Subgroup ($H$) information
| Description: | $C_2\times F_9$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Index: | \(3\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a, d, cd, b^{3}, b^{6}, b^{12}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $S_3\times F_9$ |
| Order: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_9:D_6$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $F_9:C_2^2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $F_9$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | |||
| Normalizer: | $C_2\times F_9$ | |||
| Normal closure: | $S_3\times F_9$ | |||
| Core: | $F_9$ | |||
| Minimal over-subgroups: | $S_3\times F_9$ | |||
| Maximal under-subgroups: | $F_9$ | $C_2\times C_3^2:C_4$ | $F_9$ | $C_2\times C_8$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $S_3\times F_9$ |