Subgroup ($H$) information
| Description: | $C_3\times C_9$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$b, c^{7}d^{6}$
|
| 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: | $(C_3\times C_9^2):C_6$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_9:C_6$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Automorphism Group: | $C_9:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.C_3^5.C_3.C_6^2$, of order \(236196\)\(\medspace = 2^{2} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_3\times C_6$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4374\)\(\medspace = 2 \cdot 3^{7} \) |
| $W$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $(C_3\times C_9^2):C_6$ |