Subgroup ($H$) information
| Description: | $C_{22}$ |
| Order: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Index: | \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
10 & 2 & 9 & 4 \\
1 & 2 & 10 & 9 \\
1 & 1 & 0 & 9 \\
10 & 1 & 10 & 3
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $\SL(2,11):D_6$ |
| Order: | \(15840\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(S_3\times D_4).\PSL(2,11).C_2$ |
| $\operatorname{Aut}(H)$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $W$ | $C_5$, of order \(5\) |
Related subgroups
| Centralizer: | $C_{33}:Q_8$ | ||||
| Normalizer: | $C_{132}.C_{10}$ | ||||
| Normal closure: | $\SL(2,11)$ | ||||
| Core: | $C_2$ | ||||
| Minimal over-subgroups: | $C_{11}:C_{10}$ | $C_{66}$ | $C_{44}$ | $C_{44}$ | $C_{44}$ |
| Maximal under-subgroups: | $C_{11}$ | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | not computed |