Subgroup ($H$) information
| Description: | $C_3^3.(C_6\times S_4)$ |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Index: | \(3\) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$ac^{2}e^{5}f^{4}, f^{3}, e^{2}f^{4}, f^{2}, d^{9}, bc^{2}d^{2}e^{3}f^{3}, e^{3}, d^{6}, cd^{6}e^{2}f^{4}$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^5:(C_2\times S_4)$ |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times C_6^2).C_3^4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_3^3.(C_6\times S_4)$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $W$ | $C_3^3.(C_6\times S_4)$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-1$ |
| Projective image | $C_3^5:(C_2\times S_4)$ |