Subgroup ($H$) information
| Description: | $C_3^2\times \He_3$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(3\) |
| Generators: |
$bcde^{2}fg, cef^{2}gh, eg^{2}h^{2}, g$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $(C_3^2\times \He_3).F_9$ |
| 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).
Quotient group ($Q$) structure
| Description: | $F_9$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.C_2.D_4^2$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3^6.(C_3^2:\GL(2,3)\times \GL(2,3))$, of order \(15116544\)\(\medspace = 2^{8} \cdot 3^{10} \) |
| $W$ | $C_3^2:F_9$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_3^3$ | ||
| Normalizer: | $(C_3^2\times \He_3).F_9$ | ||
| Minimal over-subgroups: | $C_3^3:\He_3$ | $C_3^4:S_3$ | |
| Maximal under-subgroups: | $C_3\times \He_3$ | $C_3^4$ | $C_3\times \He_3$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $(C_3^2\times \He_3).F_9$ |