Subgroup ($H$) information
| Description: | not computed |
| Order: | \(8748\)\(\medspace = 2^{2} \cdot 3^{7} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | not computed |
| Generators: |
$ac^{16}d^{2}, d^{6}, d^{8}, e, b^{3}c^{12}d^{9}, c^{14}d^{14}e, c^{6}d^{12}, f, b^{2}e$
|
| Derived length: | not computed |
The subgroup is nonabelian and supersolvable (hence solvable and monomial). Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $(C_3\times D_9^2).S_3^2$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_3^3.(C_3\times D_4^2)$, of order \(419904\)\(\medspace = 2^{6} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_3^4.S_3^3$, of order \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
Related subgroups
| Centralizer: | $C_1$ |
| Normalizer: | $C_3^4.S_3^3$ |
| Normal closure: | $C_3^4.S_3^3$ |
| Core: | $C_9^2.C_3^3.C_2$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $(C_3\times D_9^2).S_3^2$ |