Let $\textrm{cd}(G)=\lbrace \chi(1)\,|\, \chi\in\textrm{Irr}(G)\rbrace$ denote the set of character degrees of a finite group $G$. Similarly, denote by $\textrm{mcd}(G)$ the set of monomial character degrees. I have proved the following statement:

There exists no finite group $G$ of odd order such that:

• $\textrm{mcd}(G)=\lbrace 1,m\rbrace$
• $\textrm{cd}(G)=\lbrace 1,m,p\rbrace $ where $p$ is a prime
• $\textrm{gcd}(|G|,p^2-1)=1$

where $m$ is any positive integer and $m\neq p$. I have managed to find such groups if you remove the condition that $\textrm{gcd}(|G|,p^2-1)=1$ (with the help of Professor Mark Lewis). I would like to find an example of a group of even order satisfying the three bulleted conditions but have not suceeded.

Where would be a good place to look?