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We study Noether's problem for various subgroups H of the normalizer ot a group C 8 generated by an 8-cycle in S 8 , the symmetric group of degree 8, in three aspects according to the way they act on rational function fields, i.e., Q(X 0 ,..., X 7 ), Q(x 1 ,..., x 4 ), and Q(x, y). We prove that it has affirmative answers for those H containing C 8 properly and derive a Q-generic polynomial with four parameters for each H. On the other hand, it is known in connection to the negative answer to the same problem for C 8 /Q that there does not exist a Q-generic polynomial for C 8 . This leads us to the question whether and how one can describe, for a given field K of characteristic zero, the set of C 8 -extensions L/K. One of the main results of this paper gives an answer to this question.

Noether's problem and $\mathbb{Q}$-generic polynomials for the normalizer of the $8$-cycle in $S_8$ and its subgroups