Subgroup ($H$) information
| Description: | $C_3^4:Q_8$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,3)(2,5)(4,6)(8,10,9,13)(11,12,15,14), (1,5,4)(2,3,6)(7,15,13,8)(9,11,14,12) \!\cdots\! \rangle$
|
| 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 \(207360\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_9:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $W$ | $F_9:D_6$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $20$ |
| Möbius function | $0$ |
| Projective image | $F_9:S_6$ |