Subgroup ($H$) information
| Description: | $S_3^2:D_9^2$ |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| Index: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,32)(2,33)(3,31)(4,23,34,38,19,7,6,24,35,39,20,8,5,22,36,37,21,9)(10,44,41,14,25,28,11,43,42,13,27,29,12,45,40,15,26,30) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_9^4:(C_2\times S_5)$ |
| Order: | \(1574640\)\(\medspace = 2^{4} \cdot 3^{9} \cdot 5 \) |
| Exponent: | \(540\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9^4.(C_6\times S_5)$, of order \(4723920\)\(\medspace = 2^{4} \cdot 3^{10} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_3^3.(C_3\times D_4^2)$, of order \(419904\)\(\medspace = 2^{6} \cdot 3^{8} \) |
| $W$ | $S_3^2:D_9^2$, of order \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $135$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_9^4:(C_2\times S_5)$ |