TITLE: Are there real world applications of finite group theory?
QUESTION [112 upvotes]: I would like to know whether there are examples where finite group theory can be directly applied to solve real world problems outside of mathematics.  (Sufficiently applied mathematics such as cryptography, coding theory, or statistics still count.)
Let me clarify: I am not interested in applications of elementary group theory which happen to involve finite groups (e.g. cyclic/dihedral/easy groups as molecular symmetries).  I am interested in applications of topics specifically coming from finite group theory as a discipline, like one might see in Isaacs, Huppert, or Robinson.
"The Schur multiplier has order 2640, so we should point the laser that way." 
"Is this computer system secure?"  "No - Frobenius kernels are nilpotent."
I'm aware of this MO post, but many of the applications listed there are inside mathematics or fall in the "applications of easy groups" category.  It is entirely possible that what I'm looking for doesn't exist, and that finite group theory is still an untouchable, pure subject, like number theory in the days of G. H. Hardy.  But perhaps not.  Does anyone know of any applications of the higher level stuff?

REPLY [9 votes]: I would look in a few places.
Expander graphs, which may have originated in telephone network design and are now tied to many other areas.  A construction might use Cayley graphs of generating sets of large finite groups over a finite field.
Combinatorics of Weyl groups in types B, C and D. Applications often need constructions that come in families and can scale up beyond particular sizes and dimensions.  This field is probably too recent to have seen many applications.
Experimental designs in statistics that use highly symmetric block designs.  This is similar in flavor to coding theory and sphere packings. Amounts of symmetry used are less than in those classical application areas, but there is a large literature and who knows what might be there.  As nonparametric Big Data statistics creates new needs the math will undoubtedly get more complicated.
Closer to the continuous case, John Baez has posted in his weekly series on the Frobenius-Schur indicator as a sign of real, complex and quaternionic aspects of physics.