Subgroup ($H$) information
| Description: | $C_3\times C_3^4.C_3^2$ |
| Order: | \(2187\)\(\medspace = 3^{7} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(3\) |
| Generators: |
$a^{8}, bde^{2}fh^{2}, cg^{2}h, d^{2}efg^{2}h$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is metacyclic, monomial, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6:F_9$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3.C_2^2$, of order \(1679616\)\(\medspace = 2^{8} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | Group of order \(322333846848\)\(\medspace = 2^{6} \cdot 3^{18} \cdot 13 \) |
| $W$ | $C_3^3:S_3$, of order \(162\)\(\medspace = 2 \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | $F_5^3$ |