In over 500 hundred strike completions with which were mostly match made, I can say definitively that the group I ran with yesterday was the worst group I have ever run with. They were so bad, that it actually prompted me to write an explanation of how to run the relic in the Echo Chamber. It took us close to an hour when normally this strike should take 20 minutes.
1st: If there is a Nightstalker in the group, they should take the relic. If there is a hunter in the group and no one on Nightstalker, you should switch to Nightstalker. If there isn't a Nightstalker, I'll include some advice on general relic using, but maybe someone else can describe how they use Warlock or Titan below.
2nd: Make sure your strength is as maxed as you can with your armor. Make sure you have Lockdown, Vanish in Smoke and Triple Jump.
3rd: Melee and Grenades are supercharged while holding the relic, so use Vanish in Smoke to stay invisible. You will get a Melee back by the time your old Melee is up. Use it to stay invisible and move on without your fireteam.
4th. Jumps are mostly unaffected by the speed debuff, so use that a lot. It is quicker to jump than run, so use your jump to climb over things, instead of walking around them. Also avoid the Taken Blights, it is quicker to go around them, than to go through them.
5th: If your fireteam can't keep up, use invisibility to grab the relic. If they can, kill the major that is guarding it and move on.
6th: If you are on the cyclops run, kill the cyclops that is next to the relic and jump the length of the chasm, grab it and use invisibility to get to the end.
If you don't have a nightstalker: Grenade and Melee is super charged while holding the relic. You should always be spamming grenades to help your fire team clear the mobs. You can drop the Relic. If your fireteam is getting overrun, drop the relic for a while and start fighting them off. Whatever you do, don't just take the relic and hide in a corner and wait for the rest of the team to kill everything.
Titans
1st: Make sure you have the better control option selected for you jump. You can move very fast while handling the relic by double tapping the jump button (Titan skating). Use this to pass the spawning mobs.
2nd: Make sure to spam your grenades when you reach the next wall in order to kill the mobs that will spawn there, so you can place the relic without dieing.
Thanks to Wolfblur, TheDekuScrub, Artic_Angle10, Cmcd187, 1and AceUpThySleeve for the heads up.
Restorative Mind fight:
If you have a nightstalker in the fireteam they can use their invisibility to grab the relic. If not, one person should go to take the relic and the other two should shoot the Restorative mind in order to draw its fire away. After the person has the relic and is out of line of sight from the boss, the spotters should fall back and join them. After you get to the container, wait for the enemies to spawn. The fireteam should then kill the mobs. After the mobs are dead, the two without the relic go back to the center and the person with the relic puts it in and joins them. Now you have the entire length of the Mind's movements in order to kill it. Most people put the relic in as soon as they can and then you have to kill the mobs before attacking the Mind and you usually only have 1/2 or 1/3 of the distance.
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At the boss, DO NOT put the relic I'm until the ads are cleared, so all fire team members can do maximum DPS. You can kill him on the 2nd run, if all are running snipers.