The first Curve of Genus 7 with maximal Automorphism group was discussed in a paper by A. M. Macbeath. I have found a second Curve of genus 7 with an Automorphism group of order 144. The equations of both these curves have been computed in my paper. Now, the question is: Is there another curve of genus 7 whose Automorphism group has an order ib between 144 and 504?
Article Group action on genus 7 curves and their weierstrass points
Dear Reza. You can ask Coy L. May, Marston Conder, Emilio Bujalance or Ruben Hidalgo.
With best wishes, Recep
Thank you Recep, I know those guys, but I thought they might view my question as well. I was wondering if any of them have joined "ResearchGate" yet? I will send them an email. Thanks
Dear Reza, the entire structure of Hurwitz loci has been found by Kay Magaard and myself up to genus 16 by computational means. I'd have to look up the data to answer your question, but perhaps it might be best to switch to email. In particular, our data contains a lot of interesting 0-dimensional loci, that is, curves identified by the group and ramification type, for which it would be interesting to find an equation.
Marston Conder in https://www.math.auckland.ac.nz/~conder/BigSurfaceActions-Genus2to101-ByGenus.txt
used a Magma programme to list all large groups of automorphisms of Riemann surfaces where by large he means bigger than 4(g-1). For genus 7 he found only the (2,3,7) group of order 504 and the (2,3,12) group of order 144, which I suppose is the one you found. So there is no group of genus 7 whose orders lie between 168 and 504.
The answer is no, these curves are the only two. See the table on page 16 in
http://arxiv.org/abs/math/0205314
Thank you Sergey for the info. I hadn't seen that list, at the time that I computed the totality of Curves of Genus 7, Their Weierstrass points and their Automorphism groups. That's a great paper, it covers the genera 8, 9 and 10 as well. I have Magma on my computer, looks like you have used GAP for many of those computations. I just downloaded that paper. Once again Thanks
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