Subgroup ($H$) information
| Description: | $C_3^4:D_4$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(3,6)(5,7), (2,7,5), (5,7)(11,12), (1,3,6)(2,7,5), (2,7)(3,6)(10,11,12), (1,2,3,5)(6,7)(8,9)(10,11), (1,3,6)(2,5,7)(4,9,8)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
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: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3^4:S_3$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $(C_3^2\times S_3^2):\SD_{16}$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| $\operatorname{res}(S)$ | $S_3^4:C_2$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_3^3:(S_3\times D_4)$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $C_3^3:(S_3\times S_4)$ |