For which $m,n>1$ is $D_{mn} \cong D_n \times\Bbb Z/m\Bbb Z$?
I did a similar question before and was trying to follow the same format, but it didn't work because I don't know how to find the number of elements of order $2$ in $D_n \times\Bbb Z/m\Bbb Z$
I was thinking about doing four cases:
Case 1: $m$ and $n$ are both even
Case 2: $m$ is even and $n$ is odd
Case 3: $m$ is odd and $n$ is even (which would be very similar to Case 2)
Case 4: $m$ and $n$ are both odd
Now I am not sure where to go with these cases.