296 CHAPTER XIII. 



273. Theorem. The abstract group generated by the operators 1 ) 

 E , JEg, E 3 , B 19 W subject to the generational relations 271) and 



278) W ^ Iy 



279) WBW= 



B 2 and B ~being defined Toy 277), is Jioloedrically isomorphic with 



Writing these relations for the corresponding orthogonal sub- 

 stitutions as defined by 272), 274) and W^>w, we obtain relations 

 which reduce to identities modulo 3. The order Q of is therefore 

 ^> 25920. The holoedric isomorphism will be established when it is 

 shown that Q < 25920. To prove this statement, consider the 

 following 27 sets 2 ) of operators of 0, those of the first set being 

 the operators of G E^ 6r 960 : 



3 W f , R s3t =GW s E 3 E 2 W t 



It is shown in the next section that the generators E , E 2 , E 3 , W, 

 and therefore an arbitrary operator a of the group 0, gives rise to 

 a mere interchange of the above 27 sets when applied as a right- 

 hand multipliers. Since the first set G contains the identity 1, the 

 product la = a lies in one of the 27 sets. Hence contains at 

 most 27 960 ^ 25920 operators. In particular, it follows that the 

 27 sets form a rectangular table for with the operators 6r 960 in 

 the first row. 



We make use of the formulae derived from 271), 278), 279), 

 280), 277): 



281) 



E 2 E i W=WE 2 E 1 . 



1) For simplicity B 1 is retained. It may be dropped since 



B, = TF~X TF^r 1 = TF-X WE^. 



2) They correspond in F0(5, 3) with the 27 rows of the rectangular table. 



