Subgroup ($H$) information
| Description: | $S_3\times D_9$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$abc^{6}d^{6}e^{6}f^{6}, f^{3}, b^{2}c^{4}e, f^{4}, d^{3}e^{6}f^{3}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_9^4:(C_2\times D_4)$ |
| Order: | \(104976\)\(\medspace = 2^{4} \cdot 3^{8} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9^4.(C_6\times D_4).C_6.C_2^3$, of order \(15116544\)\(\medspace = 2^{8} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^2.S_3^2$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $W$ | $S_3\times D_9$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3888$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | not computed |
| Projective image | $C_9^4:(C_2\times D_4)$ |