Subgroup ($H$) information
| Description: | $(C_3\times C_{12}):C_4$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Index: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(2,9,11,15)(6,18,14,8), (1,17,12)(4,5,10)(7,16,13), (2,11)(4,7)(5,13)(6,14) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), and metabelian.
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 \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $D_4\times F_9:C_2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $W$ | $C_6^2:Q_8$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $36$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $(C_3^3\times C_6).Q_8^2:C_2^2$ |