Subgroup ($H$) information
| Description: | $C_2^2.D_6^2$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(9,11,14)(10,13,12), (1,5)(2,7)(3,6)(4,8)(9,11,14)(10,12,13), (1,8)(2,6) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $(C_2^3\times C_6^2):D_{12}$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.(C_2\times A_4).C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_6^2.C_2^6.C_2^6$ |
| $W$ | $D_6^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $(C_2^2\times C_6^2):D_{12}$ |