Subgroup ($H$) information
| Description: | $C_3\times \He_3$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$be^{2}f^{2}g^{2}h, cdefg^{2}, df^{2}h$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^5:F_9$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.Q_8.D_6.C_2^2$, of order \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_3^4:(S_3\times \GL(2,3))$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $W$ | $C_3:S_3$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3^3$ | ||||
| Normalizer: | $C_3^4:S_3$ | ||||
| Normal closure: | $C_3^4.C_3^3$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $C_3^2\times \He_3$ | $C_3^3:S_3$ | |||
| Maximal under-subgroups: | $C_3^3$ | $C_3^3$ | $\He_3$ | $\He_3$ | $\He_3$ |
Other information
| Number of subgroups in this autjugacy class | $432$ |
| Number of conjugacy classes in this autjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $C_3^5:F_9$ |