Find all the irreducible representations of the group given by:
$
I have 8 conjugacy classes: $\{1\}, \{z^3\}, \{x,y,xy\}, \{z,xyz,xz,yz\}, \{z^2,xz^2,yz^2,xyz^2\}, \{yz^3,xz^3,xyz^3\}, \{z^4, xz^4, yz^4\}, \{z^5,xz^5,yz^5,xyz^5\}$
and hence 8 irreducible representations. The abelianization of the group has 6 elements. Hence there are 6 irreducible representations of dimension 1. The sum of the squares of the dimensions of the irreducible representations is equal to the order of the group. Hence we have two irreducible representations with the square of the dimensions summing to 18. Hence we have two representations of dimension 3 each.
If I find one of these representations then by tensoring by a well chosen representation of dimension 1 I can hopefully find the other one.
I know that I can find a 3 dimensional representation by inducing from a subgroup of order 8. How do I do this?