$A_{n \geq 1}$ | | cyclic group $\mathbb{Z}_{n+1}$ | cyclic group $\mathbb{Z}_{n+1}$ | special unitary group $SU(n+1)$ |
A1 | | cyclic group of order 2 $\mathbb{Z}_2$ | cyclic group of order 2 $\mathbb{Z}_2$ | SU(2) |
A2 | | cyclic group of order 3 $\mathbb{Z}_3$ | cyclic group of order 3 $\mathbb{Z}_3$ | SU(3) |
A3 = D3 | | cyclic group of order 4 $\mathbb{Z}_4$ | cyclic group of order 4 $2 D_2 \simeq \mathbb{Z}_4$ | SU(4) $\simeq$ Spin(6) |
D4 | dihedron on bigon | Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$ | quaternion group $2 D_4 \simeq$ Q8 | SO(8), Spin(8) |
D5 | dihedron on triangle | dihedral group of order 6 $D_6$ | binary dihedral group of order 12 $2 D_6$ | SO(10), Spin(10) |
D6 | dihedron on square | dihedral group of order 8 $D_8$ | binary dihedral group of order 16 $2 D_{8}$ | SO(12), Spin(12) |
$D_{n \geq 4}$ | dihedron, hosohedron | dihedral group $D_{2(n-2)}$ | binary dihedral group $2 D_{2(n-2)}$ | special orthogonal group, spin group $SO(2n)$, $Spin(2n)$ |
$E_6$ | tetrahedron | tetrahedral group $T$ | binary tetrahedral group $2T$ | E6 |
$E_7$ | cube, octahedron | octahedral group $O$ | binary octahedral group $2O$ | E7 |
$E_8$ | dodecahedron, icosahedron | icosahedral group $I$ | binary icosahedral group $2I$ | E8 |