Subgroup ($H$) information
| Description: | $Q_8:S_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(3,5)(4,6), (1,7,2,8)(3,4,5,6), (1,4,2,6)(3,8,5,7), (3,7,4)(5,8,6), (4,6)(7,8), (1,2)(3,5)(4,6)(7,8), (4,7,6,8)(9,14)(10,11)(12,13)\rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $(Q_8\times C_3^2):S_4$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^4:\He_3.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^4:S_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^4:S_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2^2:S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $1$ |
| Projective image | $C_6^2:S_4$ |