Subgroup ($H$) information
| Description: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Index: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$bc^{2}e^{2}f^{2}, cef^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_3^3:F_9$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_3:F_9$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $F_9:D_6$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.C_4.C_2^2.A_6.C_2^2$, of order \(5598720\)\(\medspace = 2^{9} \cdot 3^{7} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\operatorname{res}(S)$ | $\SD_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| $W$ | $C_8$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_3^5$ | |||
| Normalizer: | $C_3^3:F_9$ | |||
| Complements: | $C_3:F_9$ | |||
| Minimal over-subgroups: | $C_3^3$ | $C_3^3$ | $C_3^3$ | $C_3:S_3$ |
| Maximal under-subgroups: | $C_3$ |
Other information
| Number of subgroups in this autjugacy class | $10$ |
| Number of conjugacy classes in this autjugacy class | $10$ |
| Möbius function | $0$ |
| Projective image | $C_3^3:F_9$ |