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In fact, there is one more with 2 in the middle, with vertices labelled (4,3,1,7,9,8).

So, recapitulating we have:
with 4 in the middle (6,2,1,3,10,5)
with 2 in the middle (4,3,1,7,9,8); (4,12,1,3,5,7); (5,3,1,8,4,11)

And I think that is!
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