Subgroup ($H$) information
| Description: | $(S_3\times C_6^2):S_4$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,7,2)(3,16,6)(4,15,11)(5,8,13)(9,18,12)(10,14,17)(19,21,22)(20,24,23) \!\cdots\! \rangle$
|
| 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 \(497664\)\(\medspace = 2^{11} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $(C_2^2\times C_6^2).D_6^2$ |
| $W$ | $C_5^2:C_8\times F_5$, of order \(4000\)\(\medspace = 2^{5} \cdot 5^{3} \) |
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 | $(C_2\times C_6^3):D_6^2$ |