Subgroup ($H$) information
| Description: | $D_6^2:C_2^2$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(2,4,6), (1,5)(2,6), (1,5)(2,6)(7,10)(8,12)(9,13)(11,14), (1,6)(2,3)(4,5) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $S_3^2:C_2\wr A_4$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.(C_2\times A_4).C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_3^2.C_2^6.C_2^5$ |
| $W$ | $C_6^2:D_4$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $D_6^2:(C_2\times A_4)$ |