436 The Substitution Groups whose Order is Pour. 



To find the number of these groups we observe that the 

 largest value of a is 3. Hence there are 6 groups of the first 

 two types. To find the number of groups of the third type, 

 i. e. of the second type of non-cyclical groups, we assign the 

 following three values to n : — 



7, 5, 3. 



Hence we have for 







n=7, 2.4-(4-3) 2 =7, 







n=5, 2.3-l-(3-2) 2 = 4, 







n=S, 1.3-(2-l) 2 = 2. 





The total number of groups is 6 + 7 + 4 + 2 = 19. 

 individual groups are given in the following list* : — 



The 



Number 

 1. 



Groups. 

 { (abcd.efgh.ijkl)cyc.(mn) )dim. 





2. 



{ (abcd.efgh) eye. (ij.kl.mn) }dim. 





3. 



{ (abed) eye. (ef.gh.ij.kl.mn) }dim. 





4. 



{ (abcd.efgh.ijkl)^(mn) }dim. 





5. 



{(abcd.efgh) ^(ij .kl.mn) }dim. 





6. 



{ (abed) ^(ef. gh.ij.kl.mn) }dim. 





7. 



(ah) (cd. ef.gh.ij.kl.mn) . 





8. 



(ab.cd) (ef.gh.ij.kl.mn). 





9. 



(ab.ed) (cd. ef.gh.ij.kl.mn). 





10. 



(ab.cd.ef) (gh.ij.kl.mn) . 





11. 



(ab.cd.ef) (ef.gh.ij.kl.mn). 





12. 



(ab.cd.ef. gh) (gh.ij.kl.mn). 





13. 



(ab.cd. ef.gh) (ef.gh.ij.kl.mn) . 





14. 



{ (abed) 4 [ (ef) (gh.ij.kl.mn)] } u . 





15. 



{ (abcd)±\_(ef.gh) (ij.kl.mn)] } hl . 





16. 



{ (abcd) 4 [(efgh) (gh.ij.kl.mn) ] } u . 





17. 



{ (abcd) 4 [ef.gh.ij) (ij.kl.mn)] \ u . 





18. 



{ (abcd.efgh)±[(ij) (kl.mn) ] } u . 





19. 



{(abcd.efgh\[(ij.kl)(kl.mn)] }j x . 





* The notation is that which Professor Cayley used and explained in 

 his articles in the Quarterly Journal of Mathematics, vol. xxv. 



