1. this is just straight-forward calculation:

H = {(2,2), (4,4), (6,6), (8,8), (10,10), (0,0)}. this has 6 elements, so so does (5,8) + H:

{(7,10), (9,0), (11,2), (1,4), (3,6), (5,8)}.

calculating powers of (5,8) + H, we get:

2[(5,8) + H] = (10,4) + H

3[(5,8) + H] = (3,0) + H

4[(5,8) + H] = (8,8) + H = H, so (5,8) + H is of order 4.

for G/H to be cyclic, it would need to have an element of order 144/6 = 24.

but G/H has no element of order > 12, since 12[(a,b) + H] = (12a, 12b) + H = (0,0) + H = H.

2. what have you tried so far?