Subgroup ($H$) information
| Description: | not computed |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Index: | \(3\) |
| Exponent: | not computed |
| Generators: |
$a^{3}f, e^{6}, b^{2}ce^{6}, de^{2}, b^{3}f, a^{2}, d, cd^{2}e^{12}, e^{9}$
|
| Derived length: | not computed |
The subgroup is maximal, 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: | $C_3^4.S_3^3$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.C_2^2\times S_3$ |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_{2382}:C_{36}$, of order \(85752\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 397 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3^4.S_3^3$ |