Subgroup ($H$) information
| Description: | $C_9:C_6$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Index: | \(3000\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{3} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$ac^{4}e^{2}, f^{10}, b^{2}cd^{13}e^{4}f^{3}, c^{2}f^{5}$
|
| Derived length: | $2$ |
The subgroup is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
Ambient group ($G$) information
| Description: | $C_3^3:D_5\wr S_3$ |
| Order: | \(162000\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^3.C_6^2.(C_{12}\times S_3^2)$ |
| $\operatorname{Aut}(H)$ | $C_9:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| $W$ | $C_9:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $600$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^3:D_5\wr S_3$ |