Let and let be the order of the largest cyclic subgroup of G. What is ? And what would be the number of distinct cyclic subgroups of order in ?
You should start by thinking about what this automorphism group looks like. Essentially, this comes down to `where can the generator, 1, be sent to by an automorphism?'. (e.g. is an automorphism. Can you think why this is?)
So, where can the generator be sent to?
The generator is sent to , as is isomorphic to ?
I think the order of the largest cyclic subgroup of G is
For an element of we know , we'll know fora any , because
I think the property that isomorphisms preserve orders would tell us that , then I think members of would be the candidates for .
The largest member of is 314. But I don't know if this helps...
Yes, you need to find out where you can send the generator too. Your choices are precisely those numbers which generate . So, which elements of your group generate it? How many of them are there?
This will give you the order of your Automorphism group (why?).