Given any finite group with multiplication $m(-,-)$ and three permutations $p,q,r$ on the underlying set of the group, we can obtain a quasigroup with binary operation $g\*h:= p(m(q(g),r(h)))$. What is an example of a finite quasigroup that provably does not arise in this way?



Finite quasigroup not coming from a scrambled finite group
The construction is called "isotopy" and you ask for quasigroups that are not "isotopic to groups". 


It's been proved a long time ago that if a loop (= a quasigroup with a unit element) is isotopic to a group G, it is actually isomorphic to G. Consequently, any non-associative loop is an answer to your question. The smallest example has five elements, and is unique (up to isotopy):



```
0 1 2 3 4          
1 0 3 4 2            
2 3 4 1 0            
3 4 0 2 1    
4 2 1 0 3

```

The same theorem gives a simple criterion to recognize quasigroups isotopic to a group. Given a quasigroup with an operation \*, pick an element e and form a new quasigroup, by x # y = x/e \* e\y. This is a loop, where e\*e is the unit element. If # is associative, 
then, obviously, \* is isotopic to a group. If # is not associative, the theorem assures no group isotope can be found.


