Encase
culture
If you're sure about those numbers, I don't know. Some near misses: sucrose (7 letters) azucar (6 letters, but the c is the fourth place)
A science word that is six letters ans starts with the letter c is convex.
The letter C curves.The letter C is a partial circle.
Tragic
choice
racily
The six letter word that can be made with those seven letters are:charmsstarch
encode
Pisces
Encase
culture
Cello is a five letter word that sounds like six letters when pronounced. The letter c is pronounced ch to make the word sound like six letters.
The letters spell the 6 letter word crouch. The letters can be used to spell the 5 letter words couch and occur. They also can be used to spell hour and ouch.
Pavane ( if you want 6 letters ) or gavotte ( for 7 letters )
If you're sure about those numbers, I don't know. Some near misses: sucrose (7 letters) azucar (6 letters, but the c is the fourth place)