My attempted solution is as follows:

1. .

Since 11 is not congruent to 1modulo7, it has one Sylow 7-subgroup and one Sylow 11-subgroup, which has an intersection at {e}. Thus, the group of order 77 is the direct product of a normal Sylow 7-subgroup and a normal Sylow 11-subgroup, which is cyclic.

I think there is only one group of order 77 up to isomorphism.