Subgroup ($H$) information
| Description: | $C_3\times C_6$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Index: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
37 & 0 \\
0 & 29
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 37
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 38
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $D_{66}:C_6$ |
| Order: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{11}:(C_2^2\times C_{10}\times S_3)$ |
| $\operatorname{Aut}(H)$ | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_6\times C_{66}$ | ||||
| Normalizer: | $C_6\times C_{66}$ | ||||
| Normal closure: | $C_6^2$ | ||||
| Core: | $C_3^2$ | ||||
| Minimal over-subgroups: | $C_3\times C_{66}$ | $C_6^2$ | |||
| Maximal under-subgroups: | $C_3^2$ | $C_6$ | $C_6$ | $C_6$ | $C_6$ |
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_{33}:D_4$ |