Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 60] GENERALIZATION OF THE OCTAHEDRON GROUP 157
2 and 8 in addition to the given subgroup of order 24. It is
the group of isomorphisms of the non-cyclic group of order 9.
If we multiply the two given generators by an operator of
order 7 and its inverse, the operator of order 7 being commutative with each of these generators, we obtain two operators of
orders 42 and 56 respectively which satisfy the conditions in
question, and hence we have the theorem:
If two operators satisfy the conditions s3 = s24, (SLS2)2= 1, the
largest group which they can generate is the direct product of the
group of order 7 and the group of isomorphisms of the non-cyclic
group of order 9. The total number of the non-abelian groups
which can be generated by two operators which satisfy these equations is six: viz., the dihedral group of order 6, the octahedral
group, the group of isomorphisms of the non-cyclic group of order
9, and the direct products of these respective groups and the group
of order 7.
The third generalization of the octahedron group to be
considered is given by the equation
S12 =S2 (s1s2) = 1.
We may again consider the commutator of sl, S2 and observe
that
(S1 - s2 - 1S1S2)3 = (s2s1s22)3s16 = S16s2 -2 (s23s)3s22
= S162-2(S14S2-11 -1)3S22 = Sl18.
As s118 is transformed into its inverse by sl, it results that the
order of sl is a divisor of 36, and hence the order of G is a divisor
of 432. It is easy to see that the group of order 48, which may
be constructed by extending the non-twelve group of order 24
by means of an operator of order 4 which has its square in this
non-twelveggroup and transforms it according to an outer
isomorphism of order 2, can be generated by two operators of
orders 4 and 8 respectively which satisfy the given conditions.
If we multiply this sl and this s2 by an operator of order 9 and
by its fifth power respectively, this operator of order 9 being
commutative with each of the operators si, S2, and having only
the identity in common with (si, s2), we obtain two operators