Subgroup ($H$) information
| Description: | $C_{12}^2:(C_2\times C_4)$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(3,6)(4,5), (9,11,10)(12,15,14), (2,7)(3,6)(12,14,15), (1,8)(2,7)(3,6)(4,5) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_6^4:(C_2\times S_4)$ |
| Order: | \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:D_4$, of order \(497664\)\(\medspace = 2^{11} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $(C_3\times C_6).C_2^6.C_2^5$ |
| $W$ | $C_2^3.D_6^2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | |
| Normalizer: | $D_4^2:S_3^2$ | |
| Normal closure: | $C_6^4:(C_2\times S_4)$ | |
| Core: | $C_2^2\times C_6$ | |
| Minimal over-subgroups: | $C_6^3.(S_3\times D_4)$ | $D_4^2:S_3^2$ |
Other information
| Number of subgroups in this autjugacy class | $27$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_6^3:(S_3\times S_4)$ |