Subgroup ($H$) information
| Description: | $(C_6\times D_{12}):C_2^2$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(8,13)(9,12), (7,10)(8,12)(9,13)(11,14), (7,8)(9,11)(10,12)(13,14), (9,12) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^2:C_2\wr S_4$ |
| Order: | \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times (C_3\times C_2^4:C_3).D_6^2$ |
| $\operatorname{Aut}(H)$ | $(C_6\times D_4^2).C_2^3$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^4:D_4\times D_6$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $W$ | $C_2^4:D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $1$ |
| Projective image | $(C_6\times D_6):S_4$ |