  Given $n$ discrete random variables, its entropy vector is the $2^n-1$
dimensional vector obtained from the joint entropies of all non-empty subsets
of the random variables. It is well known that there is a one-to-one
correspondence between such an entropy vector and a certain
group-characterizable vector obtained from a finite group and $n$ of its
subgroups [3]. This correspondence may be useful for characterizing the space
of entropic vectors and for designing network codes. If one restricts attention
to abelian groups then not all entropy vectors can be obtained. This is an
explanation for the fact shown by Dougherty et al [4] that linear network codes
cannot achieve capacity in general network coding problems. All abelian
group-characterizable vectors, and by fiat all entropy vectors generated by
linear network codes, satisfy a linear inequality called the Ingleton
inequality. It is therefore of interest to identify groups that violate the
Ingleton inequality. In this paper, we study the problem of finding nonabelian
finite groups that yield characterizable vectors which violate the Ingleton
inequality. Using a refined computer search, we find the symmetric group $S_5$
to be the smallest group that violates the Ingleton inequality. Careful study
of the structure of this group, and its subgroups, reveals that it belongs to
the Ingleton-violating family $PGL(2,q)$ with a prime power $q \geq 5$, i.e.,
the projective group of $2\times 2$ nonsingular matrices with entries in
$\mathbb{F}_q$. We further interpret this family using the theory of group
actions. We also extend the construction to more general groups such as
$PGL(n,q)$ and $GL(n,q)$. The families of groups identified here are therefore
good candidates for constructing network codes more powerful than linear
network codes, and we discuss some considerations for constructing such group
network codes.
