Showing that $GL(2,p) = SL(2,p) \rtimes H$

Question

I want to show that $GL(2, p)$ is the semidirect product of $SL(2, p)$ and the subgroup $H$ of $2 \times 2$ diagonal matrices with elements on the diagonal being $1$ and $\alpha$, where $\alpha \in \mathbb{F}_p^\times$. That is, $GL(2,p) = SL(2,p) \rtimes H$.

I know I need to verify the following conditions:

1. $SL(2, p) \cap H = \{I\}$, where $I$ is the identity matrix.
2. $SL(2, p)H = GL(2, p)$.
3. $SL(2, p) \unlhd GL(2, p)$.

Conditions 1. and 3. are fairly easy to show, but I’m having trouble with condition 2. I’m trying to decompose the matrix expression of an arbitrary element of $GL(2, p)$

$$\begin{pmatrix}a & b\\\ c & d\end{pmatrix}$$

(where $ad - bc \neq 0$) as the product of an element in $SL(2, p)$ and an element in $H$, but I didn’t succeed. If someone could give me a hint I would appreciate it.