Subgroup ($H$) information
| Description: | $C_{11}:C_5$ |
| Order: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Index: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
5 & 3 & 9 & 10 \\
1 & 6 & 10 & 9 \\
2 & 5 & 1 & 8 \\
1 & 2 & 10 & 2
\end{array}\right), \left(\begin{array}{rrrr}
9 & 7 & 1 & 8 \\
5 & 0 & 6 & 1 \\
2 & 1 & 2 & 4 \\
8 & 2 & 6 & 4
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 5$.
Ambient group ($G$) information
| Description: | $\SL(2,11):C_2^2$ |
| Order: | \(5280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 11 \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $1$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(2,11).C_2\times S_4$ |
| $\operatorname{Aut}(H)$ | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| $W$ | $C_{11}:C_5$, of order \(55\)\(\medspace = 5 \cdot 11 \) |
Related subgroups
| Centralizer: | $Q_8$ | |
| Normalizer: | $C_{44}.C_{10}$ | |
| Normal closure: | $\SL(2,11)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_{11}:C_{10}$ | |
| Maximal under-subgroups: | $C_{11}$ | $C_5$ |
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $\SL(2,11):C_2^2$ |