Subgroup ($H$) information
| Description: | $S_3^2:D_9^2$ |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$ad^{16}f^{7}, d^{6}f^{3}, d^{14}f^{4}, f, f^{3}, d^{9}e^{4}, b^{3}c^{7}d^{16}e^{7}f^{5}, e^{6}, c^{6}, c^{9}d^{6}e^{9}$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^5.S_3^2\wr C_2$ |
| Order: | \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^7.S_3\wr C_2^2$, of order \(11337408\)\(\medspace = 2^{6} \cdot 3^{11} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_3^3.(C_3\times D_4^2)$, of order \(419904\)\(\medspace = 2^{6} \cdot 3^{8} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $27$ |
| Möbius function | not computed |
| Projective image | not computed |