Certainly not a public company (6,7)

I believe the answer is:

secret society

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'certainly not a' is the definition.
I can't judge whether this defines the answer.

'public company' is the wordplay.
'public' becomes 'secret' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'company' becomes 'society' (I've seen this in another clue).
'secret'+'society'='SECRET SOCIETY'

Can you help me to learn more?

I've seen this clue in The Irish News.

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