Subgroup ($H$) information
| Description: | $D_6:S_3$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,77,80,23)(2,46,81,5)(3,65,22,31)(4,48,16,56)(6,25,47,51)(7,49,52,43) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^3:S_3.C_2^4:S_5$ |
| Order: | \(311040\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:S_3.C_2^4:S_5$, of order \(311040\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $D_6\wr C_2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| $W$ | $S_3^2:C_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $1080$ |
| Möbius function | $0$ |
| Projective image | $C_3^3:S_3.C_2^4:S_5$ |