Subgroup ($H$) information
| Description: | $C_3^2:\GL(2,\mathbb{Z}/4)$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Index: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(17,20)(18,19), (17,18)(19,20), (1,2,4,7,11,14)(3,6)(5,9,12,15,16,8)(10,13) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_2^4:A_4^2.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(221184\)\(\medspace = 2^{13} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{10}.C_3^4.C_2^5$ |
| $\operatorname{Aut}(H)$ | $S_4\times D_6^2$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| $W$ | $A_4:S_3^2$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^2:\GL(2,\mathbb{Z}/4)$ |
| Normal closure: | $C_2^4:A_4^2.\GL(2,\mathbb{Z}/4)$ |
| Core: | $C_2^3$ |
Other information
| Number of subgroups in this autjugacy class | $256$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^8.A_4:S_3^2$ |