DestinyLFG is your friend. I have beaten Crota three times already with a complete group of strangers each time. A good word of advice is to make sure the players you are with are competent. I suggest looking at their grimoire score or their play time. It is usually a good sign if they are decent or not.
In terms of taking down Crota, I did it (my fastest run yet) in 3 relic rounds (6 sword attack rounds total).
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How? Lol. My last run we did 4 hits on the first time then dropped him again after he stand up and hit him 3 times...2nd time we hit 3-2 hits and so on for 4 waves until enrage....this is a good sign yea? Meaning we can survive long enough till he rages just need to work on maximizing damage(sword was 31,random party)
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Edited by Nicewander: 12/14/2014 7:12:58 PMI timed my movements appropriately based on when I thought his shields were almost down. I did my fastest time with my Hunter with the invisibility active. If you're next to Crota right when he starts going down, you can get four swipes (RB) and two downward strikes (R1) in before he gets up.
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Heres a little tip, the person whos doing the damage with the sword, try to create orbs for him so he can get his super as he's on his way to position him, will deal more damage, so that will help out with dealing more damage.