Galois Group of Polynomial

I had this homework problem, I think I got it correct, just want a check from somebody.

Find the Galois group of over .

I get that the splitting field is where .

The next step is to calculate the dimension of this field. Here I used: let be irreducible over with degree and , then is irreducible over if . Since is minimal polynomial for it follows from above that this polynomial is irreducible over . Thus, . And thus, .

Now let , .

Thus, and , also .

Finally, is a set of distinct -automorphism.

There are all together. Thus we have found all elements of .

My remaining question is what type of group is this?

I think this is the __Frobenius group__.

And what is the subgroup diamgram of this group?

(Wikipedia did not have it).