Subgroup ($H$) information
| Description: | $(C_6\times D_6):S_4$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,16)(2,12,13,14,11,3)(4,17)(5,9)(6,15,10,8,18,7)(19,24,21,26,25,23)(20,22) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $(C_2\times C_6^3):D_6^2$ |
| Order: | \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^2:(Q_8\times F_5)$, of order \(497664\)\(\medspace = 2^{11} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $(C_2^2\times C_6^2).D_6^2$ |
| $W$ | $C_6^2.(D_6\times S_4)$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | not computed |
| Projective image | $(C_2\times C_6^3):D_6^2$ |