With a Tri Core Members of a Fireteam Triad gain the Bonus of Full Core Team. Can I create a Special Fireteam Triad with a Chaksa HMG and a Tri Core to gain the Bonus or is that canceled by the "Special" Prefix?
They’re all just Fireteam:Triad’s; the Tri Core rule doesn’t change how many of those you can make. If you see the word “Special” in a fireteam list, that just means the membership list is different. You can only make one Haris because of the Haris rules. You can have any number of Duos because of the Duo rules. You can have any number of Triads (or Special Triads) because of the Triad rules. Only one Triad can benefit from Tri-Core because the Tri-Core rules say you’re only allowed one Tri-Core trooper (restriction specified in the Tri-Core skill rules).
Unfortunately that special triad specifies one taagma of any variety but both chaska have to be the special triad version. So you can't tri core the hmg one.