Subgroup ($H$) information
| Description: | $C_{33}$ |
| Order: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Index: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Exponent: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Generators: |
$\langle(1,8,6,9,3,10,7,2,11,4,5)(12,14,13), (12,14,13)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $S_3\times \PSL(2,11)$ |
| Order: | \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times \PGL(2,11)$, of order \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(33\)\(\medspace = 3 \cdot 11 \) |
| $W$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_{33}$ | |
| Normalizer: | $C_{33}:C_{10}$ | |
| Normal closure: | $C_3\times \PSL(2,11)$ | |
| Core: | $C_3$ | |
| Minimal over-subgroups: | $C_{11}:C_{15}$ | $S_3\times C_{11}$ |
| Maximal under-subgroups: | $C_{11}$ | $C_3$ |
Other information
| Number of subgroups in this conjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $S_3\times \PSL(2,11)$ |