-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '15984.a', 'ambient_counter': 1, 'ambient_order': 15984, 'ambient_tex': 'C_{12}\\times F_{37}', 'central': False, 'central_factor': False, 'centralizer_order': 432, 'characteristic': False, 'core_order': 12, 'counter': 62, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '15984.a.74.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '74.a1', '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': 74, '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': '216.73', 'subgroup_hash': 73, 'subgroup_order': 216, 'subgroup_tex': 'C_6\\times C_{36}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '15984.a', 'aut_centralizer_order': None, 'aut_label': '74.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '37.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1', '37.a1'], 'contains': ['148.a1', '148.b1', '148.c1', '222.a1', '222.c1'], 'core': '1332.a1', 'coset_action_label': None, 'count': 37, 'diagramx': [1701, -1, 3969, -1], 'generators': [3996, 12, 7992, 18, 4, 5328], 'label': '15984.a.74.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '37.a1', 'old_label': '74.a1', 'projective_image': '1332.31', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '74.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '216.73', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 6, 6, 12], 'aut_gens': [[1, 6], [1, 102], [1, 7], [5, 174], [181, 153]], 'aut_group': '864.2629', 'aut_hash': 2629, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 15, 'aut_perms': [1484184529, 467226184446, 100602029089, 102027744150], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 2, 1], [3, 1, 6, 1], [4, 1, 4, 1], [6, 1, 2, 1], [6, 1, 4, 1], [6, 1, 6, 1], [6, 1, 12, 1], [9, 1, 18, 1], [12, 1, 8, 1], [12, 1, 24, 1], [18, 1, 18, 1], [18, 1, 36, 1], [36, 1, 72, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_6^2:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '864.2629', 'autcent_hash': 2629, 'autcent_nilpotent': False, 'autcent_order': 864, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_6^2:D_6', '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, 3], [3, 1, 8], [4, 1, 4], [6, 1, 24], [9, 1, 18], [12, 1, 32], [18, 1, 54], [36, 1, 72]], 'center_label': '216.73', 'center_order': 216, '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', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 73, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['4.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 4], [4, 1, 2, 2], [6, 1, 2, 12], [9, 1, 6, 3], [12, 1, 4, 8], [18, 1, 6, 9], [36, 1, 12, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 36, 'exponents_of_order': [3, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '36.14', 'hash': 73, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 6], [1, 6]], '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, 216]], 'label': '216.73', 'linC_count': 5184, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 252, 'linQ_dim': 10, 'linQ_dim_count': 252, 'linR_count': 1296, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6*C36', 'ngens': 6, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 216, 'number_divisions': 48, 'number_normal_subgroups': 80, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 80, 'number_subgroups': 80, 'old_label': None, 'order': 216, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 3], [3, 8], [4, 4], [6, 24], [9, 18], [12, 32], [18, 54], [36, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 6, 6, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 102], [1, 7], [5, 174], [181, 153]], 'outer_group': '864.2629', 'outer_hash': 2629, 'outer_nilpotent': False, 'outer_order': 864, 'outer_permdeg': 15, 'outer_perms': [1484184529, 467226184446, 100602029089, 102027744150], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6^2:D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 8], [6, 12], [12, 6]], 'representations': {'PC': {'code': 685828699326291, 'gens': [1, 3], 'pres': [6, -2, -3, -2, -2, -3, -3, 12, 50, 69, 118]}, 'GLFp': {'d': 2, 'p': 37, 'gens': [101307, 557210]}, 'Perm': {'d': 18, 'gens': [4097379686400, 355687428096000, 357120, 79833600, 1313901388800, 80884]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 36], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times C_{36}', 'transitive_degree': 216, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '432.200', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1332, 'aut_gen_orders': [6, 18, 36, 18, 18], 'aut_gens': [[1, 36], [12565, 7596], [12061, 8892], [1405, 12780], [7381, 14220], [1873, 4572]], 'aut_group': None, 'aut_hash': 792994677548912317, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 63936, 'aut_permdeg': 259, 'aut_perms': [8835231879832002776270782753903548331915586349395184643548388738767259861488035600078494366484401308925025550498260666085523796571635325766865672395509039815536468624972121054897191940882571013383996479325682894652523189983794352329138697016893101657259227990356646289639361265941840025389097253802059376960511598884463235211485228438681423199714111635305527970476782478929136088767766408769412106439166049796825440459595828263287308779533461347307248358883399986061556973339111112886491586448320027120090005243699, 14471575791647984824430444347105761486948463818417641567569197997696458671450580240731460260152048254138823638867168784986584586783405676245858847838253326029946191859122714292747286175399515870598500788365267323158477847431860775257332444838479998072443337009158924429795219326575105426548705237532840049587938014420753701875568966535395222465473384105611447568441540888367768920998319778157609643718701163510190693150654914684588203105260816239777956458283496599888767655798721238627792994533700059942344357762890, 10527823555870074975603631276727534930123080381938560801699659389160869335852234409932577178900313113538948572690250649972005322746659558284875105599890487823697405811984056823157631530153494368404199202483330898641848570339523739916811869104571374811127001590271973469024978055272795440842123286684618579974279080886449777152998256276891855050217669968221849657467625167670778684805935495367974547777515774327020342189569191522540700637478507554970355050961350846636141818894189029251390981618162267513135185327794, 2494953384080998100389713271128525317897264159182681066906839836006098590293590782076399526985465702503692122473021374298515031009653580840748849217419863887491973104783621411422277125261375900735563067064491624377351376511239094233435929212597465766940411165109767790649531146649557986781946211572772221444253823295693640504165620713019927399447868013821965268706270929787339917648770650800293991369319184578997326980511708899709181354642472323780314940650502378619288498475091221718714747433740647089725775665253, 1879156675783510685770034401612354623658578968299413828972656133245984566818528333391914555931981458158298920526995513526936638207474229486422367867841602675586614865559010751105449347027549666611686478864155925273061896774643491976294222588772916154432564373815869072993899188218033374062929235936780705814466848475430148462361279275037496026216791095100559916737874793888820240758477235535763428299315624584787298697532300443308221831936598450016368771330614299801680907195903782593833845207334208728537713208889], 'aut_phi_ratio': 12.333333333333334, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 37, 2, 1], [3, 1, 2, 1], [3, 37, 1, 2], [3, 37, 2, 2], [4, 1, 2, 1], [4, 37, 2, 1], [4, 37, 4, 2], [6, 1, 2, 1], [6, 37, 1, 2], [6, 37, 2, 4], [6, 37, 4, 3], [9, 37, 3, 6], [12, 1, 4, 1], [12, 37, 2, 4], [12, 37, 4, 9], [12, 37, 8, 6], [18, 37, 3, 6], [18, 37, 6, 6], [36, 37, 6, 12], [36, 37, 12, 12], [37, 36, 1, 1], [74, 36, 1, 1], [111, 36, 2, 1], [148, 36, 2, 1], [222, 36, 2, 1], [444, 36, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{222}.C_{18}.C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '48.38', 'autcent_hash': 38, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3\\times D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1332, 'autcentquo_group': '1332.31', 'autcentquo_hash': 31, 'autcentquo_nilpotent': False, 'autcentquo_order': 1332, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{37}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 37, 2], [3, 1, 2], [3, 37, 6], [4, 1, 2], [4, 37, 10], [6, 1, 2], [6, 37, 22], [9, 37, 18], [12, 1, 4], [12, 37, 92], [18, 37, 54], [36, 37, 216], [37, 36, 1], [74, 36, 1], [111, 36, 2], [148, 36, 2], [222, 36, 2], [444, 36, 4]], 'center_label': '12.2', 'center_order': 12, 'central_product': True, 'central_quotient': '1332.31', 'commutator_count': 1, 'commutator_label': '37.1', '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', '37.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1332.31', 1], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 37, 1, 2], [3, 1, 2, 1], [3, 37, 2, 3], [4, 1, 2, 1], [4, 37, 2, 5], [6, 1, 2, 1], [6, 37, 2, 11], [9, 37, 6, 3], [12, 1, 4, 1], [12, 37, 4, 23], [18, 37, 6, 9], [36, 37, 12, 18], [37, 36, 1, 1], [74, 36, 1, 1], [111, 36, 2, 1], [148, 36, 2, 1], [222, 36, 2, 1], [444, 36, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 864, 'exponent': 1332, 'exponents_of_order': [4, 3, 1], 'factors_of_aut_order': [2, 3, 37], 'factors_of_order': [2, 3, 37], 'faithful_reps': [[36, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '7992.a', 'hash': 1147931464434849270, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 1332, 'inner_gen_orders': [36, 37], 'inner_gens': [[1, 8676], [7345, 36]], 'inner_hash': 31, 'inner_nilpotent': False, 'inner_order': 1332, 'inner_split': True, 'inner_tex': 'F_{37}', 'inner_used': [1, 2], 'irrC_degree': 36, 'irrQ_degree': 144, 'irrQ_dim': 144, 'irrR_degree': 72, 'irrep_stats': [[1, 432], [36, 12]], 'label': '15984.a', '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': True, 'monomial': True, 'name': 'C12*F37', '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, 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': 90, 'number_characteristic_subgroups': 41, 'number_conjugacy_classes': 444, 'number_divisions': 86, 'number_normal_subgroups': 156, 'number_subgroup_autclasses': 140, 'number_subgroup_classes': 300, 'number_subgroups': 5484, 'old_label': None, 'order': 15984, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 75], [3, 224], [4, 372], [6, 816], [9, 666], [12, 3408], [18, 1998], [36, 7992], [37, 36], [74, 36], [111, 72], [148, 72], [222, 72], [444, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[5329, 7956], [3997, 8028], [9325, 2628]], 'outer_group': '48.38', 'outer_hash': 38, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 7, 'outer_perms': [745, 1680, 1707], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 44, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 3, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 22], [4, 24], [6, 12], [12, 18], [36, 2], [72, 3], [144, 1]], 'representations': {'PC': {'code': '16524042863852504807657951240713532000254346487170149174608922200597968863130125716179175519619787751934495769927', 'gens': [1, 5], 'pres': [8, -2, -2, -3, -3, -2, -2, -3, -37, 16, 41, 90, 347044, 259932, 82460, 14548, 116, 65669, 240205, 6069, 34877, 141, 153222, 112910, 14134, 6750, 222, 525319, 387087, 48407, 23071]}, 'GLFp': {'d': 2, 'p': 37, 'gens': [50691, 405225, 962409]}, 'Perm': {'d': 44, 'gens': [1473647718507523536228011078992636227327426857028483, 2913759652257673605104566425069958962244530419998948, 61958305861368147964452455736446518711083563906954880]}}, 'schur_multiplier': [12], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [12, 36], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}\\times F_{37}', 'transitive_degree': 444, 'wreath_data': None, 'wreath_product': False}
