Subgroup ($H$) information
| Description: | $C_3^4:C_6$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(12,13), (6,8,9), (1,9,5,4,8,7,2,6,3)(11,12,13), (1,4,2)(6,8,9), (11,13,12), (3,7,5)(6,9,8)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^3:S_3\times S_4$ |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.C_3^4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_3^2.S_3^3$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| $\operatorname{res}(S)$ | $C_3^2.S_3^3$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_3^2:S_3^2$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $1$ |
| Projective image | $C_3^3:S_3\times S_4$ |