396 REV. T. p. KIRKMAN ON THE THEORY OF 



by S go a 345612 nijklm efghcd 



ySgoab 456123 ijklmn fghcde 



Sgoa^y 561234 jklmni ghcdef 



goabjS 612345 Mmnij hcdefy 



fghcde ijklmn 123456 oabySg 



ghcdef jklmni 234561 abySqo 



hcdefg klmnij 345612 bySgoa 



cdefgh Imnijk 456123 ySgoab 



defghc mnijkl 561234 ^goaby 



efghcd nijklm 612345 goabyS 



^ Imnijk fghcde ySgoab 123456 



mnijkl ghcdef Sgoaby 234561 

 nijklm hcdefg goabyS 345612 

 ijklmn cdefgh oabySg 456123 

 jklmni defghc abySgo 561234 

 klmnij efghcd bySgoa 612345. 

 This group contains only one square root of unity, 

 d = /\.^6i2'^oabySgfghcdelmnijk, 

 with two of its cube roots of the sixth order^ six of its 

 square roots of the fourth order, and twelve of its sixth 

 roots of the twelfth order, besides two substitutions of the 

 third order. 



Or we may consider the above group of twenty-four as 

 constructed by adding to the model group of the sixth 

 order, which begins it, its three derived derangements by 

 241434 344412 ^^^ 434214 

 1234' 1234 1234' 



where i, 2, 3, 4, like 7^1^93- • in Art. 27, stand for the four 

 circular factors of the model group. And we may, by this 

 method, easily construct an equivalent group of twenty- 

 four on any model group equivalent to the one above 

 employed, whether its circular factors, as above, are, or 

 whether they are not, composed of contiguous elements of 

 unity. 



