Subgroup ($H$) information
| Description: | $(C_2^2\times C_4^2).S_4$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,14,8,12,2,13,7,11)(3,16,6,10,4,15,5,9)(17,20)(18,19), (1,2)(3,4)(5,6) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_2^8.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(24576\)\(\medspace = 2^{13} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4\wr C_2\times S_6$, of order \(393216\)\(\medspace = 2^{17} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_4^2:A_4.D_4.C_2^3$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^6.\GL(2,\mathbb{Z}/4)$ |
| Normal closure: | $C_4^2:C_2^4.S_4$ |
| Core: | $D_4^2:C_2^2$ |
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_2^2\times S_6$ |