Subgroup ($H$) information
| Description: | $C_3^2:C_{12}$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,2)(3,4,6,5), (3,6)(4,5), (1,6,3), (8,10,9), (2,5,4)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_6^2:D_{12}$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_9:C_2\times S_4$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $F_9:C_2^2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $F_9:C_2^2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $\SOPlus(4,2)$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3$ | ||
| Normalizer: | $C_3^2:D_{12}$ | ||
| Normal closure: | $C_6^2:C_{12}$ | ||
| Core: | $C_3^2:C_4$ | ||
| Minimal over-subgroups: | $C_6^2:C_{12}$ | $C_3^2:D_{12}$ | |
| Maximal under-subgroups: | $C_3^2:C_6$ | $C_3^2:C_4$ | $C_{12}$ |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $1$ |
| Projective image | $C_6^2:D_{12}$ |