Hi Mary,
I would have suggested consulting the Muster Rolls for the Regiment at TNA, but as far as I can tell these only survive for the period after 1878 when he was already dead. However it was only a perfunctory search so it might be worth doing your own search using the TNA Discovery tool. If you do find a muster roll from the relevant period there's no guarantee it will show the soldiers listed by troop. I have never accessed the cavalry muster rolls so I don't know how the details were recorded.
Failing that I would suggest asking the two museums which hold some records and artifacts relating to the 3rd Dragoons: the
Cheshire Military Museum in Chester and the
Royal Scots Dragoon Guards Museum in Edinburgh.
All that having been said, I very much doubt if there will be anything in the way of paper records relating to the troop(s) he was in. As you are probably aware the cavalry troop was roughly equivalent in size to the Infantry platoon, that is to say it consisted of one officer and around 30 - 35 other ranks. Although the composition of the troop would have remained the same for as long as was practical it is also likely that your great great grandfather was in more than one troop over his 25 year career in the Dragoons.
Troops were not employed independently away from their regiment, so wherever in the world the Regiment was stationed, you can expect him and his troop to have gone also.The only variation would be that it is possible, as he was an experienced trooper, that he might have been employed for a period of time at the Regimental depot where he could have assisted in training the recruits.
His service record indicates that he was not promoted above the rank of private and having seen the very long list of his minor offences, this is perhaps unsurprising!
