*SPOILER* I think?
Is a elite hunt you get from Montblanc when u reach the paragon of justice rank.

He appears in th Lhusu mines just West of the Orange Save crystal which is what you encountered while trying to do the anitlon infestation (if you did it) While fighting hime you are able to steal from him 5 times, the last 2 times being the extremely rare Genji Golves and Genji Shield
After you kill him and his dog He moves to the deeper area of the Lhusu mines.
Gilgamesh's stats are lv40-50 and his dog is around lv40-50
when you beat him his dogs tats are lv 65 and his stats are lv 70 his last to stances let you steal the genji helm and genji armour

After you've killed him and his dog you'll get your bounty which contains gil and the rare MASAMUNE
Battle plan is rather easy, steal from him every stance to get potions and gil and also to get the items and keep one person hacking and another person curing the stealer and the attacker, it is recommended you cast dispel on gilgamesh and enkidu at the start or with foe>any (>) dispelga and cast decoy on the attacker so the stealer/healer doesn't pull to much attetion
P.S take enkidu out first and when Gilgamesh begins his special move just quickly take out a summon like addremelech or just counter with a quickening and pull off some nasty chain!

I hope this helped you and some other people do this hunt!

:greenking