Subgroup ($H$) information
| Description: | $C_3^2\wr C_2.Q_8$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(3,11)(6,15)(8,14)(9,18), (2,8)(3,6)(4,7)(5,13)(9,11)(10,16)(12,17)(15,18) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $(C_3^3\times C_6).Q_8^2:C_2^2$ |
| Order: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_4^2.C_2^5.C_2^3$, of order \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_4^2.C_2^4$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $\card{W}$ | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $16$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $(C_3^3\times C_6).Q_8^2:C_2^2$ |