The tetrahedral group is the group of orientation-preserving symmetries
of the regular tetrahedron.  It has size 12.  Given an specific regular 
tetrahedron in 3D, it also acts as a rotation group on R^3.  This can be 
conveniently described via quaternions.  The details of the following 
explanation were taken from

    http://www.math.sunysb.edu/~tony/bintet/index.html

Identify R^3 with the space spanned by i,j,k in the space of quaternions
H.  Consider the regular tetrahedron given by the following vertices:
   A=-i-j-k, B=-i+j+k, C=i-j+k, D=i+j-k
Note that these are half the vertices of the standard cube of side length 2.
The 12 elements of the tetrahedral group are
    e: the identity element
    i,j,k: 180 rotations about the coordinate axes i,j,k
    a,b,c,d: 120 degree positive rotations about the ray from the origin through 
        A,B,C,D, respectively
    a2,b2,c2,d2: the squares of the previous elements
Quaternion representations of these elements are
    e: 1
    i,j,k: themselves
    a,b,c,d: a=(1/2)(1+A), and respectively for b,c,d
These letter names are also used as static members in the class.

There are several other groups with important relationships with the tetrahedral
group, including the binary tetrahedral group, the octahedral group, and the
alternating group on four elements.

The binary tetrahedral group is the group of quaternion representations of the
tetrahedral group elements, and forms a double covering of the tetrahedral group.
Specifically, each tetrahedral group element is represented by the quaternion
given above and its negation.

The octahedral group is the group of orientation preserving symmetries of the
octahedron or cube, or equivalently the symmetries of oriented axis-aligned
space.  It has size 24.  Since we our tetrahedron vertices are also vertices 
of the standard cube, each tetrahedral group element is naturally an element of 
the tetrahedral group.  The other elements of the octahedral group are those
that map our standard tetrahedron to the other choice of regular tetrahedron
vertices, starting with A'=i+j+k.  These include the 90 degree rotations about
any axis.

Since "most" of the structure of the octahedral group is contained in the
tetrahedral group, it can sometimes be used to represent orientations of axis
aligned objects with fewer possibilities (12 instead of 24).  For example,
the 6 possible distinguishable orientations of bcc tetrahedra can be given by
tetrahedral group rotations of a standard bcc tet.  Since there is no subgroup
of the tetrahedral group of size 6, this is the minimal group capable of
representing bcc orientations.

Finally, since the tetrahedron is also the complete graph on 4 nodes, the
tetrahedral group is naturally isomorphic to the alternating group on 4 elements.
