Subgroup ($H$) information
| Description: | $C_3^3:D_4$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(19440\)\(\medspace = 2^{4} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,80,42)(2,52,69)(3,27,15)(4,71,48)(5,43,75)(6,18,21)(7,62,36)(8,34,63) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^4:\Sp(4,3)$ |
| Order: | \(4199040\)\(\medspace = 2^{7} \cdot 3^{8} \cdot 5 \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and perfect (hence nonsolvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:S_3.\SO(5,3)$, of order \(8398080\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $D_6^2:C_2^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $W$ | $\SOPlus(4,2)$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $9720$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4:\Sp(4,3)$ |