As a non-bio-monk, there are several questions raised in my mind by your post and by BrowserUk's reply above. With luck, they are trivial questions, and you can easily direct me to answers.
BrowserUk's solution produces the 50-base palindrome shown in the OP, but also produces other sequences that seem to be palindromic (again, to this bio-naif) and that are "embedded" within the longer sequence. Are these other sequences significant? Do you want to know about all, some or none of them?
In broader terms, consider the sequence
...CGATCG...CGATCG...CGATCG...
where ... represents zero or more bases that do not participate in any palindromic relationship. CGATCG is itself palindromic, each pair CGATCG...CGATCG presents another relationship (possibly more than one?), and the entire sequence presents...? Again, which of these seeming possibilities are significant? Which do you really want to know about?
I assume you are a biologist or are involved in biological computation, so these questions are probably pretty mundane. I hope it will not be much trouble to set me on the path to enlightenment.
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