show/hide this revision's text 3 added 8 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 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 $p^3$).

show/hide this revision's text 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 $p^3$).

show/hide this revision's text 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 $p^3$).