The tetrahedral group is the finite symmetry group of a tetrahedron.
As a symmetry group of one of the Platonic solids, the tetrahedral group participates in the ADE pattern:
More in detail, there are variants of the tetrahedral group corresponding to the stages of the Whitehead tower of O(3):
the full tetrahedral group is the subgroup of O(3)
which is the stabilizer of the standard embedding of the tetrahedron into Cartesian space $\mathbb{R}^3$;
the rotational tetrahedral group $T \hookrightarrow SO(3)$ is the restriction to orientation-preserving symmetries, hence to SO(3); this is isomorphic to the alternating group $A_4$;
finally the binary tetrahedral group is the double cover, hence the lift of $T$ to Spin(3)$\simeq$ SU(2);
next there is a string 2-group lift $String_{2T} \hookrightarrow String_{SU(2)}$ of the icosahadral group (Epa 10, Epa-Ganter 16)
The full tetrahedral group is Isomorphic to the symmetric group $S_4$ of permutations of four elements (see Full tetrahedral group is isomorphic to S4).
The subgroup of orientation-preserving symmetries is isomorphic to the alternating group $A_4$.
$\vert T_d\vert = 24$
$\vert T\vert = 12$
$\vert 2T\vert = 24$
The group cohomology of the tetrahedral group is discussed in Groupprops, Kirdar 13.
Wikipedia, Tetrahedral symmetry
Groupprops, Group cohomology of symmetric group:S4
Mehmet Kirdar, On The K-Ring of the Classifying Space of the Symmetric Group on Four Letters (arXiv:1309.4238)
Narthana Epa, Platonic 2-groups, 2010 (pdf)
Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, (arXiv:1605.09192)