Subgroup ($H$) information
| Description: | $C_6^2:S_4$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(9,13)(10,11), (14,15), (1,2,3)(10,11,13), (1,2,3)(4,7,5), (9,11)(10,13), (8,12)(14,15), (4,7,5), (1,3)(5,7)(8,15,12,14)(11,13)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3:S_4\times \GL(2,\mathbb{Z}/4)$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_3.(C_3\times A_4^2).C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_3^4.Q_8.(S_3\times S_4)$ |
| $W$ | $C_6:S_3\times S_4$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2\times A_4^2.D_6$ |