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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '34992.la', 'ambient_counter': 287, 'ambient_order': 34992, 'ambient_tex': 'C_3^5.S_3^2:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 729, 'characteristic': False, 'core_order': 81, 'counter': 315, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '34992.la.144.bb1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '144.bb1', 'outer_equivalence': True, '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': 144, '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': '243.61', 'subgroup_hash': 61, 'subgroup_order': 243, 'subgroup_tex': 'C_3^3\\times C_9', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '34992.la', 'aut_centralizer_order': None, 'aut_label': '144.bb1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '48.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['48.a1', '48.h1', '48.i1', '72.k1', '72.ba1', '72.bs1', '72.ca1', '72.cx1'], 'contains': ['432.a1', '432.l1', '432.m1', '432.p1', '432.q1', '432.bc1', '432.bd1', '432.be1', '432.bf1', '432.bl1', '432.bm1'], 'core': '432.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [7183, -1, 1689, -1], 'generators': [168, 144, 3888, 12960], 'label': '34992.la.144.bb1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '48.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1', 'old_label': '144.bb1', 'projective_image': '34992.la', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '144.bb1', 'subgroup_fusion': None, 'weyl_group': '24.15'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '243.61', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 936, 'aut_gen_orders': [3, 3, 13, 3, 6, 3, 3, 3, 3, 2], 'aut_gens': [[1, 3, 9, 27], [1, 3, 9, 203], [82, 165, 90, 189], [16, 88, 82, 45], [1, 3, 9, 200], [163, 3, 9, 71], [163, 3, 171, 27], [1, 3, 9, 112], [1, 3, 9, 189], [163, 84, 90, 189], [86, 3, 94, 27]], 'aut_group': '49128768.a', 'aut_hash': 6829789377147908262, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 49128768, 'aut_permdeg': 83, 'aut_perms': [1908345068815238937354193556108802943040289821207358369935475573264099584066390679277349104248117090040053137229561290129040, 10241132782696443443314086248258889552096493367769102167529388866584357555382939086758781906325662260631567071779659212225560, 37035478869123520278377129942438887003188538286868829049428777569364336622967055678522614782155528800714990680513870212518360, 1908273499386474982262768441594129856578615409553425176453440429965773946845216973078966775339189437147906883606088029592526, 13204444168824595010106793923441791852908411783546426115544095271979631773747166353248380937167628242599290599697832280947729, 13817847732840502609743152163801392573762640848045621179000516637462697127650716857308509022251237468762506907771731735866264, 64858960289985940498755964324416063228982061190913650903130660063114833994513293279877632171543220707079198020035819792334, 1966316342552749727902267370651089539934028368898663442298455346878080420882197656983932990174027785751270680500101743172200, 7409010302580350760616116636121941329104605923302002027837948965100526332858784624136965851626547461345101456949318786112648, 268866802959050458010886843533925454282204483500860054481959047180565072333878513949201756680960901618043809561139011657808], 'aut_phi_ratio': 303264.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 1, 78, 1], [9, 1, 162, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.(C_2\\times C_3^3:\\GL(3,3))', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 936, 'autcent_group': '49128768.a', 'autcent_hash': 6829789377147908262, 'autcent_nilpotent': False, 'autcent_order': 49128768, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_3^4.(C_2\\times C_3^3:\\GL(3,3))', '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, 80], [9, 1, 162]], 'center_label': '243.61', 'center_order': 243, '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': ['3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 61, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 3], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 40], [9, 1, 6, 27]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 40, 'exponent': 9, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '81.15', 'hash': 61, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 3, 9, 27], [1, 3, 9, 27], [1, 3, 9, 27], [1, 3, 9, 27]], '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, 243]], 'label': '243.61', 'linC_count': None, 'linC_degree': 4, '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': 'C3^3*C9', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 243, 'number_divisions': 68, 'number_normal_subgroups': 396, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 396, 'number_subgroups': 396, 'old_label': None, 'order': 243, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 80], [9, 162]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 936, 'outer_gen_orders': [3, 3, 13, 3, 6, 3, 3, 3, 3, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 3, 9, 203], [82, 165, 90, 189], [16, 88, 82, 45], [1, 3, 9, 200], [163, 3, 9, 71], [163, 3, 171, 27], [1, 3, 9, 112], [1, 3, 9, 189], [163, 84, 90, 189], [86, 3, 94, 27]], 'outer_group': '49128768.a', 'outer_hash': 6829789377147908262, 'outer_nilpotent': False, 'outer_order': 49128768, 'outer_permdeg': 83, 'outer_perms': [1908345068815238937354193556108802943040289821207358369935475573264099584066390679277349104248117090040053137229561290129040, 10241132782696443443314086248258889552096493367769102167529388866584357555382939086758781906325662260631567071779659212225560, 37035478869123520278377129942438887003188538286868829049428777569364336622967055678522614782155528800714990680513870212518360, 1908273499386474982262768441594129856578615409553425176453440429965773946845216973078966775339189437147906883606088029592526, 13204444168824595010106793923441791852908411783546426115544095271979631773747166353248380937167628242599290599697832280947729, 13817847732840502609743152163801392573762640848045621179000516637462697127650716857308509022251237468762506907771731735866264, 64858960289985940498755964324416063228982061190913650903130660063114833994513293279877632171543220707079198020035819792334, 1966316342552749727902267370651089539934028368898663442298455346878080420882197656983932990174027785751270680500101743172200, 7409010302580350760616116636121941329104605923302002027837948965100526332858784624136965851626547461345101456949318786112648, 268866802959050458010886843533925454282204483500860054481959047180565072333878513949201756680960901618043809561139011657808], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_3^4.(C_2\\times C_3^3:\\GL(3,3))', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3, 9], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 40], [6, 27]], 'representations': {'PC': {'code': 16392, 'gens': [1, 2, 3, 4], 'pres': [5, -3, 3, 3, 3, -3, 78]}, 'GLZN': {'d': 2, 'p': 54, 'gens': [157483, 158437, 211285, 7272037, 2991835]}, 'GLZq': {'d': 2, 'q': 27, 'gens': [196840, 19687, 203896, 19693, 19927]}, 'Perm': {'d': 18, 'gens': [357120, 711374856192000, 174356582400, 79833600, 80884]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3, 9], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3\\times C_9', 'transitive_degree': 243, '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 72, 'aut_gen_orders': [24, 24, 24, 6], 'aut_gens': [[1, 6, 24, 72, 432, 3888], [991, 25194, 288, 12360, 19872, 3456], [4051, 33942, 168, 33096, 19440, 432], [27391, 33246, 288, 15648, 4752, 30240], [193, 19578, 48, 14376, 1728, 15552]], 'aut_group': None, 'aut_hash': 8249960914196842203, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 279936, 'aut_permdeg': 225, 'aut_perms': [12416186718417439707976130326771495478115055927079727973375458468772487975024929986429658617982209757956210450514604394351842710134237857367802128126757504513719723941385475580000041014723426668771131700651127999877170341423002699526054566849850008217013229845751476276349608308167602969608322123371665576284422899367624812227071045704384803882498586726769882504931807614602858443919810787237642700289359863348676047959844947085420134, 10259789452021271532715314977856382435876181773987130586767793934537935032493619889104865853055618230462611959198690513221526671736787331983592434728491273359144972037507133908833913675169136005946691896131222890475044319406816559447928657553977282054055217788045592825480019104410189085690867320737984113922912656708841281588389872051647099921439212159508320469494918034805962391167080882100719895360371965544073142329448736310578033, 3167868734547849240268613865311377189334110681812543809538992714410319996105431237188656427190615237871108063115150536070098577307961923953001955053736727273579650059472749965942560530841494868868224474362597997165133163872125283312040850720649059910885676903189157944722415516223461062677752646524049063129284793279574358114362564555943496184566007655962351035153732830640391801762138561314189804595074772580569283068716958888749451, 4413507588187192962059983432143892879880017238085792057338070460899171250403188558466246164259080888676791068476953092632236249848489240591673665175264059242362311253318251477465892744638796502541404299614819090993945355555028873000230374882883950313845880404770846180563161720393261037110862992317556924325557834625396708355156945693490647426899225627567661455463456800709786687068798511337354439993167187059925183529845562867181255], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 54, 4, 1], [2, 81, 1, 1], [2, 729, 1, 1], [3, 4, 2, 2], [3, 8, 4, 1], [3, 9, 1, 2], [3, 16, 2, 1], [3, 36, 2, 2], [4, 1458, 2, 1], [6, 36, 2, 1], [6, 81, 1, 4], [6, 108, 4, 2], [6, 162, 4, 2], [6, 216, 4, 1], [6, 324, 2, 3], [6, 324, 4, 2], [6, 729, 1, 2], [9, 12, 2, 1], [9, 24, 2, 1], [9, 24, 4, 1], [9, 36, 2, 2], [9, 48, 2, 1], [9, 48, 4, 2], [9, 72, 4, 2], [9, 144, 2, 2], [12, 1458, 2, 2], [18, 108, 2, 1], [18, 216, 2, 1], [18, 324, 2, 2], [18, 324, 4, 3], [18, 648, 4, 3]], 'aut_supersolvable': False, 'aut_tex': 'C_9^2.(C_3\\times C_6).C_4.C_6.C_2^3', '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': 72, 'autcentquo_group': None, 'autcentquo_hash': 8249960914196842203, 'autcentquo_nilpotent': False, 'autcentquo_order': 279936, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_3:S_3.C_4.C_6.C_2^3', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 54, 4], [2, 81, 1], [2, 729, 1], [3, 4, 4], [3, 8, 4], [3, 9, 2], [3, 16, 2], [3, 36, 4], [4, 1458, 2], [6, 36, 2], [6, 81, 4], [6, 108, 8], [6, 162, 8], [6, 216, 4], [6, 324, 14], [6, 729, 2], [9, 12, 2], [9, 24, 6], [9, 36, 4], [9, 48, 10], [9, 72, 8], [9, 144, 4], [12, 1458, 4], [18, 108, 2], [18, 216, 2], [18, 324, 16], [18, 648, 12]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '34992.la', 'commutator_count': 1, 'commutator_label': '1458.1403', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 287, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 54, 1, 4], [2, 81, 1, 1], [2, 729, 1, 1], [3, 4, 1, 4], [3, 8, 1, 4], [3, 9, 2, 1], [3, 16, 1, 2], [3, 36, 2, 2], [4, 1458, 1, 2], [6, 36, 1, 2], [6, 81, 2, 2], [6, 108, 1, 8], [6, 162, 2, 4], [6, 216, 1, 4], [6, 324, 1, 2], [6, 324, 2, 6], [6, 729, 2, 1], [9, 12, 1, 2], [9, 24, 1, 6], [9, 36, 2, 2], [9, 48, 1, 10], [9, 72, 2, 4], [9, 144, 2, 2], [12, 1458, 2, 2], [18, 108, 1, 2], [18, 216, 1, 2], [18, 324, 1, 4], [18, 324, 2, 6], [18, 648, 1, 4], [18, 648, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 47174400, 'exponent': 36, 'exponents_of_order': [7, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 8], [48, 1, 10]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '3888.da', 'hash': 5680935221992372238, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 4, 3, 6, 9, 9], 'inner_gens': [[1, 9954, 24, 2976, 30240, 7776], [9949, 6, 192, 12648, 34560, 4752], [1, 342, 24, 72, 432, 3888], [1489, 26382, 24, 72, 3456, 31104], [9073, 4758, 24, 936, 432, 3888], [31105, 3030, 24, 7848, 432, 3888]], 'inner_hash': 5680935221992372238, 'inner_nilpotent': False, 'inner_order': 34992, 'inner_split': False, 'inner_tex': 'C_3^5.S_3^2:C_2^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 24], [2, 6], [4, 48], [8, 24], [12, 8], [16, 6], [24, 12], [48, 10]], 'label': '34992.la', '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': 'C3^5.S3^2:C2^2', 'ngens': 11, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 55, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 138, 'number_divisions': 102, 'number_normal_subgroups': 51, 'number_subgroup_autclasses': 1039, 'number_subgroup_classes': 3316, 'number_subgroups': 262372, 'old_label': None, 'order': 34992, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 1035], [3, 242], [4, 2916], [6, 9414], [9, 1944], [12, 5832], [18, 13608]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [29396, 9304], 'outer_gens': [[27223, 870, 288, 16080, 15552, 4320], [27223, 30030, 144, 7320, 19872, 34560]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [2, 13], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 18], [8, 24], [12, 8], [16, 10], [24, 12], [32, 2], [48, 10]], 'representations': {'PC': {'code': '82651078456115093706539709447564173678642112364455750636321487807964986739382040437891522142129151144556620013421811680837860696018693025441040751194035368664440904363449505996417631', 'gens': [1, 3, 5, 6, 8, 10], 'pres': [11, 2, 3, 2, 2, 3, 2, 3, 3, 3, 3, 3, 22, 328484, 385024, 90, 969939, 285398, 1786, 257, 196421, 814984, 139155, 101810, 192, 25878, 2184, 1887, 2661127, 76050, 506909, 25384, 4286, 348, 1283048, 427710, 21425, 3627, 855369, 855380, 87151, 285162, 47584, 438, 2822698, 287528, 235267, 39269]}, 'Perm': {'d': 24, 'gens': [56302839416196361376666, 28312052076683211075247, 3477130667804320934402]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^5.S_3^2:C_2^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
