230 Mr. G. A. Miller on the Groups of Isomorphisms 
represented as a transitive substitution group of degree 12 
with respect to a subgroup in the given group of order 72,. 
the two sets of 6 subgroups of order 12 will correspond to 
two systems of imprimitivity. Hence there are two transitive 
groups of degree 12 and order 144 whose heads are obtained 
by making the group of degree 6 and order 72 simply iso- 
morphic with itself, while the published list of these groups 
gives only one *. As it is well known that the I of the 
alternating and the symmetric group of degree 6 may be repre- 
sented as an imprimitive group of degree 12 and order 1440 f, 
it has been proved that the substitution groups of degree 6 
which cannot be represented as substitution groups of lower 
degrees have groups of isomorphisms of the following orders : 
168, 8, 48, 64, 12, 432, 24, 72, 144, 1440. The two groups 
of order 48 are clearly distinct, as one involves opei*ators of 
order 8 while the other does not have this property. The 
13 distinct abstract groups which may be represented as 
substitution groups of degree 6, but of no lower degree, have 
therefore 11 distinct groups of isomorphisms. 
§4. The Groups of Degree Seven. 
Only eleven of the forty groups of degree 7 are simply 
isomorphic with groups of lower degrees. Three others are 
simply isomorphic with the cyclic groups of orders 7, 10, 12 
respectively, and hence they have for their Fs the cyclic 
groups of orders 6 and 4, and the four-group. Two of the 
remaining three groups of order 12 are simply isomorphic 
with the direct product of the four-group and the group of 
order 3, and hence their I is the direct product of the 
symmetric group of order 6 and the group of order 2. This 
is also the I of the other non-cyclic group of order 12 which 
is not simply isomorphic with groups of lower degrees, since 
the holomorph of the cyclic group of order 6 is the direct 
product of the symmetric group of order 6 and the group of 
order 2. We have now considered the Fs of the 5 possible 
abstract groups of order 12, and found that their orders 
are 4, 12, and 24 respectively. Three of these groups 
have the holomorph of the cyclic group of order 6 for 
their 1. 
The three groups of orders 14, 21, and 42 have the last of 
* Quarterly Journal of Mathematics, vol. xxviii. (1896) p. 223. 
t Bulletin of the American Mathematical Society, vol. i. (1895) 
p. 258. 
