I wish to computer the centraliser in $PGL(4,5)$ of $$ [\begin{pmatrix} 4 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}] $$ using MAGMA.
I found this page which I used to compute it. Matrix representation for PGL(n,q) in Magma
This code gives me a centraliser of order 1488000. I manually found the centraliser as $SL_3(5)\circ_4 C_4$. And this has order 496000. So apparently the central product is missing somehow..