Irreducible characters of the affine semidirect product of orders eleven and five
= Irreducible characters of the affine semidirect product of orders eleven and five
For $G=C_{11}\rtimes C_5$ with faithful action, the abelianization $G/C_{11}\cong C_5$ supplies five linear characters. The two orbits of size five on the nontrivial characters of $C_{11}$ induce two irreducible characters of degree five. Their squared degrees satisfy
$$
5\cdot1^2+2\cdot5^2=55,
$$
so these seven characters are all the irreducible characters of $G$.