Subgroup ($H$) information
| Description: | $D_4^2:S_3$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(8,13)(9,12), (7,10)(8,12)(9,13)(11,14), (9,12)(10,11), (3,5)(4,6)(7,8) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $(C_2\times D_6^2):S_4$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.(C_2^4\times A_4).C_2^4$ |
| $\operatorname{Aut}(H)$ | $S_3\times C_2^3.D_4^2$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $W$ | $C_2\wr C_2^2\times S_3$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $D_6^2:S_4$ |