In the paper

M.C. Tamburini and M. Vsemirnov, Irreducible $(2,3,7)$-subgroups of ${\rm PGL}_n(F)$, $n \le 7$, J. Algebra 300 (2006), 339–362

the Hurwitz groups with absolutely irreducible projective representations of degrees up to $7$ over any field are determined (although the results in the smaller dimensions (up to $5$ I think) were not new.

Here is a list of the simple groups that arise - I hope I have copied this correctly!

*Added later*: I am sorry, I misunderstod the paper. This is not a complete list - this is a list of so-called *rigid triples* - and the authors say that they will complete the classification in a leter paper, which has now appeared in J. Algebra 321 (2009), no. 8, 2119–2138. In the second paper, they have not completely identified the groups that arise in dimension $7$, but they remark that some of the groups $G_2(q)$ arise - these had been found earlier by Malle.

$n=2$: ${\rm PSL}(2,p^m)$, where $m=1$ if $p \equiv 0,\pm 1 \mod 7$, and $m=3$ otherwise.

$n=3$ and $n=4$: nothing new.

$n=5$: ${\rm PSL}(5,p^m)$ and ${\rm PSU}(5,p^m)$ with $p \ne 5$ for certain values of $m$. See the survey paper by Conder for details.

$n=6$: ${\rm PSL}(6,p^m)$, with $p \ne 3$ and $m$ odd, and ${\rm PSU}(6,p^m)$, with $p \ne 3$ and $m$ even where, in both cases, $m$ is the order of $p$ mod $9$.

$n=7$: ${\rm PSL}(7,p^m)$, with $p \ne 7$ and $m$ odd, and ${\rm PSU}(7,p^m)$, with $p \ne 7$ and $m$ even where, in both cases, $m$ is the order of $p$ mod $49$.

*Added later*: I checked with a computer calculation that none of the three groups of order less than $10^9$ that you are uncertain about are Hurwitz. These are $C_2(7) = {\rm PSp}(4,7)$, $D_4(2) = {\rm P \Omega}^+(8,2)$ and $^2D_4(4) = {\rm P \Omega}^-(8,2)$. The first of these has a projective representation of degree $4$ and is covered by known results: none of the $4$-dimensional symplectic groups are Hurwitz. As far as I know, the $8$-dimensional orthogonal groups are not covered by published results, but it is very likely that somebody has dome these calculations already! The computer checks I used are more or less brute force, and they work easily for groups of order up to $10^9$, but will start to become impractical with group orders much higher than that.

I checked also that ${\rm He}$ is not an image of $(2,3,7;10)$.

Hurwitz groupis a finite group that occurs (up to isomorphism) as the automorphism group of a Riemann surface of genus $g$, and has the maximum possible order $84(g-1)$ for such a group. (from groupprops.subwiki.org/wiki/Hurwitz_group) $\endgroup$6more comments