Functions formed on Groups. 47 



n 4 . We take for G 3 s 462531, which is 



G 3 s= 123456 i53 62 4 



462531 645231 



516324 214356 = 



341265 3 6l 542 



254613 5 26 4i3 



635*42 432165 

 and has 214356 in common with I 8 , which is now written 



18=123456x123456 =H/J/' 



^213465 214356 = 6 

 /"2213456 



/M 23465 

 We get on 



462531 = G 3 S by w^ in H 4 ", 



£3*1 = 541623, 6^2 = 641532, 03,3 = 452163. 



The groups Ga, G 2 s, G 3 s, mark out each three, thus written ; 



2 = aa/3/3yy. 



Gs =(356142), G 81 = 6352i4, G £2 = 365i24, 0^=642513; 



G 2 S = (456231), G 2£l = 532641, G 2S2 = 546132, G 2£3 = 352641; 



G 3 S = (46253i), G 3£1 =-54i623, 6^2 = 641532, 63*3 = 452163. 



We have given account of 7 standards and of 21 groups 

 marked by them. 



Gs, G 2 s and GsS have no such symmetry under ao, (3ft, and 

 yy, that any C in I 8 can make 



C6/= 0/, art. 3, 

 true of any one of them. All three have 60 values. 



n 5 . There remain 60-28 = 32 groups unmarked, which 

 have no substitution of I g , which is now to be written 

 18=123456 21345664 -Hs-Jj 



21436561 : 12436465 

 21435662 1243566s 



213465^3 12346567 



Taking at random from groups unmarked, as we require 

 them, 



G D = 234561, G SD = 245613, G 3D = 3426i5, 6^ = 354612, 



