Subgroup ($H$) information
| Description: | $(C_3\times C_6^2):A_4$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(10,12,11), (7,8)(13,15), (1,2)(3,4)(11,12)(13,15), (13,14,15), (10,12) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_2\times C_3^3:S_4^2$ |
| Order: | \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_3^3.A_4^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $S_3^3:A_4:S_4$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $W$ | $(A_4\times C_3^3):S_4$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2\times C_3^3:S_4^2$ |