282 CHAPTER XII. 



Hence d < 6. For ^ = 3, 4, 5 we find Q = 12, 24, 60 respectively. 

 But di =3, d 2 = 3, d 3 = 2, Q = 12 is excluded by 258). For d = 4, 

 ^2 = % d 3 = 2, the 24 is of the octahedral type ( 248). For ^ = 5, 

 d% =3, e? 3 = 2, the 6r 60 is of the icosahedral type ( 254). 



257. The tetrahedral and octahedral subgroups of the Gu(y A. 

 group of either type must contain a self - conjugate four- group. For 

 p > 2, the desired groups are therefore given by the theorem at the 

 end of 246. For p = 2, they contain operators of period 2 and 

 are therefore to be sought among the subgroups determined in 

 250253. But for p = 2, the dihedron 2 (i +2/ ) and the G M (fa 

 are neither of the tetrahedral and neither of the octahedral type. 

 There remain for consideration only the subgroups of the 6r%i) of 

 250. There is no octahedral subgroup of G^i) since the sub- 

 stitutions of period p = 2 in the latter are all commutative. In a 

 tetrahedral group the three substitutions of period 2 are all commu- 

 tative. Hence if there be a tetrahedral subgroup of 6r^-i), p = 2, 

 then must 2 m =4, d=3 and n even (since 3 must divide 2" 1). 

 Inversely, if m = 2, p = 2, n even, there exists a subgroup G$n d = 6r 12 

 of 6rJ( a _i). The 6r 12 is not commutative, since it would then contain 

 only operators of period p = 2 ( 241), and therefore 6r 12 has the 

 tetrahedral type ( 247). We may state the complete theorems: 



For s=p n =8hl,the G M (s) contains two systems each of M(s)/24: 

 conjugate octahedral groups 6r 24 and two systems each of Jf(s)/24 con- 

 jugate tetrahedral groups 6r 12 . Every 6r 12 is self -conjugate under a 6r 24 . 

 The two systems are conjugate under G^M(y 



For s = Sh + 3 or s = 2 n , n even, the G M (*) contains no octahedral 

 6r 24 but contains one system of Jf(s)/12 conjugate tetrahedral 6r 12 . 

 For p>2, tJie 6r 2 j/(. $ ) contains one system of M(s)/l2 conjugate octa- 

 hedral 6r 24 each containing one 6r 12 . For s 2 n , n odd, GM( S ) contains 

 no octahedral and no tetrahedral groups. 



258. Icosahedral subgroups of GM( S ) for p = 5. An icosahedral 

 6r 60 is generated by two operators F 5 , F 2 different from the identity 

 and subject to the generational relations (267) 



GM($) contains 4 (5 + 1) = 24 substitutions of period 5 and each 

 is conjugate within GM($) with one of the substitutions ( 241) 



