216 THE REV. T. P. KIRKMAN ON NON-MODULAR GROUPS. 



table with A=ab = QiRj &c., and that the first sextuplet has 

 all its elements permntable with «=6A=CJ, &c., as read 

 in the triplets AQR^ Aab, adCj &c. We see also^ looking 

 at the 8-plet and 6-plet_, that df in the third and ce in the 

 sixth septuplet are substitutions of the seventh order. 



We may take at pleasure any system of didymous factors 

 of a substitution of the eighth order. Take then 



18367452= « 



5327i846=Q 

 42615387=6 

 8675423i=N 

 37148625 = 6 

 21583764 = ^ 

 65432178=/ 

 74826513 = 8. 



We have A=«Z>=35i62487 and we have to form the 

 system 



18367452 =« 

 35162487 =A 

 a bf ff h i j k =d 

 I mn p q r s =B 

 3 7 I 48 6 25 =6 



=C 



=Ci 



where AdBbCc are all permutable with a. 



It is plain that the only possible vertical circles under 

 3 and I in A are 333 . . and 1 1 1 . . . We have then 



35162487 =A 



36 i^A i j k ^d 



'^mi p q r s =B 

 37148625 =6 



=C 



=c. 



