Actually, cannot be cyclic. It isn't even abelian! With this in mind, you can remove (a) and (b) from consideration.
To deal with (c) and (d), I would try looking at the orders at some elements in each group. Compare them with your .
Using the external direct product , the group
is isomorphic to one (and only one!) of the following groups (this question reminds me of a game show...)
I am assuming these groups are the well known groups
are the integers modulo n under addittion
is the alternating group (all even permutations of n) under function composition
is the dihedral group of order
and is the set of permutations of n
The question recommends determining which groups it is not isomorphic to and determining the answer by elimination.
I started by the question noting that is cyclic, I believe? so it must be isomorphic to a cyclic group as well...
and also it must have the same number of elements of each order of any group it is isomorphic... but even armed with those facts I am having trouble finding that any of a), b), c) or d) are isomorphic to the given group
Any help appreciated!