Subgroup ($H$) information
| Description: | $C_5\times D_{33}$ |
| Order: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Index: | \(3\) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$a, b^{165}, b^{45}, b^{99}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_5\times D_{99}$ |
| Order: | \(990\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(990\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{99}.C_{60}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_{11}:(C_2\times C_{20}\times S_3)$ |
| $\card{\operatorname{res}(S)}$ | \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $D_{33}$, of order \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $D_{99}$ |