In the meantime, you can try to get what you want in the April 1 rush. But also, if you and your friends all came to Lone Pine, I can guarantee that you would get permits to hike as a group in 2 or 3 days, and most likely on the first day you try.
So just plan on taking several days, and you are good to go. There are plenty of other places to day hike to get acclimated -- you would be best doing that anyway for several days before the Whitney hike.
As Steve points out, for those that have the luxury/flexibility of a few extra days, a permit is virtually assured. For those without that flexibility, the probably is still high that a permit will be obtained. For those who make the trip, and do not get a permit, all is NOT lost. There are plenty of alternatives to Mt. Whitney, that offer spectacular hikes with or without camping. For day hikes there are no permits typically required, and for overnight hikes permits are normally not a problem.
All along hwy 395 there are roads going west that lead to trail heads that lead to magnificent hiking and camping. One web site you might want to check out is: http://www.mammothcityconcierge.com/activities/trail-heads.asp
This web site is a bit out of date, so not all of the information is correct (toilets along the Mt. Whitney Trail), so keep that in mind.
Check it out. Even If you miss Mt. Whitney, you will still not be disappointed.