Subgroup ($H$) information
| Description: | $\SL(2,5)$ |
| Order: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Index: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rrrr}
9 & 0 & 10 & 0 \\
6 & 9 & 5 & 10 \\
5 & 0 & 2 & 0 \\
1 & 5 & 5 & 2
\end{array}\right), \left(\begin{array}{rrrr}
9 & 0 & 1 & 6 \\
5 & 1 & 6 & 1 \\
6 & 8 & 0 & 0 \\
7 & 6 & 6 & 3
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Derived length: | $0$ |
The subgroup is nonabelian and quasisimple (hence nonsolvable and perfect).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(2,11).C_2\times S_4$ |
| $\operatorname{Aut}(H)$ | $S_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $W$ | $A_5$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Related subgroups
| Centralizer: | $Q_8$ | ||
| Normalizer: | $\SL(2,5):C_2^2$ | ||
| Normal closure: | $\SL(2,11)$ | ||
| Core: | $C_2$ | ||
| Minimal over-subgroups: | $\SL(2,11)$ | $\SL(2,5):C_2$ | |
| Maximal under-subgroups: | $\SL(2,3)$ | $C_5:C_4$ | $C_3:C_4$ |
Other information
| Number of subgroups in this autjugacy class | $22$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-2$ |
| Projective image | $C_2^2\times \PSL(2,11)$ |