Subgroup ($H$) information
| Description: | $(C_2\times C_4^2).S_4$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,6,2,5)(3,8,4,7)(9,14)(10,13)(11,16)(12,15)(17,23,18,24)(19,21,20,22) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2:A_4.C_2^5.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_4^2:A_4.C_4.C_2^3$ |
| $\operatorname{res}(S)$ | $C_2^6:C_3\wr C_3$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $(C_2\times C_4^2).S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$ |