-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '960.2664', 'ambient_counter': 2664, 'ambient_order': 960, 'ambient_tex': 'C_{12}.(C_4\\times D_{10})', 'central': False, 'central_factor': False, 'centralizer_order': 40, 'characteristic': False, 'core_order': 120, 'counter': 21, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.2664.4.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '4.g1.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': 4, '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': '240.64', 'subgroup_hash': 64, 'subgroup_order': 240, 'subgroup_tex': 'C_{30}.D_4', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '960.2664', 'aut_centralizer_order': None, 'aut_label': '4.g1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '24.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.a1.a1', '8.h1.a1', '8.h1.b1', '12.c1.a1', '20.c1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': None, 'generators': [1, 640, 192, 2, 480, 16], 'label': '960.2664.4.g1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '4.g1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.g1.a1', 'subgroup_fusion': None, 'weyl_group': None}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '40.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 2, 6, 4, 2, 2], 'aut_gens': [[1, 4, 24], [135, 124, 72], [123, 140, 24], [5, 4, 24], [15, 6, 24], [121, 4, 24], [3, 4, 24]], 'aut_group': '768.327978', 'aut_hash': 327978, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 768, 'aut_permdeg': 15, 'aut_perms': [375696921622, 120, 375696923047, 375700590720, 186818607360, 87668945287], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [3, 2, 1, 1], [4, 6, 4, 1], [5, 1, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [10, 1, 4, 1], [10, 1, 8, 1], [10, 2, 8, 1], [15, 2, 4, 1], [20, 6, 16, 1], [30, 2, 4, 1], [30, 2, 8, 1], [30, 2, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times C_2^4:D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.260', 'autcent_hash': 260, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [3, 2, 1], [4, 6, 4], [5, 1, 4], [6, 2, 7], [10, 1, 12], [10, 2, 8], [15, 2, 4], [20, 6, 16], [30, 2, 28]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 64, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['48.19', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [3, 2, 1, 1], [4, 6, 2, 2], [5, 1, 4, 1], [6, 2, 1, 3], [6, 2, 2, 2], [10, 1, 4, 3], [10, 2, 4, 2], [15, 2, 4, 1], [20, 6, 8, 2], [30, 2, 4, 3], [30, 2, 8, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 60, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '60.11', 'hash': 64, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 1], 'inner_gens': [[1, 140, 24], [129, 4, 24], [1, 4, 24]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'D_6', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 50]], 'label': '240.64', 'linC_count': 384, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 16, 'linQ_dim': 10, 'linQ_dim_count': 84, 'linR_count': 144, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C30.D4', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 90, 'number_divisions': 28, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 44, 'number_subgroup_classes': 68, 'number_subgroups': 120, 'old_label': None, 'order': 240, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 24], [5, 4], [6, 14], [10, 28], [15, 8], [20, 96], [30, 56]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[3, 4, 216], [135, 124, 24], [3, 4, 72], [133, 6, 24]], 'outer_group': '64.196', 'outer_hash': 196, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [806520, 806414, 408376, 806409], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2:C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 6], [8, 8], [16, 2]], 'representations': {'PC': {'code': 144100052868803678227, 'gens': [1, 3, 5], 'pres': [6, -2, -2, -2, -3, -2, -5, 12, 2522, 50, 387, 88]}, 'GLZN': {'d': 2, 'p': 20, 'gens': [156209, 72009, 76219, 98112, 8081, 130311]}, 'Perm': {'d': 16, 'gens': [180587232009, 1313901388816, 46200, 16, 2789705318400, 43545600]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{30}.D_4', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [12, 6, 12, 20, 4, 12, 4, 30, 4], 'aut_gens': [[1, 4, 32], [641, 862, 226], [347, 716, 610], [163, 470, 224], [347, 708, 370], [11, 596, 930], [347, 918, 224], [819, 726, 752], [641, 462, 50], [25, 788, 560]], 'aut_group': None, 'aut_hash': 6668165578841517860, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 61440, 'aut_permdeg': 128, 'aut_perms': [284897376397020313331995433279707012483066223897541484931652436389521572932757715130478672808598637800924798051163442310796096291468623323218215143002121205224679903759244511011416653643407570445984435054889242156976, 233619736669342019299451946759221923217994571399186862121343386937109725307323636672794404670746033949058736826410816307839665404360375544418457364821922117684518678785388620657466989100786144315863882210209084968635, 246543535339917832560167568793310969530300729730149532820148327789615234028772265919571578427151402236524566898163427046437086134701600051600064621462322873433524366371724126529290342166116517515954930167833753867358, 233667334039881739712916667664621671190564523430784207820503984442844846613779795683421853646278313329630953076774821512699408466192337087492834170149997152347247192866031357770710613776499099755694975086377688674186, 326193297062057538107395847412769198651521518560619183546571008129950793212132073947971220722668818473647167455408653667482627933654047864056956594128911999337718678966312931130186646189480988415496527480378300678770, 234078878586106500184489372303168535109655032560525293310555068468825497454964912642290774207859963505705564244235179713852815031582528947043781841597259090437638271204844610886694271237885562801909638763520917964109, 131738843507168434128237483630466694354173095911105501514205758745817365598187893686712063478239550270394671340235899865450639851220238303153533271721117630928256232066086833430941321958443782014275600827060152892667, 285309306235349655315176057716196521125114054546481123386440340336432421573498502870647771377079164212875860198974859577229411133941378560794677127769618152730048819482069708823196296745039059935147023289883312381536, 124202556837866851392945946544590275555386878863052956334564366149266780889145700631935072117477132367464887974394725278027945819189433020542355204756460449342586382618965546582911010437301687088630354841681174714046], 'aut_phi_ratio': 240.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [3, 2, 1, 1], [4, 1, 4, 1], [4, 2, 1, 2], [4, 12, 4, 1], [4, 20, 4, 1], [5, 2, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 2, 1], [8, 30, 4, 2], [10, 2, 2, 1], [10, 2, 4, 1], [10, 4, 4, 1], [12, 2, 4, 1], [12, 4, 1, 2], [12, 20, 8, 1], [15, 4, 2, 1], [20, 2, 8, 1], [20, 4, 2, 2], [20, 12, 16, 1], [30, 4, 2, 1], [30, 4, 4, 1], [30, 4, 8, 1], [60, 4, 4, 2], [60, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{15}:((C_2^8\\times C_4).C_2^2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '480.1197', 'autcentquo_hash': 1197, 'autcentquo_nilpotent': False, 'autcentquo_order': 480, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [3, 2, 1], [4, 1, 4], [4, 2, 2], [4, 12, 4], [4, 20, 4], [5, 2, 2], [6, 2, 3], [6, 4, 2], [8, 30, 8], [10, 2, 6], [10, 4, 4], [12, 2, 4], [12, 4, 2], [12, 20, 8], [15, 4, 2], [20, 2, 8], [20, 4, 4], [20, 12, 16], [30, 4, 14], [60, 4, 16]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '120.11', 'commutator_count': 2, 'commutator_label': '60.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2664, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [3, 2, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 12, 2, 2], [4, 20, 2, 2], [5, 2, 2, 1], [6, 2, 1, 3], [6, 4, 1, 2], [8, 30, 2, 4], [10, 2, 2, 3], [10, 4, 2, 2], [12, 2, 2, 2], [12, 4, 1, 2], [12, 20, 4, 2], [15, 4, 2, 1], [20, 2, 4, 2], [20, 4, 2, 2], [20, 12, 8, 2], [30, 4, 2, 3], [30, 4, 4, 2], [60, 4, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4032, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '120.42', 'hash': 2664, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 4, 30], 'inner_gens': [[1, 28, 368], [9, 4, 928], [657, 68, 32]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 120, 'inner_split': True, 'inner_tex': 'C_{15}:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 60], [4, 44]], 'label': '960.2664', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C12.(C4*D10)', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 34, 'number_characteristic_subgroups': 44, 'number_conjugacy_classes': 120, 'number_divisions': 52, 'number_normal_subgroups': 104, 'number_subgroup_autclasses': 148, 'number_subgroup_classes': 228, 'number_subgroups': 872, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 136], [5, 4], [6, 14], [8, 240], [10, 28], [12, 176], [15, 8], [20, 224], [30, 56], [60, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4, 2, 4, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 8], 'outer_gens': [[481, 4, 416], [3, 14, 32], [499, 22, 48], [3, 12, 32], [3, 20, 48], [3, 20, 32], [25, 6, 354]], 'outer_group': '512.7532443', 'outer_hash': 3711438873667065943, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 16, 'outer_perms': [8396027152576, 40279696, 6920136120360, 4103734078087, 4103734078080, 4103734083256, 127370887], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3.C_2^6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 10], [8, 18], [16, 2], [32, 2]], 'representations': {'PC': {'code': 4325057092258164398649447448382561785961268445191, 'gens': [1, 3, 6], 'pres': [8, -2, -2, -2, -2, -2, 2, -3, -5, 16, 674, 66, 771, 91, 17669, 11157, 141, 39430, 12566, 222, 12311]}, 'Perm': {'d': 28, 'gens': [12210562039160627381364653544, 23845263204898072433768388481, 35212386116265158041470028800, 46520603436014101241193790080, 56988239863421575016275968000, 46520603436014101241066419200, 3, 6504]}}, 'schur_multiplier': [2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}.(C_4\\times D_{10})', 'transitive_degree': 480, 'wreath_data': None, 'wreath_product': False}
