Subgroup ($H$) information
| Description: | $(C_2^2\times C_4^2).S_4$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(5,6)(7,8)(13,14)(15,16)(21,22)(23,24), (1,2)(7,8)(9,16)(10,15)(11,13)(12,14) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $(C_2\times C_4^3).\GL(2,\mathbb{Z}/4)$ |
| Order: | \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^3:C_{20}.D_4$, of order \(49152\)\(\medspace = 2^{14} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_4^2:A_4.D_4.C_2^3$ |
| $W$ | $C_5^2\wr C_2:C_4^2$, of order \(20000\)\(\medspace = 2^{5} \cdot 5^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_5\wr C_2^2:C_4$ |