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



The entire non-modular group consists of 2 . 28 substitu- 

 tions of the third order, 2. 21 of the fourth, 21 of the 

 second, and 8 . 6r of the seventh, which gives 



8r. 67 + 28.23 + 21 .24 + 21 . i^+ i = 7.6.4 + 8(r— i), 

 which is no divisor of 7 . 6 . 5 . 4 . 3 . 2, if r= 7. Wherefore 

 r=i. 



The 2 1 substitutions of the second order are 



157=1643527, 261=1275463, 372=4231657, 



126=1243765, 237 = 6235417, 3i4=i734652» 

 134=1634725, 245 = 6274513, 365=4731562, 



413 = 1534276, 5i7=i462537» 612=1257364, 715 = 1326547, 

 425 = 3214576, 524=7264531, 635 = 2137564, 723 = 5236147, 

 467 = 3514267, 663 = 7432561, 647=2154367, 746 = 5324167. 



Those of the seventh order are the poAvers of 



4231657.1426735=4125736, 4731562.1426735=4176235, ^ 



6235417.1426735=6521734, 2137564.1426735=2716435, 



3214576 . 1426735=3427615, 3514267 . 1426735=3456712, 



6274513 . 1426735=6421375, 2154367 • 1426735=2416753. 



The 28 triplets and the 21 quadruplets give those of the 

 third and fourth orders. And thus the entire group of 

 7 . 6 . 4 is readily constructed. 



I have shown in the memoir above quoted, art. 93, that 

 two groups of 7 . 6 . 4 can be constructed to contain the 

 powers of 3456712 last but one above written. I have no 

 doubt that the same thing is proveable by beginning this 

 investigation with the only other system of triads that can 

 be made to exhaust the duads in 7. We ought to find that 

 the powers of 3456712 are also part of the second group of 

 7 . 6 . 4 so constructed. 



This group is maximum, i. e. it has no derived derange- 

 ments, as may be proved by finding the number of its equi- 

 valents, which is 30. 



As I have shown in the abstract of this paper printed in 

 the ^ Proceedings ' of the Literary and Philosophical Society 

 of Manchester, April 29, 1862, that we can demonstrate 



