Thanks Urbanite. It has helped. Since posting my question the answers have helped me to see things much more clearly, and have provided a way forward with this reluctance by the directos to hold an EGM.
I've been involved in a lot of 'block politics' over the years, but it is difficult to comment without knowing more. A few questions:
1) How many directors do you have and what are their term limits?
2) How many shareholders?
3) Do you have external managing agents?
4) What provisions if any do your articles make for removal of a director, e.g. by vote of no confidence?
My feeling is that if all else fails a solicitor's letter to the board on behalf of the dissident shareholder group could be the answer.