of the Groups iclwse Degree is less than Eight. 23 1 
these for their I, since the metacyclic group of order p (p 1} 
and degree p, p being any odd prime, is the group of iso- 
morphisms of all its subgroups ivhose orders are divisible by p 
and exceed p. The group of order 20 has the direct product 
of the group of this order and of degree 5 and the group of 
order 2 for its I, according to theorem I. The two groups 
of order 24 which are the direct product of the symmetric 
group of order 6 and the four-group may be obtained by 
extending the direct product of the four-group and the cyclic 
group of order 3 by means of an operator of order 2 which 
transforms each operator of this direct product into its 
inverse. Hence by the theorem given above, these groups 
have the holomorph of this direct product for their I. The 
order of this holomorph is 144. The group which is the 
direct product of the cyclic group of order 4 and the 
symmetric group of order 6 is invariant under the holomorph 
of the cyclic group of order 12, and has the group of cogre- 
dient isomorphisms of this holomorph for its I. Hence the 
order of this I is 24, and it is the direct product of the 
symmetric group of order 6 and the four-group. 
The direct product of the octic group and the group of 
order 3 has these two groups for characteristic subgroups,, 
and hence its I is the direct product of the octic and the 
group of order 2, while the I of the dihedral group of order 
24 is the holomorph of the cyclic group of order 12. The 
orders of these two Fa are 14 and 48 respectively. It 
remains to consider the I of the group denoted by 
^(abcd) 8 com. (efg) all j-dim.* 
As this is invariant under the direct product of the octic and 
the symmetric group of order 6, and only two operators of 
this product are commutative with each of its operators, its I 
is the group of cogredient isomorphisms of this direct 
product. This I is the direct product of the dihedral group 
of order 12 and the group of order 2. We have now con- 
sidered the five types of groups of order 24 which can be 
represented as substitution groups of degree 7 but not of a 
lower degree. The orders of 'their I's are 144, 24, 48, 16, 
The two I's of order 24 are simply isomorphic. 
The group of order 36 contains three invariant tetra- 
hedral groups. As each of these may be transformed into 
itself by the symmetric group of order 24, the I of the group 
of order 36 contains this symmetric group invariantly and is 
* American Journal of Mathematics, vol. xxi. (1899) p. 32£. 
