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Let $p$ be an odd prime. Look at the group $P$ generated by the following elements of $GL_p(\mathbb{C})$: $\left(\begin{smallmatrix} 1&&&\\&w&&\\&&\ddots&\\&&&w^{p-1}\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}&1&&\\&&1&\\&&&\ddots\\1&&&&\end{smallmatrix}\right)$. Then $P$ is an extraspecial group of order $p^3$ and exponent $p$. Its image in $PGL_p(\mathbb{C})$ is elementary abelian $E$ of order $p^2$, and the normaliser modulo centraliser is $SL_2(p)$, acting as the group of all automorphisms of determinant one of $E$. There are $p-1$ representations like this, permuted by $GL_2(p)$.

I believe that what you are looking at is a direct sum of some copies of this representation plus some copies of the regular representation of $E$. If the numbers of copies of each of the $p-1$ representations above are equal, the normaliser modulo centraliser is $GL_2(p)$. Otherwise, something between $SL_2(p)$ and $GL_2(p)$.

Sorry to edit this again... but actually you can't mix different ones of these representations if the central element of $P$ has to act as a scalar. So there are only two possibilities: Either your representation consists of $k$ copies of one of the $p-1$ representations above, in which case the normaliser modulo centraliser is $SL_2(p)$; or $k$ is divisible by $p$, and your representation is a direct sum of copies of the regular representation of the elementary abelian $E$, in which case the normaliser modulo centraliser is $GL_2(p)$.

Let $p$ be an odd prime. Look at the group $P$ generated by the following elements of $GL_p(\mathbb{C})$: $\left(\begin{smallmatrix} 1&&&\\&w&&\\&&\ddots&\\&&&w^{p-1}\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}&1&&\\&&1&\\&&&\ddots\\1&&&&\end{smallmatrix}\right)$. Then $P$ is an extraspecial group of order $p^3$ and exponent $p$. Its image in $PGL_p(\mathbb{C})$ is elementary abelian $E$ of order $p^2$, and the normaliser modulo centraliser is $SL_2(p)$, acting as the group of all automorphisms of determinant one of $E$. There are $p-1$ representations like this, permuted by $GL_2(p)$; in other words, related by automorphisms of $E$. You can distinguish between these $p-1$ representations by the scalar representing a non-identity central element.

You can't mix different ones of these representations if the central element of $P$ has to act as a scalar on the entire space. So there are only two possibilities: Either your representation consists of $k$ copies of one of the $p-1$ representations above, in which case the normaliser modulo centraliser is $SL_2(p)$; or $k$ is divisible by $p$, and your representation is a direct sum of $p/k$ copies of the regular representation of the elementary abelian $E$, in which case the normaliser modulo centraliser is $GL_2(p)$.

Note I have edited this answer several times, for the sake of both correctness and readability. Apologies for any confusion caused.

Let $p$ be an odd prime. Look at the group $P$ generated by the following elements of $GL_p(\mathbb{C})$: $\left(\begin{smallmatrix} 1&&&\\&w&&\\&&\ddots&\\&&&w^{p-1}\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}&1&&\\&&1&\\&&&\ddots\\1&&&&\end{smallmatrix}\right)$. Then $P$ is an extraspecial group of order $p^3$ and exponent $p$. Its image in $PGL_p(\mathbb{C})$ is elementary abelian $E$ of order $p^2$, and the normaliser modulo centraliser is $SL_2(p)$, acting as the group of all automorphisms of determinant one of $E$. There are $p-1$ representations like this, permuted by $GL_2(p)$.

I believe that what you are looking at is a direct sum of some copies of this representation plus some copies of the regular representation of $E$. If the numbers of copies of each of the $p-1$ representations above are equal, the normaliser modulo centraliser is $GL_2(p)$. Otherwise, something between $SL_2(p)$ and $GL_2(p)$.

Let $p$ be an odd prime. Look at the group $P$ generated by the following elements of $GL_p(\mathbb{C})$: $\left(\begin{smallmatrix} 1&&&\\&w&&\\&&\ddots&\\&&&w^{p-1}\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}&1&&\\&&1&\\&&&\ddots\\1&&&&\end{smallmatrix}\right)$. Then $P$ is an extraspecial group of order $p^3$ and exponent $p$. Its image in $PGL_p(\mathbb{C})$ is elementary abelian $E$ of order $p^2$, and the normaliser modulo centraliser is $SL_2(p)$, acting as the group of all automorphisms of determinant one of $E$. There are $p-1$ representations like this, permuted by $GL_2(p)$.

I believe that what you are looking at is a direct sum of some copies of this representation plus some copies of the regular representation of $E$. If the numbers of copies of each of the $p-1$ representations above are equal, the normaliser modulo centraliser is $GL_2(p)$. Otherwise, something between $SL_2(p)$ and $GL_2(p)$.

Sorry to edit this again... but actually you can't mix different ones of these representations if the central element of $P$ has to act as a scalar. So there are only two possibilities: Either your representation consists of $k$ copies of one of the $p-1$ representations above, in which case the normaliser modulo centraliser is $SL_2(p)$; or $k$ is divisible by $p$, and your representation is a direct sum of copies of the regular representation of the elementary abelian $E$, in which case the normaliser modulo centraliser is $GL_2(p)$.

Let $p$ be an odd prime. Look at the group $P$ generated by the following elements of $GL_p(\mathbb{C})$: $\left(\begin{smallmatrix} 1&&&\\&w&&\\&&\ddots&\\&&&w^{p-1}\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}&1&&\\&&1&\\&&&\ddots\\1&&&&\end{smallmatrix}\right)$. Then $P$ is an extraspecial group of order $p^3$ and exponent $p$. Its image in $PGL_p(\mathbb{C})$ is elementary abelian $E$ of order $p^2$, and the normaliser modulo centraliser is $GL_2(p)$, acting as the group of all automorphisms of $E$. I believe that what you are looking at is a direct sum of some copies of this representation plus some copies of the regular representation of $E$, which does not change the normaliser modulo centraliser.

Let $p$ be an odd prime. Look at the group $P$ generated by the following elements of $GL_p(\mathbb{C})$: $\left(\begin{smallmatrix} 1&&&\\&w&&\\&&\ddots&\\&&&w^{p-1}\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}&1&&\\&&1&\\&&&\ddots\\1&&&&\end{smallmatrix}\right)$. Then $P$ is an extraspecial group of order $p^3$ and exponent $p$. Its image in $PGL_p(\mathbb{C})$ is elementary abelian $E$ of order $p^2$, and the normaliser modulo centraliser is $SL_2(p)$, acting as the group of all automorphisms of determinant one of $E$. There are $p-1$ representations like this, permuted by $GL_2(p)$.

I believe that what you are looking at is a direct sum of some copies of this representation plus some copies of the regular representation of $E$. If the numbers of copies of each of the $p-1$ representations above are equal, the normaliser modulo centraliser is $GL_2(p)$. Otherwise, something between $SL_2(p)$ and $GL_2(p)$.