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



I26; nor has {617 .126} = (157) a permutable of (647); 

 neither has {647 .126} = (612) a permutable of 



126.517.126 = 751. 

 Therefore 647 • 5i7 . I26 is irreducible. 



Precisely in the same way it is proved that all the eight 

 following values of D, 



635. 647. 425, 467, 372, 356, 273. 245, 

 render D .517 .I26 irreducible. And there are no more 

 values^ because we cannot have D = 6i2 permutable with 

 I26, nor D=43i permutable with I34 in (I57), nor D=3i4 

 for the same reason, nor D = 2i6 permutable with I26. 



We can thus demonstrate, what is indeed sufficiently 

 evident from symmetry, that there are eight irreducible 

 triplets D . E,Q, whatever substitution of the fourth order 

 RQ may be 



We cannot proceed further without a closer definition of 

 our 21 triads. We define I57, I26, I34 as the mutually 

 permutables 



1643527. 1243765, 1634725, 



all of the same form and of the second order. In like 

 manner 



517=1462537, 524=7264351, 536 = 7432561, 



which mutually determine each other, 



517 . I26 = 1462537 . 1243765 = 1426735, 



of which the circular factors are i, 75, 2463. None of the 

 eight values has three figures of the circle 2463, and none 

 has I. Therefore each of them exchanges i for one of 

 2463, and either 7 or 5 for another of 2463 ; that is, each 

 first makes the three circles of 1426735 into two circles of 

 five and two, and then unites these two into one of seven. 

 Hence it appears that the eight irreducible triplets are all 

 substitutions of the seventh order. 



I know not whether the following theorem has ever been 



