[jjklxjsr] When is group theory practically useful?

Primality proving: Examine the group structure of the multiplicative group (Z/nZ)* for the number n being tested (Lucas primality test, requiring factoring n-1).

To determine whether a polynomial of degree 5 or greater is solvable in radicals, examine the structure of its Galois group.  But there are many tricky details: General Formulas for Solving Solvable Sextic Equations, Thomas R. Hagedorn.

Examine the subgroups of a twisty puzzle like the Rubik's cube to find an admissable heuristic for A* search.

We do not include instances of group theory merely being used to prove a theorem, though the distinction is hazy ("prove that 17 is prime", "prove this algorithm is correct").