I know there are 12 rotational symmetries of a regular tetrahedron ${T}$ (including the identity $e$), but I can only find 9 of them.
The first 8 are the $r = \pi/3$ rad rotations with 1 vertex fixed at a time. There are 4 vertices of interest, and there can only be 2 angles of rotation for each case ($r=\pi/3, r^2 = 2\pi/3$, with $r^3=e$), so then there are $2\cdot 4 = 8$ reflectional symmetries, plus the identity $e$, this gives $8+1=9$ symmetries. What are the other 3?
And I know there are 12 reflectional symmetries of $T$ as well. I know I can reflect about the axes through the top vertex and the midpoints of bottom edge (there are 3 such axes). I can reflect about the axes that pass through a bottom vertex to the midpoint of the opposite face (3 more axes). Each reflection can only be done once, so this gives me 6 reflections. What could the other six be?
The regular tetrahedron has $6$ edges in $3$ opposite pairs. There is a $2$-fold symmetry (rotation through angle $\pi$) about the axis that pierces the midpoints of a pair of opposite edges. Look at the center image here.