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?
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?).