"Are there any groups of primes larger than 3?"

Well, it depends on what you mean by "group".

The sequence 3, 5, 7 is the only "triple prime" sequence. (You can prove that for any 6 consecutive integers, 3 must divide one even and one odd number, meaning that only for the sequences of integers from 2 to 7 or from 3 to 8 will allow a triple prime.)

If you take the "blasphemous" position that 1 is prime, then you could technically have 4 odd prime numbers in a sequence, and 5 of 7 numbers in that interval be prime.

Other than that, the best interval of twin primes I can think of is the 101-103, 107-109 "group": it is the third pair of twin primes that occurs within a span of nine integers. (The earlier ones being 5-7, 11-13 and also 11-13, 17-19.) (Note that in a sequence of 5 consecutive odd numbers, one of them must be divisble by 5. It also just so happens that $3, 5, 7 | 105$, leaving the rest of the odd numbers in that sequence off the hook.

[edit: typo; 15 is not prime, as I had accidentally stated!]