Direct product and isomorphism problem

Let G = Z_3 direct product Z_3 direct product Z_3 and let H be the subgroup of SL(3, Z_3) consisting of 1 a b the matrix H = { 0 1 c with a, b, c in Z_3 } 0 0 1 What is the order of G and H and are G and H isomorphic?

Permutation problem

Let b be the permutation (1 2 3)(4 5 6 7)(8 9 10 11 12 13) what is b^99 as a product of disjoint cycles. -I know b^99=b^3 but I'm a little confused on the disjoint cycles part.

Permutation problem

How many elements of order 5 are there in S_8? - I think there are 8! / 3!5! = 56 ways to order the elements in the cycle but how many of order 5 are there?

Group problem

1 3 1 4 Let the matrix A = 0 2 and the matrix B = 5 1 be elements in GL(2, Z_7). Find (A^-1 * B^-1)^-1. - I am unsure of when to perform the operation mod 7.

Group elements problem

List all of the elements of order 15 in Z_600. - I think the elements are 40*1, 40*2, 40*4, 40*6, 40*7, 40*8, 40*10, 40*11, 40*13