Subgroup ($H$) information
| Description: | $C_2\times \He_3$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Index: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$a^{3}, a^{2}, d^{22}, c$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3\times D_{11}\times \He_3$ |
| Order: | \(1782\)\(\medspace = 2 \cdot 3^{4} \cdot 11 \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.Q_8.C_{33}.C_{30}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| $W$ | $C_3^2$, of order \(9\)\(\medspace = 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $99$ |
| Number of conjugacy classes in this autjugacy class | $9$ |
| Möbius function | $1$ |
| Projective image | $D_{11}\times C_3^3$ |