Do colleges see office referrals?

Currently a junior - my chemistry teacher caught around 16/17 people who had the exact same answers and “wrong” work on a homework assignment. He’s giving us zeroes, an office referral, and a call home. I know that he dislikes me in particular. I was wondering if colleges would be able to see this on my permanent record and how it would affect my chances of getting in. If they can see it, how would I go about explaining?

How were you involved in this?
Do you understand why this is wrong?

Talk to your guidance counselor about what will be shown on your transcript/references.

Understand that colleges take cheating/plagiarism seriously.