A few tidbits from the guidebook "Western Whitewater" (Cassidy, Cross, & Calhoun, 1994), on the Desolation and Gray Canyons section of the Green River:
Eleven Class II +/- rapids, with one Class III- rapid (Coal Creek Rapid 69.5 miles below put-in)
Recommended levels: 1,000-20,000 cfs, shallow and rocky below 2,000.
Scenery: Excellent. Contrasting desert canyons.
Solitude: Excellent (except mid-May through July and in October)
Wildlife: Bighorn sheep, deer, birds
Water: Warm and murky in summer, purify and filter. Carry drinking water.
Camping: Many broad beaches, especially in Desolation Canyon.
Side hikes: Many excellent side canyons.
Selected quotes from the text:
"Today, the ramparts of Desolation and Gray Canyons look much as they did when Major Powell and his men first ran the river in 1869. These stark chasms remain virtually untouched by civilization."
"By any name, the Desolation-Gray float is one of the Canyon Country's finest, with excellent scenery, superb side hikes, expansive campsites abundant wildlife (especially in spring), and above all, sereneity and solitude."
You might want to read a guidebook to help you decide. Listed in Western Whitewater:
"Desolation River Guide", Evans & Belknap.
"River Guide to Desolation and Gray Canyons", Rampton
Also, call and chat with the BLM rangers, they are usually more than happy to help.
Weather Service flash flood outlook for that area Sunday-Friday. Be careful camping and hiking in the side canyons.
I've never floated Deso-Gray, but it's on my list. If it's like other Canyon Country rivers, the side hikes will be at least half the fun. Hope your group plans on lots of side hikes – maybe you could lead some?
A river trip in the most awesome setting can be much less fun if the people don't get along. People are stressed by the weather, river, sand in their PB&J, whatever, so try to cut everyone a lot of slack. To much alcohol and other recreational substances can amplify the conflicts, so worry if you lose count of the cases of beer loaded onto rafts. Might be good to ask in advance what trip policies are around alcohol etc.
I say: go for it.