# Semidirect product.

I have a problem with representation of this : $(D_n \times D_n) \rtimes \mathbb{Z}_2$, where $\mathbb{Z}_2$ acts on $D_n \times D_n$ by exchanging the two components. $D_n = \langle x, \ y \ | \ x^n = y^2 = (xy)^2 = e \rangle$ - dihedral group with order $2n$. I want to find all subgroups of $(D_n \times D_n) \rtimes \mathbb{Z}_2$. But first, I want to find representation of $(D_n \times D_n) \rtimes \mathbb{Z}_2$. We have five generated elements : $x_1,\ y_1, \ x_2, \ y_2, \ z$, where $x_1, \ y_1$ and $x_2, \ y_2$ generated elements of two dihedral groups, $z$ generated element of $\mathbb{Z}_2$. We have some conditions : $x_1^n = y_1^2= (x_1 y_1)^2=e,\ x_2^n = y_2^2 = (x_2 y_2)^2 = e, \ x_1 x_2 = x_2 x_1, \ y_1 y_2 = y_2 y_1$. Which conditions on $z$ ?

• Since $z$ has order $2$, you'll need the relation $z^2 = e$. But you also need relations describing the action of $z$ on the direct product of the dihedral groups: $z^{-1}x_1z = x_2, z^{-1}y_1z = y_2$. – James Apr 16 '15 at 16:30