  If a number field has a large degree and discriminant, the computation of the
class number becomes quite difficult, especially without the assumption of GRH.
In this article, we will unconditionally show that a certain nonabelian number
field of degree 120 has class number one. This field is the unique $A_5 \times
C_2$ extension of the rationals that is ramified only at 653 with ramification
index 2. It is the largest degree number field unconditionally proven to have
class number 1.
  The proof uses the algorithm of Gu\`ardia, Montes and Nart to calculate an
integral basis and then finds integral elements of small prime power norm to
establish an upper bound for the class number; further algebraic arguments
prove the class number is 1. It is possible to apply these techniques to other
nonabelian number fields as well.
