Subgroup ($H$) information
| Description: | $C_3^3:Q_8$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(7,12)(8,11)(9,10)(13,15), (1,5,4)(2,6,3)(7,15,13,8)(9,11,14,12), (7,14,12) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $F_9:S_6$ |
| Order: | \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSU(3,2):C_2.A_6.C_2^2$, of order \(207360\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_3^2:\GL(2,3)$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| $W$ | $F_9:C_2^2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $60$ |
| Möbius function | $0$ |
| Projective image | $F_9:S_6$ |