This property is not preserved under extension. The discrete Heisenberg group $H_3(\mathbb{Z})$, which consists of integer matrices of the form
$$\left[ \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array} \right],$$
is an a central extension of $\mathbb{Z}$ by $\mathbb{Z} \times\mathbb{Z}$. times \mathbb{Z}$by$\mathbb{Z}$. It has finite quotients$H_3(\mathbb{F}_p)$with irreducible representations of dimension$p$(this follows from the fact that$H_3(\mathbb{F}_p)$is non-abelian and has order$p^3$). 2 added 4 characters in body This property is not preserved under extension. The discrete Heisenberg group$H_3(\mathbb{Z})$, which consists of integer matrices of the form $$\left[ \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array} \right],$$ is an extension of$\mathbb{Z}$by$\mathbb{Z} \times\mathbb{Z}$. It has finite quotients$H_3(\mathbb{F}_p)$with irreducible representations of order dimension$p$(this follows from the fact that$H_3(\mathbb{F}_p)$is non-abelian and has order$p^3$). 1 This property is not preserved under extension. The discrete Heisenberg group$H_3(\mathbb{Z})$, which consists of integer matrices of the form $$\left[ \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array} \right],$$ is an extension of$\mathbb{Z}$by$\mathbb{Z} \times\mathbb{Z}$. It has finite quotients$H_3(\mathbb{F}_p)$with irreducible representations of order$p$(this follows from the fact that$H_3(\mathbb{F}_p)$is non-abelian and has order$p^3\$).