Subgroup ($H$) information
| Description: | not computed |
| Order: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Index: | \(3\) |
| Exponent: | not computed |
| Generators: |
$a^{3}, ce^{2}f, def, b^{9}, b^{6}f, f, e, a^{2}$
|
| 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: | $C_3^5.S_3^2$ |
| Order: | \(8748\)\(\medspace = 2^{2} \cdot 3^{7} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_3^4.C_2^2$ |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $C_3^5.S_3^2$ |