Council secretary's exterior show of support (5)

I believe the answer is:

synod

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'council' is the definition.
(I know that synod is a type of council)

'secretary's exterior show of support' is the wordplay.
'exterior' means to remove the middle letters.
'show of support' becomes 'nod' (I am not sure about this - if you are sure you should believe this answer much more).
'secretary' with its middle removed is 'sy'.
'sy'+'nod'='SYNOD'

Can you help me to learn more?

(Other definitions for synod that I've seen before include "Ecclesiastical gathering" , "Forum for discussing" , "Council of bishops" , "Church council" , "Ecumenical council" .)

I've seen this clue in The Independent.

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