The troika is composed of the European Commission(EC), the European Central Bank(ECB) and the International Monetary Fund(IMF). The first two are representatives of a 27 member European unions. There are 26, 25 and 17 member unions that are prominent. Basically it's two bureaucratic manifestations of the full EU.
The composition of the Greek troika:
Toirka = EU+IMF (3=2+1)
EU= ECB + EC (2=1+1)
Europe = All
The IMF is and international, of course, but it's last 10 directors, have been Europeans, and four of them, including the current one - Christine Lagarde - were French. The IMF has committed over $500 committed for a great EU financial firewall including $109, which the U.S.committed in 2009 to the IMF. No government has ever lost money investing in IMF (according to the IMF - so I'll have to check). The developing nations are also unhappy with having their money committed to stench the bleeding in Europe.
The Greek troika views itself as three European entities dragging a slacker, at least according to the German standards.
The Greeks, at least, see this differently. The experience three riders (of the apocalypse ?) EC, ECB, and IMF.
I read this as three impatient riders running an inherently slow horse as hard as they can -- although, I don't know how to draw it. It wouldn't work with horses, much less with people.