Subgroup ($H$) information
| Description: | $C_3^3:C_{12}:C_8$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,4,7)(2,5,8)(3,6,9)(10,14,15)(11,12,16)(13,17,18), (10,15,14)(11,16,12) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $\SOPlus(4,2)^2.D_4$ |
| Order: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SOPlus(4,2)^2.D_4$, of order \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_4^2.C_2^4$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $\card{W}$ | \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $\SOPlus(4,2)^2.D_4$ |