The best dates in terms of availability are, unfortunately, not the best dates for a group of relatively novice hikers - unless you get some snow and ice practice around LA this winter with ice axe and crampons. The Whitney trail can have abundant snow and ice on it well into June, and then be back to semi-winter conditions by September or early October. These are radically different hiking conditions than a clear trail in August - but much more demanding and dangerous. You would probably have no trouble getting an overnight permit for 6 in May or October - but you'd likely need winter climbing gear and the skills to use it.
The highest demand for trail permits is, of course, during the summer months when the trail is at its clearest and folks are burning vacation time - June through August. It's Grand Central Station at the Portal and on the mountain those months. If you can get everyone to agree on going mid-week, that's your best bet for getting lottery dates. Remember, you can enter as many dates on one lottery application as you want. As Steve suggested, if you get skunked in the lottery, look at the past year's actual permit usage (conveniently located here at the Zone) and map a strategy to target calls to the Inyo around those similar time frames for 2011. People begin cancelling permits about 10 days out from their start date, so you will almost certainly wind up a week out - or less - from your scheduled group day calling INFS frantically seeking 6 cancellations for the same day.
Even though I flew 2000 miles out there this past July with no permit in hand, and succeeded in snagging two cancellations within a day of my target date, I wouldn't bet the house that 6 will pop open for a single day in advance. That's more likely (in the summer) to happen with no-shows the day of, and you'd have to have your group out there with gear and ready to go the same day - or not, if you're unlucky. That's a tough call with 6 different people. If you press ahead and plan this, and get to this point, be sure you have a Plan B alternate trail in mind. September is probably your best bet to have this happen successfully, and still have a decent chance with the weather.
San Jacinto is in Palm Springs - the peak with the tram to near the top. It's a bit over 11,000 feet and has several really demanding trails. My hiking partner from the Inland Empire this past July spent most of his training time here or on Baldy. San Gorgonio is - I believe - in the San Bernardino mountains, in the San Gorgonio Wilderness. It's over 10K' and has several trails that exceed 4000 feet elevation gain.
