 1, 5], [2, 6, 2, 1], [2, 24, 2, 2], [2, 48, 1, 2], [2, 48, 2, 1], [2, 48, 4, 2], [2, 96, 1, 1], [2, 96, 2, 2], [2, 96, 4, 1], [2, 192, 2, 1], [3, 2048, 1, 1], [4, 12, 4, 1], [4, 16, 2, 1], [4, 24, 2, 1], [4, 24, 4, 1], [4, 32, 1, 1], [4, 48, 2, 2], [4, 48, 4, 2], [4, 48, 8, 1], [4, 96, 1, 4], [4, 96, 4, 2], [4, 96, 8, 2], [4, 192, 2, 6], [4, 192, 4, 4], [4, 768, 8, 1], [6, 2048, 1, 1], [6, 2048, 2, 1], [8, 768, 2, 2], [8, 768, 4, 2], [8, 1536, 4, 1], [12, 2048, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5.C_2^6.C_6.C_2^6.C_2^3', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 1657002828416672943, 'autcentquo_nilpotent': False, 'autcentquo_order': 1572864, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^4.C_2^6.C_6.C_2^4.C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 3, 6], [2, 6, 7], [2, 24, 4], [2, 48, 12], [2, 96, 9], [2, 192, 2], [3, 2048, 1], [4, 12, 4], [4, 16, 2], [4, 24, 6], [4, 32, 1], [4, 48, 20], [4, 96, 28], [4, 192, 28], [4, 768, 8], [6, 2048, 3], [8, 768, 12], [8, 1536, 4], [12, 2048, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': None, 'commutator_count': 2, 'commutator_label': None, 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 15, 'conjugacy_classes_known': True, 'counter': 364, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 6], [2, 6, 1, 7], [2, 24, 1, 4], [2, 48, 1, 12], [2, 96, 1, 9], [2, 192, 1, 2], [3, 2048, 1, 1], [4, 12, 1, 4], [4, 16, 2, 1], [4, 24, 1, 6], [4, 32, 1, 1], [4, 48, 1, 20], [4, 96, 1, 24], [4, 96, 2, 2], [4, 192, 1, 28], [4, 768, 1, 8], [6, 2048, 1, 1], [6, 2048, 2, 1], [8, 768, 1, 12], [8, 1536, 1, 4], [12, 2048, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 24, 'exponents_of_order': [14, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 32], [48, 1, 2]], 'familial': False, 'frattini_label': '64.267', 'frattini_quotient': '768.1090235', 'hash': 1410416440151032101, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [4, 6, 4], 'inner_gens': [[324406735719760038223312, 242089631080154105851441, 134021651038289450980417], [540592881021807489815632, 432475093854571140547200, 108118421355137293521343], [133970438453813304969479, 242089752728044219627446, 540643971961189621313278]], 'inner_hash': 7803420404517911634, 'inner_nilpotent': False, 'inner_order': 24576, 'inner_split': False, 'inner_tex': 'C_2^5.C_2^6.D_6', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 4], [2, 2], [3, 28], [4, 4], [6, 14], [8, 8], [12, 68], [24, 26], [48, 10]], 'label': '49152.nz', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'C2^8.C2^3:S4', 'ngens': 3, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 64, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 164, 'number_divisions': 158, 'number_normal_subgroups': 58, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 49152, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 1983], [3, 2048], [4, 15424], [6, 6144], [8, 15360], [12, 8192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 122000787877207687, 0, 0, 0, 0, 0], 'outer_gens': [[404210921837720710849777, 458378079536141476605487, 620448280086831229151863], [404210921837720710849777, 432475093854571140547200, 594545293696489010704056], [324406614430347057487319, 432474971855084750966400, 540643971962503522702073], [324406614429045649699192, 432474971855084750966400, 540643971961189581028438], [324406614430347057487319, 432474971855084750971567, 540643971961189581033593], [350207540225758877340119, 458378201535627866186287, 566444897757902848228433], [404210787743566546025022, 432475093854571140547200, 540593137473750908865222]], 'outer_group': '256.55643', 'outer_hash': 55643, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 14, 'outer_perms': [6267305934, 289, 288, 367921, 6267744415, 13412851200, 33265209601], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5:D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 4], [2, 2], [3, 28], [4, 4], [6, 14], [8, 2], [12, 68], [16, 1], [24, 24], [32, 1], [48, 9], [96, 1]], 'representations': {'PC': {'code': '2319815593751020901071564699007412310409578987935505079196972227715248488370494090811814336314800143153624793454078308928055930110114661544227790296150143155469567843905100964503833176390106686919072216157650366881847670531640413954676875898938432929677786889440043810755505817664947406105963979799760896', 'gens': [1, 2, 4, 5, 7, 8, 10, 12, 13, 14, 15], 'pres': [15, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 379260, 227101, 76, 1347122, 22172, 1869843, 887058, 530133, 3314704, 1025569, 620359, 89299, 214, 1416965, 708500, 380195, 1515786, 1743861, 713196, 178971, 6816, 2139847, 528502, 719317, 159412, 33667, 25777, 352, 421208, 236543, 105338, 4780809, 1039524, 606189, 48054, 24069, 13299, 11514, 444, 2280970, 1172185, 522760, 2764811, 1382426, 641561, 2096652, 1984347, 374442, 529228, 909763, 7354814, 3808379, 716219, 374474, 72104, 52319, 10949]}, 'Perm': {'d': 24, 'gens': [324406735719760038223312, 432475093854571140547200, 540643971961189621313278]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 32, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^8.C_2^3:S_4', 'transitive_degree': 24, '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': '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}
