Subgroup ($H$) information
| Description: | $(C_3\times \He_3):C_4$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a^{6}, dgh^{2}, dfg^{2}h, a^{12}, bcde^{2}, cd^{2}e^{2}f^{2}g$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable.
Ambient group ($G$) information
| Description: | $C_3^6:F_9$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3.C_2^2$, of order \(1679616\)\(\medspace = 2^{8} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $F_9:D_6$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| $W$ | $C_3^3:C_4$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $324$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | $C_3^6:F_9$ |