Substitution Groups of Order 8p. 123 



Gr 17 contains one system of subgroups of each of the three 

 orders 2p, p, 1, and two systems of each of the two orders 

 4, 2 that do not contain an invariant subgroup except identity. 

 It is therefore simply isomorphic to the following transitive 

 groups : — 



Degree of groups 4 8 24 6 12 



Number „ 1112 2 



These seven simply isomorphic transitive groups, in order, 

 have been denoted as follows * : — (abed) all, (abed . efgh) pos. 

 (ae . bg . cf. dh) , {am .bn .ep .do .ex ,fw . gu . hv . is .jt . kr.lq . 

 H 4 , ( + abedef)^ ( ± abedef)^, (abe'def.ghijkl)^ {ahej . bidg) 

 eh .fl), (abed . efgh . ijkl)± (afk . bgi . cej . did) (af . be .eg .dh . 



v)- 



G 18 contains one system of subgroups of each of the three 

 orders, p, 4, 1, and two systems of order 2 that do not con- 

 tain any invariant subgroup except identity. It is therefore 

 simply isomorphic to the following transitive groups : — 



Degree of groups 8 6 24 12 



Number „ 1112 



These five simply isomorphic transitive groups, in order, have 

 been denoted as follows : — 



(abed . efgh) pos. (ae .bf.cg. dh), (abcdef) 2 ^, (am .bn.co.dp. 

 eq .fr .gs.ht, iu .jv . kw . Z#)H 4 , (abedef . ghijkl) n (ag . bh . ei . 

 dj . ek ./0, (abed . efgh . ijkl) A (afk . bgi . cej . did) (ab . cd . il . 

 jk) t. 



G 19 contains only two systems of subgroups that do not 

 contain an invariant subgroup except identity. The orders of 

 these subgroups are 3 and 1 respectively. The simply iso- 

 morphic transitive groups have been denoted by 



(ab ,cd .ef . gh) (ABCD) pos., (aceg . bdjh . ikmo .jlnp . qsaw . 

 rtvx) (abef . chgd . ijmn . lepol . qruv . socwi) (akr . bis . cjq . dlt . 

 eov .fmw . gnu . hpx) . 



Hence there are 32 transitive substitution groups of order 24. 

 15 of these groups are regular. The group of degree 4 is 

 primitive. All the others are nonprimitive. 



We have thus far excluded the special case when p = 2. 



* Cf. Cayley, ' Quarterly Journal of Mathematics/ vol. xxv. p. 71. 



t This group is not included in the list of transitive substitution 

 groups of decree 12 recently published in the 'Quarterly Journal of 

 Mathematics. It seems to be the only transitive group of oider 24 that 

 has not yet been published. It contains three systems of nonprimitivity. 



