of the Groups whose Degree is less than Eight. 229 
symmetric group of order 6 and the group o£ order 3, has for 
its I the direct product of the symmetric group of order 6 
and the group of order 2, according to the theorem ; if a 
group is generated by two characteristic subgroups which 
have only the identity in common, its I is the direct product 
of the Fs of these characteristic subgroups. Since the other 
group of order 18 is obtained by extending the group of 
order 9 and of type (1, 1) by means of an operator of order 
2 which transforms each of its operators into its inverse, its 
I is the holomorph of this group of order 9 and hence is the 
doubly transitive group of degree 9 and order 432, according 
to the theorem given above. The two groups of order 24 
which are not simply isomorphic with the groups of lower 
degrees are the direct products of the alternating group of 
order 12 and the group of order 2. As these factors are 
characteristic subgroups of this direct product, its I is the 
symmetric group of order 24. 
Two of the groups of order 36 are simply isomorphic with 
the square of the symmetric group of order 6, and hence 
have the double holomorph of this symmetric group for 
their I. This double holomorph is the group of order 72 
and degree 6. The other group of order 36 is a characteristic 
subgroup of this double holomorph and the order of its I 
cannot be less than 144, which is the order of the I of this 
double holomorph, as we shall soon prove. Since this group 
of order 36 is generated by two operators of orders 3 and 4 
respectively, and as the former of these operators could not 
be selected in more than 8 ways while the latter may be 
selected in no more than 18 ways, it follows that the order 
of this I is 144 if the order of the I of the given double holo- 
morph is 144. When we prove the latter fact we shall also 
prove that this group of order 36 and the group of order 72 
.and of degree 6 have the same L 
Since the groups of order 48 are the direct product of the 
symmetric group of order 24 and the group of order 2, they 
are simply isomorphic with their I. The group of order 72 
contains subgroups of order 12 and degree 6 which give rise 
to a transitive representation, and hence it admits outer 
isomorphisms according to the given theorem. Moreover, 
it contains a characteristic subgroup of order 36 which does 
not admit more than 144 holomorphisms. The holomorphism 
of the group of order 72 is fixed by that of this subgroup, 
since the former contains only one operator of order 2 which 
is commutative with every operator of a subgroup of order 6 
contained in the latter. Hence each of these groups has the 
same I and its order is 144. If this group of order 144 is 
