Q:

Trirectangular tetrahedron

Looking at http://mathworld.wolfram.com/TrirectangularTetrahedron.html
I wonder what the symmetry group of a trirectangular tetrahedron is? 

A:

Any symmetry operations of the trirectangular tetrahedron will send vertices to vertices, edges to edges and faces to faces.
Since the vertex with three right angles is distinguished, any symmetry operation will leave that vertex unchanged. 

If the lengths of the edges attached to that vertex are all different, any symmetry operation will leave them fixed. This fixes the symmetry operation to the identity operation. i.e the symmetry group will be trivial.
If two of the edges is the same, a symmetry operation will either leave the two edges unchanged or flip them. The corresponding symmetry operation can either be the identity operation or a reflection with respect to the plane perpendicular bisect the two corresponding vertices. The symmetry group is the
crystallographic point group $C_s = C_{1h}$ which is isomorphic to $S_2$, the group of permutations of $2$ symbols.
If all three edges are the same, a symmetry operation can either be

a multiple of a 3 fold rotation along an axis which passes through the distinguished vertex and the center of its opposite face.
or one of the three reflections with respect to a plane like that in case 2.

There are total six of them and there is an one to one correspondence between them and the permutations among the $3$ non-distinguished vertices. The symmetry group is the point group $C_{3v}$ which is isomorphic to the $S_3$, the group of permutations of $3$ symbols.

Summary

If $a \ne b \ne c$,  the symmetry group is trivial.
If $a = b \ne c$ or similar, the symmetry group is $C_{1h} \approx S_2$,
If $a = b = c$, the symmetry group is $C_{3v} \approx S_3$.

