I would like to create groups with $n$ elements for a given $n$.
I know that I can take some product of $\mathbb Z / p_i \mathbb Z$ for some primes $p_i$. But I want to find less obvious (=more interesting) ways to construct groups.
My ideas so far:
For example, let $n=105$ or $n=44$.
My first idea was to use groups of units, $U(n)$. Then I found out that this cannot work for odd $n$ because the $U(n)$ always have even number of elements. So this already fails for $n=105$.
My second idea was that one could consider matrix groups, like the special linear group. But the number of elements, say over $\mathbb K = \mathbb Z / p \mathbb Z$, is a square number. Of course, neither $105$ or $44$ are square numbers. So this method is no good either.
My third idea was to use some cyclic subgroup of some group, possible the group of units, but this idea is a bit fuzzy as it's not clear to me to construct the right number of elements in the subgroup. Is there a method to define a subgroup that has $n$ elements?
What methods are there to construct groups of a given number $n$ of elements?