Subgroup ($H$) information
| Description: | $D_6:S_3^2$ |
| Order: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Index: | \(3\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, c^{4}, b^{2}, b^{3}, e, c^{3}, c^{6}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_6.S_3^3$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times S_3\times C_2^2\times C_3.S_3^2$ |
| $\operatorname{Aut}(H)$ | $D_6\times D_6\wr C_2$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $D_6\times D_6^2$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $S_3^3$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $C_3.S_3^3$ |